text:
Italiana, 29, 465
  * Schmidt (1974) Schmidt E. G., 1974, MNRAS, 167, 613
  * Schmidt (2015) Schmidt E. G., 2015, ApJ, 813, 29
  * Schmidt & Reiswig (1993) Schmidt E. G., Reiswig D. E., 1993, AJ, 106, 2429
  * Schmidt et al. (1995) Schmidt E. G., Chab J. R., Reiswig D. E., 1995, AJ, 109, 1239
  * Schmidt et al. (2004) Schmidt E. G., Johnston D., Langan S., Lee K. M., 2004, AJ, 128, 1748
  * Schmidt et al. (2005) Schmidt E. G., Johnston D., Langan S., Lee K. M., 2005, AJ, 130, 832
  * Shappee et al. (2014) Shappee B., Prieto J., Stanek K. Z., et al., 2014, in American Astronomical Society Meeting Abstracts 223. p. 236.03
  * Shobbrook (1992) Shobbrook R. R., 1992, MNRAS, 255, 486
  * Spreckley & Stevens (2008) Spreckley S. A., Stevens I. R., 2008, MNRAS, 388, 1239
  * Sterken (2005) Sterken C., 2005, in Sterken C., ed., Astronomical Society of the Pacific Conference Series Vol. 335, The Light-Time Effect in Astrophysics: Causes and cures of the O-C diagram. p. 3
  * Stobie (1970) Stobie R. S., 1970, MNRAS, 148, 1
  * Stobie & Balona (1979) Stobie R. S., Balona L. A., 1979, MNRAS, 189, 641
  * Storm et al. (2011) Storm J., et al., 2011, A&A, 534, A94
  * Strohmeier (1968) Strohmeier W., 1968, Information Bulletin on Variable Stars, 304, 1
  * Szabados (1976) Szabados L., 1976, Information Bulletin on Variable Stars, 1107, 1
  * Szabados (1977) Szabados L., 1977, Communications of the Konkoly Observatory Hungary, 70, 1
  * Szabados (1980) Szabados L., 1980, Communications of the Konkoly Observatory Hungary, 76, 1
  * Szabados (1981) Szabados L., 1981, Communications of the Konkoly Observatory Hungary, 77, 1
  * Szabados (1983) Szabados L., 1983, Ap&SS, 96, 185
  * Szabados (1988) Szabados L., 1988, PASP, 100, 589
  * Szabados (1989) Szabados L., 1989, Communications of the Konkoly Observatory Hungary, 94, 1
  * Szabados (1991) Szabados L., 1991, Communications of the Konkoly Observatory Hungary, 96, 123
  * Szabados (1992) Szabados L., 1992, in McAlister H. A., Hartkopf W. I., eds, Astronomical Society of the Pacific Conference Series Vol. 32, IAU Colloq. 135: Complementary Approaches to Double and Multiple Star Research. p. 255
  * Szabados (1993) Szabados L., 1993, in IAU Colloq. 139: New Perspectives on Stellar Pulsation and Pulsating Variable Stars. p. 406
  * Szabados (1996) Szabados L., 1996, A&A, 311, 189
  * Szabados et al. (2013) Szabados L., Anderson R. I., Derekas A., Kiss L. L., Szalai T., Székely P., Christiansen J. L., 2013, MNRAS, 434, 870
  * Szabó et al. (2007) Szabó R., Buchler J. R., Bartee J., 2007, ApJ, 667, 1150
  * Turner & Berdnikov (2004) Turner D. G., Berdnikov L. N., 2004, A&A, 423, 335
  * Turner et al. (2005) Turner D. G., Savoy J., Derrah J., Abdel-Sabour Abdel-Latif M., Berdnikov L. N., 2005, PASP, 117, 207
  * Turner et al. (2006) Turner D. G., Abdel-Sabour Abdel-Latif M., Berdnikov L. N., 2006, PASP, 118, 410
  * Turner et al. (2009) Turner D. G., Percy J. R., Colivas T., Berdnikov L. N., Abdel-Latif M. A.-S., 2009, in Guzik J. A., Bradley P. A., eds, American Institute of Physics Conference Series Vol. 1170, Stellar Pulsation: Challenges for Theory and Observation. pp 167–169 (arXiv:0907.2818), doi:10.1063/1.3246435
  * Turner et al. (2010) Turner D. G., Berdnikov L. N., Percy J. R., Abdel-Sabour Abdel-Latif M., 2010, Odessa Astronomical Publications, 23, 129
  * Udalski et al. (2015) Udalski A., Szymański M. K., Szymański G., 2015, Acta Astron., 65, 1
  * Usenko (1990) Usenko I. A., 1990, Kinematika i Fizika Nebesnykh Tel, 6, 91
  * Usenko (1992) Usenko I. A., 1992, Information Bulletin on Variable Stars, 3784, 1
  * Usenko & Klochkova (2015) Usenko I. A., Klochkova V. G., 2015, Astronomy Letters, 41, 351
  * Vinkó (1991) Vinkó J., 1991, Ap&SS, 183, 17
  * Vinkó (1993) Vinkó J., 1993, MNRAS, 260, 273
  * Virtanen et al. (2020) Virtanen P., et al., 2020, Nature Methods, 17, 261
  * Wachmann (1976) Wachmann A. A., 1976, A&AS, 23, 249
  * Walker et al. (1991) Walker W. S. G., Herdman G. C. D., Marino B. F., 1991, Proceedings of the Astronomical Society of Australia, 9, 127
  * Walraven et al. (1958) Walraven T., Muller A. B., Oosterhoff P. T., 1958, Bull. Astron. Inst. Netherlands, 14, 81
  * Walraven et al. (1964) Walraven J. H., Tinbergen J., Walraven T., 1964, Bull. Astron. Inst. Netherlands, 17, 520
  * Walter (1943) Walter K., 1943, Astronomische Nachrichten, 273, 277
  * Weaver et al. (1960) Weaver H. F., Steinmetz D., Mitchell R., 1960, Lowell Observatory Bulletin, 5, 30
  * Welch & Evans (1989) Welch D. L., Evans N. R., 1989, AJ, 97, 1153
  * Winkler et al. (2003) Winkler C., Courvoisier T. J. L., Di Cocco G., et al., 2003, Astronomy & Astrophysics, 411, L1
  * Woźniak et al. (2004) Woźniak P. R., et al., 2004, AJ, 127, 2436

##  Appendix A Summary table for Cepheids showing evolutionary changes

In Table A.1 we present the period change rates calculated for the Cepheids
showing evolutionary period changes along with the data from Gaia DR3
necessary for the CMD.

Cepheid | P [d] |  $\Delta$P|${}_{100\textrm{yr}}$ | $g$ | $BP-RP$ | $E(B-V)$ |  $\log_{10}\epsilon$ [d] |  $\pi$ [mas] | References
---|---|---|---|---|---|---|---|---
U Aql | 7.024 | 2.820e-05 | 9.443 | 1.02 | 0.397 |  | 0.96 | 1,2,3,13,53,58,60,64,75,81,A28
SZ Aql | 17.141 | 2.267e-04 | 7.986 | 1.755 | 0.614 |  | 0.421 | 2,3,12,13,55,58,74,75,81,96,A29
TT Aql | 13.755 | -1.223e-05 | 6.945 | 1.744 | 0.4858 |  | 0.888 | 1,2,3,12,13,24,53,55,58,60,75,81,97,A29
EV Aql | 38.671 | -3.213e-03 | 11.191 | 1.931 | 0.705 | -0.285 | 0.029 | 2,3,13,17,49,55,58,81,89,A1
FF Aql | 4.471 | 2.582e-05 | 5.143 | 1.148 | 0.221 | -2.731 | 1.81 | 2,3,13,53,58,60,74,75,81,85,97,98,A5
KL Aql | 6.108 | -3.729e-05 | 10.025 | 1.312 | 0.24 | -2.212 | 0.228 | 2,3,13,43,53,55,58,74,79,90,A25
V336 Aql | 7.304 | 5.232e-05 | 9.321 | 1.666 | 0.67 |  | 0.47 | 2,3,13,53,55,58,74,81,96,A25
V493 Aql | 2.988 | 1.407e-05 | 10.572 | 1.7 | 0.777 |  | 0.4 | 2,3,13,30,53,55,58,74
V496 Aql | 6.807 | 3.075e-05 | 7.351 | 1.438 | 0.442 |  | 0.944 | 2,3,13,53,58,74,75,81,A29
V916 Aql | 13.443 | 1.045e-04 | 10.044 | 2.186 | 1.089 |  | 0.478 | 2,3,13,17,49,55,58,89
… | … | … | … | … | … | … | … | …
Table A.1: Summary table compiled for the Cepheids showing evolutionary changes. The $\epsilon$ parameter shown in the table corresponds to the period-fluctuation parameter of the Eddington-Plakidis method, which is summarized in Sect. 3.2. The complete table along with the list of photometry references and estimated uncertainties is available in the online supplements of the article. Cepheid | $F_{\textrm{H0:lin H1:par}}$ | $F_{\textrm{H0:lin H1:par+wave}}$ | $F_{\textrm{H0:par H1:par+wave}}$ | Amplitude [days]
---|---|---|---|---
U Aql | 22.61 | 6.12 | 0.72 |
SZ Aql | 60.94 | 17.12 | 1.77 |
TT Aql | 6.79 | 10.48 | 11.08 |
EV Aql | 91.78 | 315.39 | 205.72 | 6.997
FF Aql | 63.1 | 24.2 | 7.02 |
KL Aql | 109.7 | 70.61 | 22.26 | 0.027
V336 Aql | 18.55 | 4.79 | 0.39 |
V493 Aql | 5.59 | 9.02 | 8.8 |
V496 Aql | 9.98 | 3.54 | 1.33 |
V916 Aql | 158.75 | 92.54 | 23.69 | 0.175
… | … | … | … |
Table B.1: Summary table of the F-test model comparison conducted for Cepheids
showing evolutionary changes: columns 2-3 show the F-statistic and p-value for
the case when we compare the residuals of the linear and parabolic fit;
columns 4-5 show the results for the linear vs. parabolic plus wave model;
columns 6-7 show the results for the parabolic vs. parabolic plus wave model.
The last column shows the amplitude of the found wave (if the F-statistic was
higher than 10 in the parabolic vs. parabolic plus wave model comparison).
When fitting the parabolic plus wave model, we simultaneously optimized for
both the parameters of the parabolic change and the wavelike modulation as
well. To accept the parabolic plus wave model as a refinement of the simple
parabolic fit, we also set the requirement that the found wave should have a
period longer than 3000 days (about ten times the folding time used for the
$O-C$ diagram calculations) but has to be shorter than five quarters of the
timespan covered by observations (to avoid overfitting). Therefore, some of
the F-test results were rejected, even though the parabolic plus wave fit
seemed more optimal (for example, CH Cas, XX Cen or U Vul). We highlighted the
Cepheids that match the criteria with bold fonts. The complete table is
available in the online supplement of the article.

# | Reference | # | Reference | # | Reference
---|---|---|---|---|---
[1] | AAVSO (Kafka, 2021) | [48] | Harris (1980) | [95] | Szabados (1977)
[2] | ASAS (Pojmanski, 2001) | [49] | DASCH (Grindlay et al., 2012) | [96] | Szabados (1980)
[3] | ASAS-SN (Shappee et al., 2014) | [50] | Hellerich (1935) | [97] | Szabados (1981)
[4] | Abaffy (private comm.) | [51] | Henden (1980) | [98] | Szabados (1991)
[5] | Arellano Ferro (1984) | [52] | Henden (1996) | [99] | TESS (Ricker et al., 2015)
[6] | Arellano Ferro et al. (1998) | [53] | Hipparcos (Perryman et al., 1997) | [100] | Tabur (private comm.)
[7] | Arp et al. (1959) | [54] | Hoffmeister (1960) | [101] | Bahner & Mavridis (1971)
[8] | Asarnova (1957) | [55] | IOMC (Winkler et al., 2003) | [102] | Usenko (1992)
[9] | Babel et al. (1989) | [56] | Irwin (1961) | [103] | Wachmann (1976)
[10] | Bahner et al. (1962) | [57] | Kepler K2 (Howell et al., 2014) | [104] | Walraven et al. (1958)
[11] | Barnes et al. (1987) | [58] | KWS (Morokuma et al., 2014) | [105] | Walraven et al. (1964)
[12] | Barnes et al. (1997) | [59] | KELT (Pepper et al., 2007) | [106] | Walter (1943)
[13] | Berdnikov (2008) | [60] | Kiss (1998) | [107] | Weaver et al. (1960)
[14] | Berdnikov et al. (2009) | [61] | Koukarkina (1954) | [108] | Williams (private comm.)
[15] | Berdnikov et al. (2011) | [62] | Kovacs & Szabados (1979) | [A1] | Berdnikov & Pastukhova (1994)
[16] | Berdnikov et al. (2014) | [63] | Kox (1935) | [A2] | Berdnikov & Turner (2001)
[17] | Berdnikov et al. (2015) | [64] | Krebs (1936) | [A3] | Berdnikov & Caldwell (2001)
[18] | Berdnikov et al. (2019) | [65] | Kurochkin (1954) | [A4] | Berdnikov & Turner (2004)
[19] | Bersier (2002) | [66] | Kwee (1967) | [A5] | Berdnikov et al. (2014)
[20] | Bochum (Hackstein et al., 2015) | [67] | Landolt (1971) | [A6] | Berdnikov (2019c)
[21] | Buchancowa et al. (1972) | [68] | Laur et al. (2017) | [A7] | Erleksova (1978)
[22] | Burnashev & Burnasheva (2009) | [69] | Madore (1975) | [A8] | Erleksova & Irkaev (1982)
[23] | Connolly et al. (1983) | [70] | Malik (1965) | [A9] | Fernie (1990a)
[24] | Coulson & Caldwell (1985) | [71] | Mauder & Schöffel (1968) | [A10] | Hacke (1989)
[25] | Cousins & Lagerweij (1968) | [72] | Millis (1969) | [A11] | Hacke & Richert (1990)
[26] | Cousins & Lagerweij (1971) | [73] | Mitchell et al. (1961) | [A12] | Heiser (1996)
[27] | Dean (1977) | [74] | Mitchell et al. (1964) | [A13] | Kiehl & Hopp (1977)
[28] | Dean (1981) | [75] | Moffett & Barnes (1984) | [A14] | Klawitter (1971)
[29] | Detre (1935) | [76] | NSVS (Woźniak et al., 2004) | [A15] | Mahmoud & Szabados (1980)
[30] | Diethelm & Tammann (1982) | [77] | Udalski et al. (2015) | [A16] | Meyer (2006)
[31] | Eggen (1951) | [78] | Oja (2011) | [A17] | Miller & Wachmann (1973)
[32] | Eggen et al. (1957) | [79] | Oosterhoff (1960) | [A18] | Nijland (1935)
[33] | Eggen (1983) | [80] | Parkhurst (1908) | [A19] | Oosterhoff (1935)
[34] | Eggen (1985) | [81] | Pel (1976) | [A20] | Oosterhoff (1943)
[35] | Feinstein & Muzzio (1969) | [82] | Pingsdorf (1935) | [A21] | Parenago (1956)
[36] | Fernie et al. (1965) | [83] | Present paper | [A22] | Romano (1958)
[37] | Fernie (1970) | [84] | Reed (1968) | [A23] | Strohmeier (1968)
[38] | Fernie (1979) | [85] | SMEI (Eyles et al., 2003) | [A24] | Szabados (1977)
[39] | Fernie & Garrison (1981) | [86] | SWASP (Pollacco et al., 2006) | [A25] | Szabados (1980)
[40] | Fernie et al. (1995) | [87] | Schmidt & Reiswig (1993) | [A26] | Szabados (1981)
[41] | Filatov (1957) | [88] | Schmidt et al. (1995) | [A27] | Szabados (1988)
[42] | Floria & Kukarkina (1953) | [89] | Schmidt et al. (2004) | [A28] | Szabados (1989)
[43] | Gaia DR2 (Gaia Collaboration, 2018) | [90] | Schmidt et al. (2005) | [A29] | Szabados (1991)
[44] | Gaposchkin (1958) | [91] | Shobbrook (1992) | [A30] | Szabados et al. (2013)
[45] | Gieren (1981) | [92] | Stobie (1970) | [A31] | Vinkó (1991)
[46] | Gieren (1985) | [93] | Stobie & Balona (1979) | [A32] | Walker et al. (1991)
[47] | Grayzeck (1978) | [94] | Szabados (1976) |  |
Table A.2: Complete list of references used for the calculation of the $O-C$
diagrams along with the abbreviations.

license: Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/


text:
# The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type
theorem

Erika Hausenblas  Montanuniversität Leoben
Department Mathematik und Informationstechnologie
Franz Josef Straße 18
8700 Leoben
Austria<EMAIL_ADDRESS>and  Jonas M. Tölle  University of
Helsinki
Department of Mathematics and Statistics
P.O. Box 68 (Pietari Kalmin katu 5)
FI-00014 University of Helsinki
Finland<EMAIL_ADDRESS>
###### Abstract.

On the one hand, we investigate the existence and pathwise uniqueness of a
nonnegative martingale solution to the stochastic evolution system of
nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian
multiplicative noise. On the other hand, we present and verify a general
stochastic version of the Schauder-Tychonoff fixed point theorem, as its
application is an essential step for showing existence of the solution to the
stochastic Klausmeier system. The analysis of the system is based both on
variational and semigroup techniques. We also discuss additional regularity
properties of the solution.

###### Key words and phrases:

stochastic Klausmeier evolution system, stochastic Schauder-Tychonoff type
theorem, pattern formation in ecology, nonlinear stochastic partial
differential equation, flows in porous media, nonnegative solution, pathwise
uniqueness.

###### 2010 Mathematics Subject Classification:

Primary 35K57, 60H15; Secondary 37N25, 47H10, 76S05, 92C15

E.H. acknowledges partial funding by FWF (Austrian Science Foundation) grant P
28010 “Mathematische Analyse von Flüssigkeitskristallen mit stochastischer
Störung”. J.M.T. was supported by the Academy of Finland and the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 818437).

##  1\. Introduction

Pattern formation at the ecosystem level has recently gained a lot of
attention in spatial ecology and its mathematical modeling. Theoretical models
are a widely used tool for studying e.g. banded vegetation patterns. One
important model is the system of advection-diffusion equations proposed by
Klausmeier [36]. This model for vegetation dynamics in semi-deserted areas is
based on the “water redistribution hypothesis”, using the idea that rain water
in dry regions is eventually infiltrated into the ground. Water falling down
onto bare ground mostly runs off downhill towards the next patch of vegetation
which provides a better infiltration capacity. The soil in such regions of the
world as Australia, Africa, and Southwestern North America is prone to
nonlocality of water uptake due to the semi-arid environment. Studies of the
properties of the system and further developments can be found in e.g. [58,
65, 35, 55, 56, 57].

The Klausmeier system is a generalization of the so-called Gray-Scott system
[29] (see also [54, 59] for earlier accounts employing similar models) which
already exhibits effects similar to Turing patterns [64, 22, 44, 20], see for
instance the discussion in [66]. We refer to [46, 47, 39] for further reading
on pattern formation in biology. A discussion of reaction-diffusion type
equations with motivation from biology can be found in [50].

The underlying mathematics of this model is given by a pair of solutions
$(u,v)$ to a partial differential equation system coupled by a nonlinearity.
The function $u$ represents the surface water content and $v$ represents the
biomass density of the plants. In order to model the spread of water on a
terrain without a specific preference for the direction in which the water
flows, the original models were extended by replacing the diffusion operator
by a nonlinear porous media operator, which represents the situation that the
ground is partially filled by interconnected pores conveying fluid under an
applied pressure gradient.

To this end, let ${{\mathcal{O}}}\subset{\mathbb{R}}^{d}$ be a bounded domain,
$d=1,2,3$, having $C^{2}$-boundary or ${{\mathcal{O}}}=[0,1]^{d}$. Consider
the following problem

(1.3) $\displaystyle\left\\{\begin{array}[]{rcl}\dot{u}(t)&=&r_{u}\Delta
u^{[\gamma]}(t)-\chi u(t)\,v^{2}(t)+k-fu(t),\quad t>0,\quad
u(0)=u_{0},\phantom{\Big{|}}\\\ \dot{v}(t)&=&r_{v}\Delta
v(t)+u(t)\,v^{2}(t)-gv(t),\quad
t>0,\quad{v}(0)=v_{0},\phantom{\Big{|}}\end{array}\right.$

with Neumann (or periodic when ${{\mathcal{O}}}$ is a rectangular domain)
boundary conditions and initial conditions $u(0)=u_{0}$ and $v(0)=v_{0}$.
Here, $z^{[\gamma]}:=|z|^{\gamma-1}z$, $\gamma>1$, $z\in\mathbb{R}$, and
further, $r_{u}$, $r_{v}$, $\chi$, $k$, $f$ and $g$ denote positive constants.

The deterministic or macroscopic model is derived from the limiting behavior
of interacting diffusions — the so-called microscopic model. When applying the
strong law of large numbers and passing from the microscopic to the
macroscopic equation, one is neglecting the random fluctuations. In order to
get a more realistic model, it is necessary to add noise, which represents the
randomness of the natural environment or the fluctuation of parameters in the
model. Due to the Wong-Zakai principle, this leads to the representation of
the noise as a Gaussian stochastic perturbation with Stratonovich increment,
see [25, Section 3.4] and [43, 72]. One important consequence is the
preservation of energy in the noisy system.

In practice, we are investigating the system (1.3) driven by a multiplicative
infinite dimensional Wiener process. Under suitable regularity assumptions on
the initial data $(u_{0},v_{0})$, on $\gamma$ and on the perturbation by noise
which we shall specify later, we find that there exists a nonnegative solution
to the system (1.6) in dimensions $d=1,2,3$, see our main result111In fact, we
prove existence of nonnegative solutions to the noisy systems (3.6)–(3.7) (Itô
noise), (3.1)–(3.2) (Stratonovich noise) respectively. Theorem 3.6. Thus, we
are actually considering the following system

(1.6) $\displaystyle\left\\{\begin{array}[]{rcl}\dot{u}(t)&=&r_{u}\Delta
u^{[\gamma]}(t)-\chi u(t)\,v^{2}(t)+u(t)\,\Xi_{1}(t),\quad t>0,\quad
u(0)=u_{0},\\\ \dot{v}(t)&=&r_{v}\Delta
v(t)+u(t)\,v^{2}(t)+v(t)\,\Xi_{2}(t),\quad t>0,\quad
v(0)=v_{0}.\end{array}\right.$

Here, $\Xi_{1}$ and $\Xi_{2}$ denote independent random Gaussian noises
specified to be certain Banach space valued linear Stratonovich Wiener noises
later on. Similar stochastic equations with Lévy noise have been studied e.g.
in [17, 9]. We note that due to the cross diffusion coupling and the
nonlinearity, the solutions are not expected to be stochastically independent.

Nonlinear diffusion systems perturbed by noise exhibit certain improved well
posedness and regularity properties when compared to their deterministic
counterparts, see, among others, [4, 14, 25]. Another feature of linear
multiplicative noise is that it preserves the nonnegativity of the initial
condition [62, 1].

The challenging problem in the system given by (1.3), respective the noisy
system (1.6), is the nonlinearity appearing once with a negative sign and once
with a positive sign. Choosing different speeds of diffusion seems rather
natural, as it well known that the characteristic pattern formation may not
take place if $r_{u}=r_{v}$, cf. [31]. The stochastic perturbation, however,
does not restrict pattern formation [11, 73], noting that our choice of noise
exhibits rather small intensity and damped high frequency modes. On the other
hand, the nonlinearity is not of variational structure, such that energy
methods are not available for the analysis, and neither the maximum principle
nor Gronwall type arguments work. More direct deterministic (pathwise) methods
may also fail in general as the equation for $v$ is non-monotone.

Another difficulty is posed by the nonlinear porous media diffusion operator
in the equation for $u$. Typically, it is studied with the help of variational
monotonicity methods, see e.g. [42, 2]. To the nonlinear diffusion, neither
semigroup methods nor Galerkin approximation can be applied directly without
greater effort. Other approaches are given for instance in [27, 68, 49], to
mention a few.

Our motivation to prove a probabilistic Schauder-Tychonoff type fixed point
theorem originates from the aim to show existence of a solution to the
stochastic counterpart (1.6) of the system (1.3). As standard methods for
showing existence and uniqueness of solutions to stochastic partial
differential equations cannot be applied directly, we perform a fixed point
iteration using a mix of so-called variational and semigroup methods with
nonlinear perturbation. Here, a precise analysis of regularity properties of
nonnegative solutions to regularized and localized subsystems with truncated
nonlinearities needs to be conducted. Together with the continuous dependence
of each subsystem on the other one, we obtain the fixed point, locally in
time, by weak compactness and an appropriate choice of energy spaces. The
result is completed by a stopping time gluing argument.

Apart from the probabilistic structure, the main novelty is that we construct
the fixed point in the nonlinearity and not in the noise coefficient. This
works well because our system is coupled precisely in the nonlinearity. The
main task consists then in the analysis of regularity and invariance
properties of a regularized system with “frozen” nonlinearity (not being fully
linearized, however, as the porous media operator remains).

It may be possible to apply this method in the future to other nonlinear
systems, as for example, systems with nonlinear convection-terms as systems
with transport or Navier-Stokes type systems. It is certainly possible to
apply the method to linear cross diffusion systems. See [19] for a recent work
proving existence of martingale solutions to stochastic cross diffusion
systems, however, their approach is using another methods and is not covering
the porous media case. See [45] for a previous work using the classical
Schauder theorem for stochastic evolution equations with fractional Gaussian
noise.

Given higher regularity of the initial data, we also show the pathwise
uniqueness of the solution to (1.6) for spatial dimension $d=1$. As a
consequence of the celebrated result by Yamada and Watanabe [74, 69], we
obtain existence and uniqueness of a strong solution, see Corollary 3.11. We
refer to [53, 12, 13, 10] for previous works that employ a similar strategy
for proving the existence of a unique strong solution.

### Structure of the paper

The stochastic Schauder-Tychonoff type Theorem 2.1 is presented in Section 2.
In the subsequent section, i.e. Section 3, we apply this fixed point theorem
to show the existence of a martingale solution to the stochastic counterpart
(1.6) of the system (1.3). Section 3 contains our main result Theorem 3.6.
Section 4 is devoted to the proof of Theorem 2.1. In Section 5, we prove
several technical propositions that are need for the main result in Section 3.
Section 6 contains the proof of pathwise uniqueness, that is, Theorem 3.9.
Some auxiliary results are collected in Appendices A, B and C.

##  2\. The stochastic Schauder-Tychonoff type theorem

Let us fix some notation. Let $U$ be a Banach space. Let
${\mathcal{O}}\subset\mathbb{R}^{d}$ be an open domain, $d\geq 1$. Let
$\mathbb{X}\subset\\{\eta:[0,T]\to
E\subset{\mathcal{D}}^{\prime}({\mathcal{O}})\\}$ be a Banach function
space222Here, ${\mathcal{D}}^{\prime}({\mathcal{O}})$ denotes the space of
Schwartz distributions on ${\mathcal{O}}$, that is, the topological dual space
of smooth functions with compact support
${\mathcal{D}}({\mathcal{O}})=C_{0}^{\infty}({\mathcal{O}})$., let
$\mathbb{X}^{\prime}\subset\\{\eta:[0,T]\to
E\subset{\mathcal{D}}^{\prime}({\mathcal{O}})\\}$ be a reflexive Banach
function space embedded compactly into $\mathbb{X}$. In both cases, the
trajectories take values in a Banach function space $E$ over the spatial
domain ${\mathcal{O}}$. Let
${\mathfrak{A}}=(\Omega,{\mathcal{F}},\mathbb{F},\mathbb{P})$ be a filtered
probability space with filtration $\mathbb{F}=({\mathcal{F}}_{t})_{t\in[0,T]}$
satisfying the usual conditions. Let $H$ be a separable Hilbert space and
$(W_{t})_{t\in[0,T]}$ be a Wiener process333That is, a $Q$-Wiener process, see
e.g. [15] for this notion. in $H$ with a linear, nonnegative definite,
symmetric trace class covariance operator $Q:H\to H$ such that $W$ has the
representation

$W(t)=\sum_{i\in\mathbb{I}}Q^{\frac{1}{2}}\psi_{i}\beta_{i}(t),\quad
t\in[0,T],$

where $\\{\psi_{i}:i\in\mathbb{I}\\}$ is a complete orthonormal system in $H$,
$\mathbb{I}$ a suitably chosen countable index set, and
$\\{\beta_{i}:i\in\mathbb{I}\\}$ a family of independent real-valued standard
Brownian motions on $[0,T]$ modeled in
${\mathfrak{A}}=(\Omega,{\mathcal{F}},\mathbb{F},\mathbb{P})$. Due to [15,
Proposition 4.7, p. 85], this representation does not pose a restriction.

For $m\geq 1$, define the collection of processes

(2.1) $\displaystyle{\mathcal{M}}_{{\mathfrak{A}}}^{m}(\mathbb{X}):=$
$\displaystyle\Big{\\{}\xi:\Omega\times[0,T]\to E\;\colon\;$
$\displaystyle\qquad\text{$\xi$ is $\mathbb{F}$-progressively
measurable}\;\text{and}\;\mathbb{E}|\xi|_{\mathbb{X}}^{m}<\infty\Big{\\}}$

equipped with the semi-norm

$|\xi|_{{\mathcal{M}}_{{\mathfrak{A}}}^{m}(\mathbb{X})}:=(\mathbb{E}|\xi|_{\mathbb{X}}^{m})^{1/m},\quad\xi\in{\mathcal{M}}_{{\mathfrak{A}}}^{m}(\mathbb{X}).$

For fixed ${\mathfrak{A}}$, $W$, $m>1$, we define the operator
${\mathcal{V}}={\mathcal{V}}_{{\mathfrak{A}},W}:{\mathcal{M}}_{{\mathfrak{A}}}^{m}(\mathbb{X})\to{\mathcal{M}}_{{\mathfrak{A}}}^{m}(\mathbb{X})$
for $\xi\in{\mathcal{M}}_{\mathfrak{A}}^{m}(\mathbb{X})$ via

${\mathcal{V}}(\xi):=w,$

where $w$ is the solution of the following Itô stochastic partial differential
equation (SPDE)

(2.2) $\displaystyle dw(t)$ $\displaystyle=$
$\displaystyle\left(Aw(t)+F(\xi(t),t)\right)\,dt+\Sigma(w(t))\,dW(t),\quad
w(0)=w_{0}\in E.$

Here, we implicitly assume that (2.2) is well posed and a unique strong
solution (in the stochastic sense)
$w\in{\mathcal{M}}_{\mathfrak{A}}^{m}(\mathbb{X})$ exists for
$\xi\in{\mathcal{M}}_{\mathfrak{A}}^{m}(\mathbb{X})$. Here, we shall also
assume that $A:D(A)\subset\mathbb{X}\to\mathbb{X}$ is a possibly nonlinear and
measurable (single-valued) operator and $F:\mathbb{X}\times[0,T]\to\mathbb{X}$
a (strongly) measurable map such that

${\mathbb{P}}\left(\int_{0}^{T}\left(\left|Aw(s)\right|_{E}+\left|F(\xi(s),s)\right|_{E}\right)\,ds<\infty\right)=1,$

and assume that $\Sigma:E\to\gamma(H,E)$ is measurable. Here, $\gamma(H,E)$
denotes the space of $\gamma$-radonifying operators from $H$ to $E$, as
defined in the beginning of Section 5, which coincides with the space of
Hilbert-Schmidt operators $L_{\textup{HS}}(H,E)$ if $E$ is a separable Hilbert
space.

We are ready to formulate our main tool for proving the existence of
nonnegative martingale solutions to (1.6), that is, a stochastic variant of
the (deterministic) Schauder-Tychonoff fixed point theorem(s) from [28, §
6–7].

######  Theorem 2.1.

Let $H$ be a Hilbert space, $Q:H\to H$ such that $Q$ is linear, symmetric,
nonnegative definite and of trace class, let $U$ be a Banach space, and let us
assume that we have a compact embedding
$\mathbb{X}^{\prime}\hookrightarrow\mathbb{X}$ as above. Let $m>1$. Suppose
that for any filtered probability space
${\mathfrak{A}}=(\Omega,{\mathcal{F}},\mathbb{F},\mathbb{P})$ and for any
$Q$-Wiener process $W$ with values in $H$ that is modeled on ${\mathfrak{A}}$
the following holds.

There exists a nonempty, sequentially weak∗-closed, measurable and bounded
subset444Here, the notation ${\mathcal{X}}({\mathfrak{A}})$ means that
$\text{Law}(u)=\text{Law}(\tilde{u})$ on $\mathbb{X}$ for
$u\in{\mathcal{X}}({\mathfrak{A}})$ and
$\tilde{u}\in{\mathcal{M}}_{\tilde{{{\mathfrak{A}}}}}^{m}$ implies
$\tilde{u}\in{\mathcal{X}}(\tilde{{\mathfrak{A}}})$.
${\mathcal{X}}({\mathfrak{A}})$ of
${\mathcal{M}}_{\mathfrak{A}}^{m}(\mathbb{X})$ such that the operator
${\mathcal{V}}_{{\mathfrak{A}},W}$, defined by (2.2), restricted to
${\mathcal{X}}({\mathfrak{A}})$ satisfies the following properties:

  1. (i)

${\mathcal{V}}_{{\mathfrak{A}},W}({\mathcal{X}}({\mathfrak{A}}))\subset{\mathcal{X}}({\mathfrak{A}})$,

  2. (ii)

the restriction
${\mathcal{V}}_{{\mathfrak{A}},W}\big{|}_{{\mathcal{X}}({\mathfrak{A}})}$ is
continuous w.r.t. the strong topology of
${\mathcal{M}}_{{\mathfrak{A}}}^{m}(\mathbb{X})$,

  3. (iii)

there exist constants $R>0$, $m_{0}\geq m$ such that

$\mathbb{E}|{\mathcal{V}}_{{\mathfrak{A}},W}(v)|_{\mathbb{X}^{\prime}}^{m_{0}}\leq
R\quad\text{for every}\quad v\in{\mathcal{X}}({\mathfrak{A}}),$

  4. (iv)

${\mathcal{V}}_{{\mathfrak{A}},W}({\mathcal{X}}({\mathfrak{A}}))\subset\mathbb{D}([0,T];U)$
$\mathbb{P}$-a.s.555Here, $\mathbb{D}([0,T);U)$ denotes the Skorokhod space of
càdlàg paths in $U$.

Then, there exists a filtered probability space
$\tilde{{\mathfrak{A}}}=(\tilde{\Omega},\tilde{{\mathcal{F}}},\tilde{\mathbb{F}},\tilde{\mathbb{P}})$
(that satisfies the usual conditions) together with a $Q$-Wiener process
$\tilde{W}$ modeled on $\tilde{{\mathfrak{A}}}$ and an element
$\tilde{v}\in{\mathcal{M}}_{\tilde{{\mathfrak{A}}}}^{m}(\mathbb{X})$ such that
for all $t\in[0,T]$, $\tilde{\mathbb{P}}$-a.s.

${\mathcal{V}}_{\tilde{{\mathfrak{A}}},\tilde{W}}(\tilde{v})(t)=\tilde{v}(t)$

for any initial datum $v_{0}:=w_{0}\in
L^{m}(\Omega,{\mathcal{F}}_{0},\mathbb{P};E)$.

The proof of Theorem 2.1 is postponed to Section 4. We note that by
construction, we get that $v^{\ast}$ solves

(2.3) $\displaystyle dv^{\ast}(t)$ $\displaystyle=$
$\displaystyle\left(Av^{\ast}(t)+F(v^{\ast}(t),t)\right)\,dt+\Sigma(v^{\ast}(t))\,d\tilde{W}(t),\quad
v^{\ast}(0)=v_{0}.$

on $\tilde{\mathfrak{A}}$.

######  Remark 2.2.

We note that ${\mathcal{X}}({\mathfrak{A}})$ is sequentially weak∗-closed,
measurable and bounded subset of
${\mathcal{M}}_{\mathfrak{A}}^{m}(\mathbb{X})$ if there exists a constants
$K_{i}\geq 0$, $i=1,\ldots,N$ for some $N\in\mathbb{N}$ and sequentially
weak∗-lower semicontinuous666If $E$ is reflexive, this is for instance
satisfied by Mazur’s lemma if the maps $\Theta_{i}$, $i=1,\ldots,N$, are
convex and strongly (lower semi-)continuous. measurable mappings

$\Theta_{i}:L^{1}(0,T;E)\to[0,+\infty],\quad i=1,\ldots,N,$

with bounded sublevel sets such that $\mathcal{X}({\mathfrak{A}})$ has the
structure

$\mathcal{X}(\mathfrak{A})=\Big{\\{}\xi\in{\mathcal{M}}_{\mathfrak{A}}^{m}(\mathbb{X})\;\colon\;\mathbb{E}[\Theta_{i}(\xi)]\leq
K_{i},\;i\in\\{1,\ldots,N\\}\Big{\\}}.$

Then, of course, the set ${\mathcal{X}}({\mathfrak{A}})\cap A$ is still
sequentially weak∗-closed and bounded for any sequentially weak∗ closed and
measurable set $A$, where we shall use this fact later for the convex cone of
nonnegative processes
$A=\\{\xi\in{\mathcal{M}}_{\mathfrak{A}}^{m}(\mathbb{X})\;\colon\;\xi\geq
0\;\mathbb{P}\otimes\operatorname{Leb}\text{-a.e.}\\}$.

##  3\. Existence of a solution to the stochastic Klausmeier system

In this section, we shall prove the existence of a nonnegative solution to the
stochastic Klausmeier system. First, we will introduce some notation and the
definition of a (martingale) solution. After fixing the function spaces and
the main hypotheses on the parameters of those, we present our main result
Theorem 3.6. The following proof consists mainly of a verification of the
conditions for Theorem 2.1 and a stopping time localization procedure. As
pointed out before, we are using compactness arguments to show the existence,
which leads to the loss of the initial stochastic basis.

Let $H_{1}$ and $H_{2}$ be a pair of separable Hilbert spaces, let
$\mathfrak{A}=(\Omega,{{\mathcal{F}}},\mathbb{F},{\mathbb{P}})$ be a filtered
probability space and let $W_{1}$, $W_{2}$ be a pair of independent Wiener
process modeled on $\mathfrak{A}$ taking values in $H_{1}$ and $H_{2}$,
respectively, with covariance operators $Q_{1}$ and $Q_{2}$, respectively. We
are interested in the solution to the following reduced Klausmeier system for
$x\in{{\mathcal{O}}}$ and $t>0$,

(3.1) $d{u}(t,x)=(r_{u}\Delta u^{[\gamma]}(t,x)-\chi
u(t,x)\,v^{2}(t,x))\,dt+\sigma_{1}u(t,x)\circ dW_{1}(t,x),$

and,

(3.2) $d{v}(t,x)=(r_{v}\Delta
v(t,x)+u(t,x)\,v^{2}(t,x))\,dt+\sigma_{2}v(t,x)\circ dW_{2}(t,x),$

with Neumann (or periodic if ${\mathcal{O}}=[0,1]^{d}$) boundary conditions
and initial conditions $u(0)=u_{0}$ and $v(0)=v_{0}$. Let $r_{u},r_{v},\chi>0$
be positive constants. Here, we use the abbreviation
$x^{[\gamma]}:=|x|^{\gamma-1}x$ for $\gamma>1$. We note that due to Sobolev
embedding techniques, we actually have to choose $\gamma>2$. The hypotheses on
the linear noise coefficient maps $\sigma_{1},\sigma_{2}$ are specified below.

Due to the nonlinear porous media term, we do neither use solutions in the
strong stochastic sense, nor mild solutions, that is, solutions in the sense
of stochastic convolutions. Let us define what we mean with a solution on a
fixed stochastic basis.

######  Definition 3.1.

A couple $(u,v)$ of stochastic processes on ${\mathfrak{A}}$ is called
_solution to the system_ (3.1)–(3.2) for initial data $(u_{0},v_{0})$ if there
exists $\rho\in\mathbb{R}$ such that

$u\in L^{2}(\Omega;C([0,T];H_{2}^{-1}({{\mathcal{O}}})))\cap
L^{\gamma+1}(\Omega\times(0,T)\times{\mathcal{O}})$

and

$v\in L^{2}(\Omega;C([0,T];H^{\rho}_{2}({{\mathcal{O}}})))\cap
L^{2}(\Omega\times(0,T);H_{2}^{\rho+1}({{\mathcal{O}}}))$

such that $u$ and $v$ are
$\mathbb{F}=({\mathcal{F}}_{t})_{t\in[0,T]}$-adapted, and satisfy

(3.3) $u(t)=u_{0}+\int_{0}^{t}(r_{u}\Delta{u}^{[\gamma]}(s)-\chi
u(s)v^{2}(s))\,ds+\int_{0}^{t}\sigma_{1}u(s)\circ dW_{1}(s)$

for every $t\in[0,T]$, $\mathbb{P}$-a.s. in $H^{-1}_{2}({\mathcal{O}})$, and

(3.4) $v(t)=v_{0}+\int_{0}^{t}(r_{v}\Delta
v(s)+u(s)v^{2}(s))\,ds+\int_{0}^{t}\sigma_{2}v(s)\circ dW_{2}(s)$

for every $t\in[0,T]$, $\mathbb{P}$-a.s. in $H_{2}^{\rho}({\mathcal{O}})$.

Note that the above $ds$-integral for $u$ in (3.3) is initially a
$V^{\ast}$-valued777Here, $V:=L^{\gamma+1}({\mathcal{O}})$ is dualized in a
Gelfand triple over ${\mathcal{H}}:=H^{-1}_{2}({\mathcal{O}})$ which, in turn,
is identified with its own dual by the Riesz isometry. Thus $V^{\ast}$ is not
to be mistaken to be equal to be the usual Banach space dual of $V$, namely
$L^{\frac{\gamma+1}{\gamma}}({\mathcal{O}})$. See the proof of Theorem 5.3 for
details. Bochner integral, where $V^{\ast}$ is defined as in the proof of
Theorem 5.3, however seen to be in fact $H^{-1}_{2}({\mathcal{O}})$-valued,
see the discussion in [42, Section 4.2] for further details.

As mentioned before, due to the loss of the original probability space, we are
considering solutions in the weak probabilistic sense.

######  Definition 3.2.

A martingale solution to the problem (3.1)–(3.2) for initial data
$(u_{0},v_{0})$ is a system

(3.5)
$\left(\Omega,{{\mathcal{F}}},{\mathbb{F}},\mathbb{P},(W_{1},W_{2}),(u,v)\right)$

such that

  1. (i)

the quadruple $\mathfrak{A}:=(\Omega,{{\mathcal{F}}},{\mathbb{F}},\mathbb{P})$
is a complete filtered probability space with a filtration
${\mathbb{F}}=(\mathcal{F}_{t})_{t\in[0,T]}$ satisfying the usual conditions,

  2. (ii)

$W_{1}$ and $W_{2}$ are independent $H_{1}$-valued, respectively,
$H_{2}$-valued Wiener processes over the probability space $\mathfrak{A}$ with
covariance operators $Q_{1}$ and $Q_{2}$, respectively,

  3. (iii)

both $u:[0,T]\times\Omega\to H^{-1}_{2}({{\mathcal{O}}})$ and
$v:[0,T]\times\Omega\to H_{2}^{\rho}({{\mathcal{O}}})$ are
${\mathbb{F}}$-adapted processes such that the couple $(u,v)$ is a solution to
the system (3.1) and (3.2) over the probability space $\mathfrak{A}$ in the
sense of Definition 3.1 for some $\rho\in\mathbb{R}$.

######  Remark 3.3.

For our purposes, instead of the Stratonovich formulation the system
(3.1)–(3.2), it is convenient to consider the equations in Itô form:

(3.6) $d{u}(t,x)=(r_{u}\Delta u^{[\gamma]}(t,x)-\chi
u(t,x)\,v^{2}(t,x))\,dt+\sigma_{1}u(t,x)\,dW_{1}(t,x),$

and,

(3.7) $d{v}(t,x)=(r_{v}\Delta
v(t,x)+u(t,x)\,v^{2}(t,x))\,dt+\sigma_{2}v(t,x)\,dW_{2}(t,x),$

with initial data $u(0)=u_{0}$ and $v(0)=v_{0}$. One reason for this is that
the stochastic integral then becomes a local martingale. In order to show the
existence of a solution to the Stratonovich system, one would have to
incorporate the Itô-Stratonovich conversion term (cf. [24]), which, due to the
linear multiplicative noise, is a linear term being just a scalar multiple of
$u$, $v$, respectively. If one is interested in the exact form of the
correction term, we refer to [21]. We also refer to the discussion in [31],
where the constant accounting for the correction term is computed explicitly.

From now on, we shall consider the system (3.1)–(3.2) in Itô form, that is, as
in (3.6)–(3.7). Definitions 3.1 and 3.2 have to be adjusted in the obvious
way. We note that we can solve (3.1)–(3.2) by a straightforward modification
of the proof given here.

Before presenting our main result, we will first introduce the hypotheses on
$d$, $\gamma$, $\rho$ and the initial conditions $u_{0}$ and $v_{0}$, and on
$\sigma_{1}$, $\sigma_{2}$.

######  Hypothesis 3.4 (Existence).

Let $d\in\\{1,2,3\\}$. Let $\gamma>2$, and $m>2$,
$m_{0}>\frac{2(\gamma+1)}{\gamma}$ such that

$\frac{d}{2}-\rho\leq\frac{2}{m}+\frac{d}{m_{0}}\quad\rho<1-\frac{d}{2}$

and $p^{\ast}\geq 2$ and $p_{0}^{\ast}\geq 2$ such that

$\frac{1}{p^{\ast}}+\frac{2}{m}<1\quad\mbox{and}\quad\frac{2}{m_{0}}+\frac{1}{p_{0}^{\ast}}<\frac{\gamma}{\gamma+1}.$

Let

$l>1+\left(1-\frac{d}{2}-\rho\right)^{-1}$

Let us assume that the initial conditions $u_{0}$, $v_{0}$ are
${\mathcal{F}}_{0}$ measurable and satisfy

$\mathbb{E}|u_{0}|_{L^{2}}^{l}\vee\mathbb{E}|u_{0}|_{L^{p^{\ast}}}^{p^{\ast}_{0}}\vee\mathbb{E}|u_{0}|_{L^{\gamma+1}}^{\gamma+1}<\infty,\quad\mbox{and}\quad\mathbb{E}|v_{0}|_{H^{\rho}_{2}}^{m_{0}}\vee\mathbb{E}|v_{0}|_{H^{-\delta_{0}}_{m}}^{m_{0}}<\infty,\quad\delta_{0}<\frac{1}{m_{0}},$

and that $u_{0}$ and $v_{0}$ are a.e. nonnegative functions (nonnegative Borel
measures that are finite on compact subsets, respectively).

######  Hypothesis 3.5 (Noise).

Let $\\{\psi_{k}:k\in\mathbb{Z}\\}$ be the eigenfunctions of $-\Delta$ and
$\\{\nu_{k}:k\in\mathbb{Z}\\}$ the corresponding eigenvalues. Let $W_{1}$ and
$W_{2}$ be a pair of Wiener processes given by

${W}_{j}(t,x)=\sum_{k\in\mathbb{Z}}\lambda_{k}^{(j)}\psi_{k}(x)\beta^{(j)}_{k}(t),\quad
t\in[0,T],\quad j=1,2,\quad x\in{\mathcal{O}},$

where $\\{\lambda_{k}^{(j)}\;:\;k\in\mathbb{Z}\\}$, $j=1,2$, is a pair of
nonnegative sequences belonging to $\ell^{2}(\mathbb{Z})$,
$\\{\beta_{k}^{(j)}:[0,T]\times\Omega\to\mathbb{R}\;:\;(k,j)\in\mathbb{Z}\times\\{1,2\\}\\}$
is a family of independent standard real-valued Brownian motions. We assume
that $\lambda^{(j)}_{k}\leq C_{j}|\nu_{k}|^{-\delta_{j}}$, $j=1,2$, where
$\delta_{1}>\frac{1}{2}$, $\delta_{2}>\frac{1}{2}$, and $C_{1}>0$, $C_{2}>0$.

Compare also with Subsection 5.1 for details.

Under these hypotheses, the existence of a martingale solution can be shown.

######  Theorem 3.6.

Assume that Hypotheses 3.4–3.5 hold. Then there exists a martingale solution
to system (3.6)–(3.7) satisfying the following properties

  1. (i)

$u(t,x)\geq 0$ and $v(t,x)\geq 0$ ${\mathbb{P}}$-a.s., for a.e. $t\in[0,T]$
and a.e. $x\in{\mathcal{O}}$,

  2. (ii)

Let $p\geq 1$ and $\mathbb{E}|u_{0}|^{p+1}_{L^{p+1}}<\infty$. Then there
exists a constant $C_{0}(p,T)>0$ such that

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
T}|u(s)|_{L^{p+1}}^{p+1}\right]+\gamma
p(p+1)r_{u}\mathbb{E}\int_{0}^{T}\int_{{\mathcal{O}}}|u(s,x)|^{p+\gamma-2}|\nabla
u(s,x)|^{2}\,dx\,ds$
$\displaystyle{}+(p+1)\chi\mathbb{E}\int_{0}^{T}\int_{{\mathcal{O}}}|u(s,x)|^{p+1}|v(s,x)|^{2}\,dx\,ds\leq
C_{0}(p,T)\,\left(\mathbb{E}|u_{0}|_{L^{p+1}}^{p+1}+1\right).$

  3. (iii)

for any choice of parameters $\rho$, $m_{0}$, $l$ as in Hypothesis 3.4, there
exists a constant $C_{2}(T)>0$ such that

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
T}|v(s)|_{H^{\rho}_{2}}^{m_{0}}\right]+\mathbb{E}\left(\int_{0}^{T}|v(s)|_{H^{\rho+1}_{2}}^{2}\,ds\right)^{m_{0}/2}$
$\displaystyle\leq$ $\displaystyle
C_{2}(T)\left(1+\mathbb{E}|v_{0}|_{H^{\rho}_{2}}^{{m_{0}}}+\mathbb{E}|u_{0}|_{L^{2}}^{l}\right).$

We shall also collect a standard notion on uniqueness.

######  Definition 3.7.

The system (3.6)–(3.7) is called _pathwise unique_ if, whenever
$(u_{i},v_{i})$, $i=1,2$ are martingale solutions to the system (3.6)–(3.7) on
a common stochastic basis
$(\Omega,{{\mathcal{F}}},{{\mathbb{F}}},{\mathbb{P}},(W_{1},W_{2}))$, such
that ${\mathbb{P}}(u_{1}(0)=u_{2}(0))=1$ and
${\mathbb{P}}(v_{1}(0)=v_{2}(0))=1$, then

${\mathbb{P}}(u_{1}(t)=u_{2}(t))=1\quad\mbox{and}\quad{\mathbb{P}}(v_{1}(t)=v_{2}(t))=1,\quad\mbox{for
every }t\in[0,T].$

The theorem of Yamada and Watanabe [74] asserts that (weak) existence and
pathwise uniqueness of the solution of a stochastic equation is equivalent to
the existence of a unique strong solution. Therefore, showing pathwise
uniqueness is a fundamental step for obtaining the existence of a unique
strong solution. Under certain additional conditions, as collected below, we
are able to prove pathwise uniqueness.

######  Hypothesis 3.8.

Let $\delta_{0}\in(0,\frac{1}{\gamma})$, and
$\rho\geq\frac{d}{2}-\frac{1}{2}$.

######  Theorem 3.9.

Let $T>0$ and let
$\mathfrak{A}=(\Omega,{{\mathcal{F}}},({{\mathcal{F}}}_{t})_{t\in[0,T]},{\mathbb{P}})$
a probability space satisfying the usual conditions, let $(W_{1},W_{2})$ be a
Wiener process over $\mathfrak{A}$ on $H_{1}$ and $H_{2}$ and satisfying
hypothesis 3.5. Let initial data $u_{0}\in H^{-1}_{2}({{\mathcal{O}}})$ and
$v_{0}\in L^{2}({{\mathcal{O}}})$ together with the parameters
$m^{\ast},m_{0},p^{\ast},p^{\ast}_{0},l,\delta_{0}$ and $\rho$ satisfying
Hypothesis 3.4 and Hypothesis 3.8. Assume further that Hypothesis 3.5 holds.
Let $(u_{i},v_{i})$, $i=1,2$, be two solutions to the system (3.6)–(3.7) with
initial data $(u_{0},v_{0})$ and belonging ${\mathbb{P}}$-a.s. to
$C([0,T];H^{-1}_{2}({{\mathcal{O}}}))\times
C([0,T];H^{-\delta_{0}}_{2}({{\mathcal{O}}}))$. Then, the processes
$(u_{1},v_{1})$ and $(u_{2},v_{2})$ are indistinguishable in
$H^{-1}_{2}({{\mathcal{O}}})\times H^{-\delta_{0}}_{2}({{\mathcal{O}}})$.

######  Remark 3.10.

It can be shown that for dimension $d=1$ such indices
$m^{\ast},m_{0},p^{\ast},p^{\ast}_{0},l,\delta_{0}$ and $\rho$ satisfying
Hypothesis 3.4 and Hypothesis 3.8 can be found. Because of our condition
$\rho\geq\frac{d}{2}-\frac{1}{2}$, we cannot handle dimensions $d\in\\{2,3\\}$
as Hypothesis 3.4 then implies that $\rho<0$.

###### Proof.

The proof is postponed to Section 6. ∎

By this Theorem at hand, the following corollary is a consequence of the
Theorem of Yamada-Watanabe.

######  Corollary 3.11.

There exists a unique strong solution (in the stochastic sense) to the system
(3.6)–(3.7).

###### Proof.

This follows from Theorem 3.6 in combination with Theorem 3.9 and the Yamada-
Watanabe theorem, see [42, Appendix E] and [51, 38, 48]. ∎

The proof of Theorem 3.6 an application of Theorem 2.1 and consists of five
steps. However, to keep the proof itself simple, we will postpone several
technical results which are collected in Section 5.

###### Proof of Theorem 3.6.

In the first step, we are specifying the underlying Banach spaces. In the
second step, we shall construct the operator $\mathcal{V}$ for a truncated
system and show that the operator $\mathcal{V}$ satisfies the assumptions of
Theorem 2.1. In the third step, we localize via stopping times and glue the
fixed point solutions together, which exist by Theorem 2.1. In the fourth
step, we prove that the stopping times are uniformly bounded. In the fifth
step, we show that we indeed yield a martingale solution satisfying the above
properties.

  * Step (I)

The underlying space(s). Here we define the spaces on which the operator
$\mathcal{V}$ will act. Let the probability space
$\mathfrak{A}=(\Omega,{{\mathcal{F}}},\mathbb{F},{\mathbb{P}})$ be given and
let $W_{1}$ and $W_{2}$ be two independent $H_{1}$ and $H_{2}$-valued Wiener
processes defined over $\mathfrak{A}$ with covariances $Q_{1}$ and $Q_{2}$.
Let $W=(W_{1},W_{2})$, $H=H_{1}\times H_{2}$, with covariance operator

$Q=\left(\begin{matrix}Q_{1}&0\\\ 0&Q_{2}\end{matrix}\right).$

Let us define the Banach space

$\mathbb{Y}=L^{\gamma+1}(0,T;{L^{\gamma+1}}({{\mathcal{O}}}))\cap
L^{\infty}(0,T;{H^{-1}_{2}}({{\mathcal{O}}}))$

equipped with the norm

$\|\eta\|_{\mathbb{Y}}:=\|\eta\|_{L^{\gamma+1}(0,T;{L^{\gamma+1}})}+\|\eta\|_{L^{\infty}(0,T;{H^{-1}_{2}})},\quad\eta\in\mathbb{Y},$

and the reflexive Banach space

$\mathbb{Z}:=L^{m_{0}}(0,T;L^{m}({{\mathcal{O}}})),$

equipped with the norm

$\|\xi\|_{\mathbb{Z}}:=\|\xi\|_{L^{m_{0}}(0,T;L^{m})},\quad\xi\in\mathbb{Z}.$

Finally, let us fix an auxiliary Banach space
$\mathbb{H_{\rho}}:=L^{2}(0,T;H^{\rho+1}_{2}({{\mathcal{O}}}))\cap
L^{\infty}(0,T;{H^{\rho}_{2}}({{\mathcal{O}}}))$ equipped with the norm

(3.8)
$\|\xi\|_{\mathbb{H_{\rho}}}:=\|\xi\|_{L^{2}(0,T;H^{\rho+1}_{2})}+\|\xi\|_{L^{\infty}(0,T;{H^{\rho}_{2}})},\quad\xi\in\mathbb{H_{\rho}}.$

######  Remark 3.12.

If

$\frac{d}{2}-\rho\leq\frac{2}{m}+\frac{d}{m_{0}},$

then one can show by Sobolev embedding and interpolation theorems (see
Proposition A.4 in the appendix) that there exists a constant $C>0$ such that

$\|\xi\|_{\mathbb{Z}}\leq
C\|\xi\|_{\mathbb{H_{\rho}}},\quad\xi\in\mathbb{H_{\rho}}.$

Let us fix the compactly embedded reflexive Banach subspace of $\mathbb{Z}$ by

$\mathbb{Z}^{\prime}:=L^{m_{0}}(0,T;H^{\sigma}_{m}({{\mathcal{O}}}))\cap\mathbb{W}^{\alpha}_{m_{0}}(0,T;H^{-\delta}_{m}({{\mathcal{O}}}))$

equipped with the norm

$\|\xi\|_{\mathbb{Z}^{\prime}}:=\left(\|\xi\|_{L^{m_{0}}(0,T;H^{\sigma}_{m})}^{m_{0}}+\|\xi\|_{\mathbb{W}^{\alpha}_{m_{0}}(0,T;H^{-\delta}_{m})}^{m_{0}}\right)^{1/m_{0}},$

compare with Appendix B, where the compact embedding
$\mathbb{Z^{\prime}}\hookrightarrow\mathbb{Z}$ is discussed. The parameters
$\sigma>0$, $\alpha\in(0,1)$, $\delta>0$ are specified in Proposition 5.8.

Now, denote the space of progressively measurable (pairs of) processes
$\mathcal{M}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$ by

$\displaystyle\mathcal{M}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$
$\displaystyle:=$
$\displaystyle\Big{\\{}(\eta,\xi)\;\colon\;\eta,\xi:[0,T]\times\Omega\to{\mathcal{D}}^{\prime}({\mathcal{O}})\;\mbox{such
that}$ $\eta$ and $\xi$ are progressively measurable on $\mathfrak{A}$ and
$\displaystyle\qquad\mathbb{E}\|\eta\|^{2}_{\mathbb{Y}}<\infty\;\mbox{and}\;\mathbb{E}\|\xi\|^{m_{0}}_{\mathbb{Z}}<\infty\Big{\\}}$

equipped with the norm

(3.9)
$\|(\eta,\xi)\|_{\mathcal{M}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})}:=\left(\mathbb{E}\|\eta\|^{2}_{\mathbb{Y}}\right)^{\frac{1}{2}}+\left(\mathbb{E}\|\xi\|^{m_{0}}_{\mathbb{Z}}\right)^{\frac{1}{m_{0}}},\quad(\eta,\xi)\in\mathcal{M}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z}).$

Note that here, progressive measurability is meant relative to the Borel
$\sigma$-fields of the target spaces $H_{2}^{-1}({\mathcal{O}})$ and
$H^{\rho}_{2}({\mathcal{O}})$ respectively.

Finally, for fixed $R_{0},R_{1},R_{2}>0$, let us define the subspace
$\mathcal{X}_{\mathfrak{A}}=\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$ by

$\displaystyle\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$ $\displaystyle:=$
$\displaystyle\Bigg{\\{}(\eta,\xi)\in{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})\;\colon$
$\displaystyle\qquad\mathbb{E}\left[\sup_{0\leq s\leq
T}|\eta(s)|_{L^{p^{\ast}}}^{p_{0}^{\ast}}\right]^{\frac{1}{{p_{0}^{\ast}}}}\leq
R_{0},\;\mathbb{E}\|\xi\|^{m_{0}}_{\mathbb{Z}}\leq
R_{1},\;\mathbb{E}\|\xi\|_{\mathbb{H_{\rho}}}^{m_{0}}\leq R_{2},\;\mbox{and}$
$\displaystyle\qquad\quad\eta\;\mbox{and}\;\xi\;\mbox{are
nonnegative}\;\mathbb{P}\otimes\mbox{Leb-a.e.
in}\;{\mathcal{D}}^{\prime}({\mathcal{O}})\Bigg{\\}}.$

At this point, we note that by Remark 2.2, $\mathcal{X}_{\mathfrak{A}}$ is a
sequentially weak∗-closed and bounded subset of
$\mathcal{M}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$. Every
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}$ is a pair of a (equivalence class of
a) nonnegative function (or a nonnegative Borel measure that is finite on
compact subsets of ${\mathcal{O}}$).

######  Remark 3.13.

Note that in order to apply Theorem 2.1 formally, we will assume the obvious
modification (or extension) of its statement and proof, such that we can treat
pairs of spaces with different exponents (here,
$\mathcal{M}^{m}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$ would not make sense,
as in (3.9) either $m=2$ or $m=m_{0}$). Another easy yet not so elegant
workaround would be to replace $2$ by $m_{0}$ in (3.9).

  * Step (II)

The truncated system.

Let $\phi\in{\mathcal{D}}(\mathbb{R})$ be a smooth cutoff function that
satisfies

$\phi(x)\begin{cases}=0,&\mbox{ if }|x|\geq 2,\\\ \in[0,1],&\mbox{ if
}1<|x|<2,\\\ =1,&\mbox{ if }|x|\leq 1,\end{cases}$

and let $\phi_{\kappa}(x):=\phi(x/\kappa)$,
$x\in{\mathbb{R}},\,\kappa\in\mathbb{N}$. In addition, for any progressively
measurable pair of processes
$(\eta,\xi)\in{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$ let us
define for $t\in[0,T]$

(3.10) $\displaystyle h(\eta,\xi,t)$ $\displaystyle:=$
$\displaystyle\left(\int_{0}^{t}|\eta(s)|_{L^{\gamma+1}}^{\gamma+1}\,ds\right)^{\nu}+\left(\int_{0}^{t}|\xi(s)|_{L^{m}}^{m_{0}}\,ds\right)^{\nu},$

where $\nu\in(0,1]$ is chosen such that $\frac{1}{p_{0}^{\ast}}+\frac{\nu
m_{0}}{\gamma+1}\leq 1$ and $\frac{1}{p_{0}^{\ast}}+\frac{\nu}{m_{0}}\leq 1$

Let us consider the truncated system given by

(3.11)
$\left\\{\begin{array}[]{rcl}d{u_{\kappa}}(t)&=&\left[r_{u}\Delta(u_{\kappa}(t))^{[\gamma]}-\chi\phi_{\kappa}(h(u_{\kappa},v_{\kappa},t))u_{\kappa}(t)v_{\kappa}^{2}(t)\right]\,dt+\sigma_{1}u_{\kappa}(t)dW_{1}(t),\\\
u_{\kappa}(0)&=&u_{0},\phantom{\Big{|}}\end{array}\right.$

and

(3.12) $\left\\{\begin{array}[]{rcl}d{v_{\kappa}}(t)&=&\left[r_{v}\Delta
v_{\kappa}(t)+\phi_{\kappa}(h(u_{\kappa},v_{\kappa},t))u_{\kappa}(t)v_{\kappa}^{2}(t)\right]\,dt+\sigma_{2}v_{\kappa}(t)dW_{2}(t).\\\
{v_{\kappa}}(0)&=&v_{0}.\phantom{\Big{|}}\end{array}\right.$

We shall show the existence of a martingale solution to system (3.11)–(3.12).

######  Proposition 3.14.

For any $\kappa\in\mathbb{N}$, there exists constants $R_{0}>0$, $R_{1}>0$,
$R_{2}>0$, depending on $\kappa$, such that there exists a martingale solution
$(u_{\kappa},v_{\kappa})$ to system (3.11)–(3.12) contained in
$\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$.

###### Proof of Proposition 3.14.

The proof consists of several steps. First, we shall define an operator
denoted by $\mathcal{V}_{\kappa}$ which satisfies the assumptions of Theorem
2.1, yielding the existence of a martingale solution.

    * Step (a)

Definition of the operator ${\mathcal{V}}_{\kappa}$.

First, define

${\mathcal{V}}_{\kappa}:=\mathcal{V}_{\kappa,{\mathfrak{A}}}:\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})\to{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$

by

$\mathcal{V}_{\kappa}(\eta,\xi):=(u_{\kappa},v_{\kappa}),\quad\text{for}\quad(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$

where $u_{\kappa}$ is a solution to

(3.13)
$\left\\{\begin{array}[]{rcl}d{u_{\kappa}}(t)&=&\left[r_{u}\Delta(u_{\kappa}(t))^{[\gamma]}-\chi\phi_{\kappa}(h(\eta,\xi,t))\;\eta(t)\,\xi^{2}(t)\right]\,dt+\sigma_{1}u_{\kappa}(t)\,dW_{1}(t),\\\
u_{\kappa}(0)&=&u_{0},\phantom{\Big{|}}\end{array}\right.$

and $v_{\kappa}$ is a solution to

(3.14) $\left\\{\begin{array}[]{rcl}d{v_{\kappa}}(t)&=&\left[r_{v}\Delta
v_{\kappa}(t)+\phi_{\kappa}(h(\eta,\xi,t))\;\eta(t)\,\xi^{2}(t)\right]\,dt+\sigma_{2}v_{\kappa}(t)\,dW_{2}(t),\\\
{v_{\kappa}}(0)&=&v_{0}.\phantom{\Big{|}}\end{array}\right.$

The operator $\mathcal{V}_{\kappa}$ is well defined for
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$. In fact, by
Theorem 5.3, given such a pair of processes $(\eta,\xi)$, the existence of a
unique solution $u_{\kappa}$ to (3.13) with

(3.15) $\displaystyle\mathbb{E}\|u_{\kappa}\|_{\mathbb{Y}}^{2}$
$\displaystyle\leq$ $\displaystyle C_{1}(\kappa,T)$

follows. By Proposition 5.6 and Proposition 5.7, the existence of a unique
solution $v_{\kappa}$ to (3.14) with

$\mathbb{E}\|v_{\kappa}\|_{\mathbb{H_{\rho}}}^{m_{0}}\leq\mathbb{E}|v_{0}|_{H^{\rho}_{2}}^{m_{0}}+C(\kappa,T)R_{0},$

follows.

    * Step (b)

The ${\mathcal{V}}_{\kappa}$-invariant set ${\mathcal{X}}$.

Let

$\displaystyle
C_{0}(T)\Big{(}\mathbb{E}|u_{0}|^{p^{\ast}_{0}}_{L^{p^{\ast}}}+1\Big{)}\leq
R_{0},$

where the constant $C_{0}(\gamma,T)$ is as in (5.5). Then, due to Proposition
5.5, we know that

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq
T}|u_{\kappa}(s)|^{p^{\ast}_{0}}_{L^{p^{\ast}}}\right]\vee\mathbb{E}\left[\sup_{0\leq
s\leq T}|u_{\kappa}(s)|_{L^{\gamma+1}}^{\gamma+1}\right]\leq R_{0}.$

Furthermore, by Proposition 5.6, the existence of a unique solution to (3.14)
such that

(3.16) $\displaystyle\mathbb{E}\|v_{\kappa}\|_{\mathbb{H_{\rho}}}^{m_{0}}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}|v_{0}|^{m_{0}}_{H^{\rho}_{2}}+R_{0}C_{2}(\kappa,T)$

follows.

Let $R_{2}:=\mathbb{E}|v_{0}|^{m_{0}}_{H^{\rho}_{2}}+R_{0}C_{2}(\kappa,T)$.
Finally, by Proposition 5.8 we can show that for $R_{1}\geq
C(T)\left\\{\mathbb{E}|v_{0}|_{H_{m}^{-\delta_{0}}}^{m_{0}}+C(\kappa)\right\\}$
(here the constants are given by Proposition 5.8)

(3.17) $\displaystyle\mathbb{E}\|v_{\kappa}\|_{\mathbb{Z}}^{m_{0}}$
$\displaystyle\leq$ $\displaystyle R_{1}.$

Then, ${\mathcal{V}}_{\kappa}$ maps
$\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$ into itself. Hence, assumption
(i) of Theorem 2.1 is satisfied.

    * Step (c)

Continuity of ${\mathcal{V}}_{\kappa}$ on ${\mathcal{X}}$.

Next we show that assumption (ii) of Theorem 2.1 is satisfied. We need to show
that the restriction of the operator $\mathcal{V}_{\kappa}$ to
$\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$ is continuous in the strong
topology of ${{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$. Let
$\\{(\eta^{(n)},\xi^{(n)}):n\in\mathbb{N}\\}\subset\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$
be a sequence converging strongly to $(\eta,\xi)$ in
${{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$.

Firstly, due to Proposition 5.11 and Remark 3.12, we know that the sequence
$\\{u_{\kappa}^{(n)}:n\in\mathbb{N}\\}$ where $u_{\kappa}^{(n)}$ solves

(3.18)
$\left\\{\begin{array}[]{rcl}d{u_{\kappa}^{(n)}}(t)&=&\left[r_{u}\Delta(u_{\kappa}^{(n)}(t))^{[\gamma]}-\chi\phi_{\kappa}(h(\eta^{(n)},\xi^{(n)},t))\,\eta^{(n)}(t)\,(\xi^{(n)}(t))^{2}\right]\,dt\\\
&&\quad+\sigma_{1}u_{\kappa}^{(n)}(t)\,dW_{1}(t),\\\
u_{\kappa}^{(n)}(0)&=&u_{0}.\phantom{\Big{|}}\end{array}\right.$

converges in ${{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$ to
$u_{\kappa}$ which solves (3.13).

Secondly, due to Proposition 5.10 and keeping Remark 3.12 in mind, it follows
that the sequence $\\{v_{\kappa}^{(n)}:n\in\mathbb{N}\\}$, where
$v_{\kappa}^{(n)}$ solves

(3.19) $\left\\{\begin{array}[]{rcl}d{v_{\kappa}^{(n)}}(t)&=&\left[r_{v}\Delta
v_{\kappa}^{(n)}(t)+\phi_{\kappa}(h(\eta^{(n)},\xi^{(n)},t))\,\eta^{(n)}(t)\,(\xi^{(n)}(t))^{2}\right]\,dt\\\
&&\quad+\sigma_{2}v_{\kappa}^{(n)}(t)\,dW_{2}(t),\\\
v_{\kappa}^{(n)}(0)&=&v_{0}.\phantom{\Big{|}}\end{array}\right.$

converges in ${{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$ to
$v_{\kappa}$, where $v_{\kappa}$ solves (3.14).

This gives that the operator

${\mathcal{V}}_{\kappa}|_{\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})}:\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})\to{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$

is continuous.

    * Step (d)

Tightness.

Next we show that assumption (iii) of Theorem 2.1 is satisfied.

First, note that the embedding
${\mathbb{Z}^{\prime}}\hookrightarrow{\mathbb{Z}}$ is compact by the Aubin-
Lions-Simon Lemma B.1.

Second, by Proposition 5.8 it follows that for any $\kappa\in\mathbb{N}$ there
exists a constant $C=C(\kappa,T)>0$ such that for all
$(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$

$\mathbb{E}\,|v_{\kappa}|^{m_{0}}_{\mathbb{Z}^{\prime}}\leq C,$

where $v_{\kappa}$ solves (3.14). Observe that we have the compact embedding
$L^{\gamma+1}({{\mathcal{O}}})\hookrightarrow H^{-1}_{2}({{\mathcal{O}}})$.
Hence, by Proposition 5.12 and by standard arguments (cf. [23]), we know that
the laws of the set
$\\{u_{\kappa}:(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\\}$
are tight on $C([0,T];H^{-1}_{2}({{\mathcal{O}}}))$. By Proposition 5.12 and
the Aubin-Lions-Simon Lemma B.1, the laws of
$\\{u_{\kappa}:(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\\}$
are tight on $L^{\gamma+1}(0,T;L^{\gamma+1}({{\mathcal{O}}}))$. Therefore, it
follows that the set
$\\{\mathcal{V}_{\kappa}(\eta,\xi):(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\\}$
is tight on $\mathbb{Y}\times\mathbb{Z}$, which implies assumption (iii) of
Theorem 2.1 for

$\mathbb{X}^{\prime}:=L^{\gamma+1}(0,T;L^{\gamma+1}({{\mathcal{O}}}))\times\mathbb{Z}^{\prime}\hookrightarrow\mathbb{Y}\times\mathbb{Z}=:\mathbb{X}$

    * Step (e)

Continuity of paths.

To show assumption (iv) of Theorem 2.1, we notice that by choosing

$U:=H^{-1}_{2}({\mathcal{O}})\times H_{2}^{\rho}({{\mathcal{O}}}),$

it follows from Proposition 5.12 and standard arguments that

$\mathcal{V}_{\kappa}(\eta,\xi)\in C([0,T];U)\quad{\mathbb{P}}\text{-a.s.}$

for all $(\eta,\xi)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$.

    * Step (f)

Existence of a fixed point.

Finally, by Theorem 2.1, for each $\kappa\in\mathbb{N}$, there exists a
probability space
$\tilde{\mathfrak{A}}_{\kappa}=(\tilde{\Omega}_{\kappa},\tilde{{{\mathcal{F}}}}_{\kappa},\tilde{\mathbb{F}}_{\kappa},\tilde{{\mathbb{P}}}_{\kappa})$,
a Wiener process
$\tilde{W}^{\kappa}=(\tilde{W}^{\kappa}_{1},\tilde{W}^{\kappa}_{2})$, modeled
on $\tilde{\mathfrak{A}}_{\kappa}$, and elements

$(\tilde{u}_{\kappa},\tilde{v}_{\kappa})\in\mathcal{X}_{\tilde{\mathfrak{A}}_{\kappa}}(R_{0},R_{1},R_{2})\cap\mathbb{D}([0,T);U)$

such that we have a $\tilde{\mathbb{P}}$-a.s. fixed point

$\mathcal{V}_{\kappa,\tilde{{\mathfrak{A}}}_{\kappa}}(\tilde{u}_{\kappa}(t),\tilde{v}_{\kappa}(t))=(\tilde{u}_{\kappa}(t),\tilde{v}_{\kappa}(t))$

for every $t\in[0,T]$. Due to the construction of
$\mathcal{V}_{\tilde{{\mathfrak{A}}}}$, the pair
$(\tilde{u}_{\kappa},\tilde{v}_{\kappa})$ solves the system (3.13)–(3.14) over
the stochastic basis $\tilde{{\mathfrak{A}}}_{\kappa}$ with the Wiener noise
$\tilde{W}^{\kappa}$, see Section 4 for details.

∎

  * Step (III)

Here, we will construct a family of solutions
$\\{(\bar{u}_{\kappa},\bar{v}_{\kappa}):\kappa\in\mathbb{N}\\}$ following the
solution to the original problem until a stopping time
${\bar{\tau}}_{\kappa}$. In particular, we will introduce for each
$\kappa\in\mathbb{N}$ a new pair of processes
$({\bar{u}}_{\kappa},{\bar{v}}_{\kappa})$ following the Klausmeier system up
to the stopping time ${\bar{\tau}}_{\kappa}$. Besides, we will have
$({\bar{u}}_{\kappa},{\bar{v}}_{\kappa})|_{[0,{\bar{\tau}}_{\kappa})}=({\bar{u}}_{\kappa+1},{\bar{v}}_{\kappa+1})|_{[0,{\bar{\tau}}_{\kappa})}$.

Let us start with $\kappa=1$. From Proposition 3.14, we know there exists a
martingale solution consisting of a probability space
$\mathfrak{A}_{1}=(\Omega_{1},{{\mathcal{F}}}_{1},\mathbb{F}_{1},{\mathbb{P}}_{1})$,
two independent Wiener processes $({W}^{1}_{1},{W}^{1}_{2})$ defined over
$\mathfrak{A}_{1}$, and a couple of processes $(u_{1},v_{1})$ solving
${\mathbb{P}}_{1}$–a.s. the system

$\displaystyle\left\\{\begin{array}[]{rcl}du_{1}(t)&=&\left[r_{u}\Delta
u_{1}^{[\gamma]}(t)-\chi\,\phi_{1}(h(u_{1},v_{1},t))\,u_{1}(t)\,v_{1}^{2}(t)\right]dt+\sigma_{1}u_{1}(t)\,d{W}^{1}_{1}(t),\\\
dv_{1}(t)&=&\left[r_{v}\Delta
v_{1}(t)+\,\phi_{1}(h(u_{1},v_{1},t))\,u_{1}(t)\,v_{1}^{2}(t)\right]\,dt+\sigma_{2}v_{1}(t)\,d{W}^{1}_{2}(t),\\\
(u_{1}(0),v_{1}(0))&=&(u_{0},v_{0}).\end{array}\right.$

Let us define now the stopping time

$\displaystyle\tau_{1}^{\ast}:=\inf\\{s\geq 0\;\colon\;h(u_{1},w_{1},s)\geq
1\\}.$

Observe, on the time interval $[0,\tau_{1}^{\ast})$, the pair $(u_{1},v_{1})$
solves the system given in (1.6). Now, we define a new pair of processes
$(\bar{u}_{1},\bar{v}_{1})$ following $(u_{1},v_{1})$ on $[0,\tau_{1}^{\ast})$
and extend this processes to the whole interval $[0,T]$ in the following way.
First, we put $\overline{\mathfrak{A}}_{1}:=\mathfrak{A}_{1}$ and
$\bar{{W}}_{j}^{1}:={W}_{j}^{1}$, $j=1,2$, and let us introduce the processes
$y_{1}$ and $y_{2}$ being a strong solution over $\overline{\mathfrak{A}}_{1}$
to

$\displaystyle dy_{1}(t,u_{1}(\tau^{\ast}_{1}),\sigma)$ $\displaystyle=$
$\displaystyle r_{u}\Delta
y_{1}^{[\gamma]}(t,u_{1}(\tau^{\ast}_{1}),\sigma)\,dt+\sigma_{1}y_{1}(t,u_{1}(\tau^{\ast}_{1}),\sigma)\,d(\theta_{\sigma}\bar{{W}}^{1}_{1})(t)$
(3.21) $\displaystyle y_{1}(0,u_{1}(\tau^{\ast}_{1}))$ $\displaystyle=$
$\displaystyle u_{1}(\tau^{\ast}_{1}){},\quad t\geq 0,$ $\displaystyle
y_{2}(t,v_{1}(\tau^{\ast}_{1}),\sigma)$ $\displaystyle=$ $\displaystyle
e^{r_{v}\Delta t}v_{1}(\tau^{\ast}_{1})$ (3.22)
$\displaystyle{}+\int_{0}^{t}e^{(r_{v}\Delta)(t-s)}\sigma_{2}y_{2}(s,v_{1}(\tau^{\ast}_{1}),\sigma)\,d(\theta_{\sigma}\bar{{W}}^{1}_{2})(s),\quad
t\geq 0.$

Here, $\theta_{\sigma}$ is the shift operator which maps ${W}_{j}(t)$ to
${W}_{j}(t+\sigma)-W_{j}(\sigma)$. Since the couple $(u_{1},v_{1})$ is
continuous in $H_{2}^{-1}({{\mathcal{O}}})\times
H^{\rho}_{2}({{\mathcal{O}}})$, we know that $u_{1}(\tau^{\ast}_{1})$ and
$v_{1}(\tau^{\ast}_{1})$ are well defined random variables belonging
${\mathbb{P}}_{1}$-a.s. to $L^{2}({{\mathcal{O}}})$ and
$H^{\rho}_{2}({{\mathcal{O}}})$, respectively. By [2, Theorem 2.5.1] the
existence of a unique solution $y_{1}$ over $\overline{\mathfrak{A}}_{1}$ to
(3.21) in $H^{-1}_{2}({{\mathcal{O}}})$ is given. Since
$(e^{t(r_{v}\Delta-\tilde{\alpha}\operatorname{Id})})_{t\geq 0}$ is an
analytic semigroup on $H^{\rho}_{2}({{\mathcal{O}}})$ for $\tilde{\alpha}>0$,
the existence of a unique solution $y_{2}$ over $\overline{\mathfrak{A}}_{1}$
to (LABEL:eq2) in $H^{\rho}_{2}({{\mathcal{O}}})$ can be shown by standard
methods, cf. [15]. Furthermore, it is straightforward to verify that the
assumptions on the initial conditions are satisfied. Now, let us define two
processes $\bar{u}_{1}$ and $\bar{v}_{1}$ being identical to $u_{1}$ and
$v_{1}$, respectively, on the time interval $[0,\tau^{\ast}_{1})$ and
following the porous media, respective, the heat equation (with lower order
terms) with noise and without nonlinearity, i.e.,
$y_{1}(\cdot,u_{1}(\tau^{\ast}_{1}),\tau^{\ast}_{1})$ and
$y_{2}(\cdot,v_{1}(\tau^{\ast}_{1}),\tau^{\ast}_{1})$, afterwards. In
particular, let

$\bar{u}_{1}(t)=\begin{cases}u_{1}(t)&\mbox{ for }0\leq t<\tau^{\ast}_{1},\\\
y_{1}(t,u_{1}(\tau^{\ast}_{1}),\tau^{\ast}_{1})&\mbox{ for
}\tau^{\ast}_{1}\leq t\leq T,\end{cases}$

and

$\bar{v}_{1}(t)=\begin{cases}v_{1}(t)&\mbox{ for }0\leq t<\tau_{1}^{\ast},\\\
y_{2}(t,v_{1}(\tau^{\ast}_{1}),\tau_{1}^{\ast})&\mbox{ for
}\tau_{1}^{\ast}\leq t\leq T.\end{cases}$

Let us now construct the probability space and the processes for the next time
interval. First, let $(u_{1}(\tau^{\ast}_{1}),v_{1}(\tau^{\ast}_{1}))$ have
probability law $\mu_{1}$ on $H_{2}^{-1}({{\mathcal{O}}})\times
H^{\rho}_{2}({{\mathcal{O}}})$. Again, from Proposition 3.14, we know there is
a martingale solution consisting of a probability space
$\mathfrak{A}_{2}=(\Omega_{2},{{\mathcal{F}}}_{2},\mathbb{F}_{2},{\mathbb{P}}_{2})$,
a pair of independent Wiener processes $({W}_{1}^{2},{W}_{2}^{2})$, a couple
of processes $(u_{2},v_{2})$ solving ${\mathbb{P}}_{2}$-a.s. the system

$\displaystyle\left\\{\begin{array}[]{rcl}du_{2}(t)&=&\left[r_{u}\Delta
u^{[\gamma]}_{2}(t)-\chi\,\phi_{2}(h(u_{2},v_{2},t))\,u_{2}(t)\,v_{2}^{2}(t)\right]dt+\sigma_{2}u_{2}(t)\,d{W}^{2}_{1}(t),\\\
dv_{2}(t)&=&\left[r_{v}\Delta
v_{2}(t)+\phi_{2}(h(u_{2},v_{2},t))\,u_{2}(t)\,v_{2}^{2}(t)\right]\,dt+\sigma_{2}v_{2}(t)\,d{W}^{2}_{2}(t),\\\
(u_{2}(0),v_{2}(0))&\sim&\mu_{1}.\end{array}\right.$

with initial condition $(u_{2}(0),v_{2}(0))$ having law $\mu_{1}$. Let us
define now the stopping times on $\mathfrak{A}_{2}$,

$\displaystyle\tau_{2}^{\ast}:=\inf\\{s\geq 0:h(u_{2},v_{2},s)\geq 2\\}.$

Let
$\overline{\mathfrak{A}}_{1}:=(\overline{\Omega}_{1},\overline{{{\mathcal{F}}}}_{1},\overline{\mathbb{F}}_{1},\overline{{\mathbb{P}}}_{1}):=\mathfrak{A}_{1}$,
with
$\overline{\mathbb{F}}_{1}:=(\overline{{{\mathcal{F}}}}^{1}_{t})_{t\in[0,T]}$.
Let $\overline{\Omega}_{2}:=\overline{\Omega}_{1}\times\Omega_{2}$,
$\overline{{{\mathcal{F}}}}_{2}:=\overline{{{\mathcal{F}}}}_{1}\otimes{{\mathcal{F}}}_{2}$,
$\overline{{\mathbb{P}}}_{2}:=\overline{{\mathbb{P}}}_{1}\otimes{\mathbb{P}}_{2}$
and let
$\overline{\mathbb{F}}_{2}:=(\overline{{{\mathcal{F}}}}^{2}_{t})_{t\in[0,T]}$,
where

$\overline{{{\mathcal{F}}}}^{2}_{t}:=\begin{cases}\overline{{{\mathcal{F}}}}^{1}_{t},&\mbox{if}\quad
t<\tau_{1}^{\ast},\\\ {{\mathcal{F}}}^{2}_{t-\tau_{1}^{\ast}},&\mbox{if}\quad
t\geq\tau_{1}^{\ast}.\end{cases}$

Let
$\overline{\mathfrak{A}}_{2}:=(\overline{\Omega}_{2},\overline{{{\mathcal{F}}}}_{2},\overline{\mathbb{F}}_{2},\overline{{\mathbb{P}}}_{2})$.
Finally, let us set for $j=1,2$

$\bar{{W}}_{j}^{2}(t):=\begin{cases}\bar{{W}}^{1}_{j}(t),&\mbox{if}\quad
t<\tau_{1}^{\ast},\\\
{W}^{2}_{j}({t-\tau_{1}^{\ast}})+\bar{{W}}^{1}_{j}(\tau_{1}^{\ast}),&\mbox{if}\quad
t\geq\tau_{1}^{\ast}.\end{cases}$

which give independent Wiener processes for $j=1,2$, w.r.t. the filtration
$\overline{\mathbb{F}}_{2}$.

Now, let us define two processes $\bar{u}_{2}$ and $\bar{v}_{2}$ being
identical to $\bar{u}_{1}$ and $\bar{v}_{1}$, respectively, on the time
interval $[0,\tau^{\ast}_{1})$, being identical to $u_{2}$ and $v_{2}$ on the
time interval $[\tau^{\ast}_{1},\tau^{\ast}_{1}+\tau_{2}^{\ast})$ and
following the porous media, respective, the heat equation (with lower order
terms) with multiplicative noise, afterwards. Let us note, for any initial
condition having distribution equal to $u_{2}(\tau_{2}^{\ast})$ and
$v_{2}(\tau^{\ast}_{2})$ that there exists a strong solutions
$y_{1}(\cdot,\cdot,\tau^{\ast}_{2}+\tau^{\ast}_{1})$ and
$y_{2}(\cdot,\cdot,\tau^{\ast}_{2}+\tau^{\ast}_{1})$ of the systems (3.21) and
(LABEL:eq2), respectively, on $\overline{\mathfrak{A}}_{2}$. Let for
$t\in[0,T]$

$\bar{u}_{2}(t)=\begin{cases}\bar{u}_{1}(t)&\mbox{ for }0\leq
t<\tau^{\ast}_{1},\\\ u_{2}(t-\tau_{1}^{\ast})&\mbox{ for }\tau^{\ast}_{1}\leq
t\leq\tau^{\ast}_{1}+\tau_{2}^{\ast},\\\
y_{1}(t-(\tau^{\ast}_{1}+\tau^{\ast}_{2}),u_{2}(\tau^{\ast}_{2}),\tau^{\ast}_{1}+\tau^{\ast}_{2})&\mbox{
for }\tau^{\ast}_{2}+\tau^{\ast}_{1}\leq t\leq T,\end{cases}$
$\bar{v}_{2}(t)=\begin{cases}v_{1}(t)&\mbox{ for }0\leq t<\tau_{1}^{\ast},\\\
v_{2}(t-\tau_{1}^{\ast})&\mbox{ for }\tau^{\ast}_{1}\leq
t\leq\tau^{\ast}_{1}+\tau_{2}^{\ast},\\\
y_{2}(t-(\tau^{\ast}_{1}+\tau^{\ast}_{2}),v_{2}(\tau^{\ast}_{2}),\tau_{1}^{\ast}+\tau^{\ast}_{2})&\mbox{
for }\tau_{1}^{\ast}+\tau^{\ast}_{2}\leq t\leq T.\end{cases}$

In the same way we will construct for any $\kappa\in\mathbb{N}$ a probability
space $\overline{\mathfrak{A}}_{\kappa}$, a pair of independent Wiener
processes $(\bar{{W}}^{1}_{\kappa},\bar{{W}}^{2}_{\kappa})$, over
$\overline{\mathfrak{A}}_{\kappa}$ and a pair of processes
$(\bar{u}_{\kappa},\bar{v}_{\kappa})$ starting at $(u_{0},v_{0})$ and solving
system (3.11)-(3.12) up to time

(3.24)
$\displaystyle\bar{\tau}_{\kappa}:=\tau_{1}^{\ast}+\cdots+\tau_{\kappa}^{\ast}$

and following the porous media, respective, the heat equation afterwards.
Besides, we know that
$(\bar{u}_{\kappa},\bar{v}_{\kappa})|_{[0,\bar{\tau}_{\kappa-1})}=(\bar{u}_{\kappa-1},\bar{v}_{\kappa-1})|_{[0,\bar{\tau}_{\kappa-1})}$.

  * Step (IV)

Uniform bounds on the stopping time.

Let us consider the family
$\\{(\bar{u}_{\kappa},\bar{v}_{\kappa})\;\colon\;\kappa\in\mathbb{N}\\}$. The
next aim is to show that there exists $R_{0},R_{1}$ and $R_{2}>0$ independent
of $\kappa\in\mathbb{N}$ such that
$\\{(\bar{u}_{\kappa},\bar{v}_{\kappa})\;\colon\;\kappa\in\mathbb{N}\\}\subset\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})$.

First, that due to Proposition 5.13 there exists a constant $C_{0}(1,T)>0$
such that

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq\bar{\tau}_{\kappa}\wedge
T}|\bar{u}_{\kappa}(s)|_{L^{2}}^{2}\right]+2\gamma
r_{u}\mathbb{E}\int_{0}^{\bar{\tau}_{\kappa}\wedge
T}\int_{{\mathcal{O}}}|\bar{u}_{\kappa}(s,x)|^{\gamma-1}|\nabla\bar{u}_{\kappa}(s,x)|^{2}\,dx\,ds$
$\displaystyle{}+2\chi\mathbb{E}\int_{0}^{\bar{\tau}_{\kappa}\wedge
T}\int_{{\mathcal{O}}}|\bar{u}_{\kappa}(s,x)|^{2}|\bar{v}_{\kappa}(s,x)|^{2}\,dx\,ds\leq
C_{0}(1,T)\,\left(\mathbb{E}|u_{0}|_{L^{2}}^{2}+1\right).$

From above we can conclude that there exists a constant $C>0$ such that

$\displaystyle\mathbb{E}\|\bar{u}_{\kappa}\mathbbm{1}_{[0,\bar{\tau}_{\kappa}\wedge
T]}\|_{\mathbb{Y}}^{2}\leq C.$

Here, it is important that $\bar{u}_{\kappa}\geq 0$. This estimate can be
extended to the time interval $[0,T]$ by standard results (see e.g. [2] and
Proposition 5.13). Next, let us assume $R_{1}>0$ is that large that

(3.26) $\displaystyle
C(T)\left(\mathbb{E}|v_{0}|_{H^{-\delta_{0}}_{m}}^{m_{0}}+R_{0}^{\delta_{1}}\,R_{2}^{\delta_{1}}\right)\leq
R_{2},$

where the constants $C(T)$, $\delta_{0}$, $\delta_{1}$ are given in
Proposition 5.14. Observe, by Proposition 5.14, we have for $l$ as in
Hypothesis 3.4

(3.27) $\displaystyle\hskip
25.83325pt\mathbb{E}\|\bar{v}_{\kappa}\|_{\mathbb{Z}}^{m_{0}}\leq
C(T)\left(\mathbb{E}|v_{0}|_{H^{-\delta_{0}}_{m}}^{m_{0}}+\Bigg{(}\mathbb{E}\left(\int_{0}^{\bar{\tau}_{\kappa}\wedge
T}u_{\kappa}^{2}(s)v_{\kappa}^{2}(s)\,ds\right)^{l}\Bigg{)}^{\delta_{1}}\,R_{1}^{\delta_{1}}\right).$

Then, by Proposition 5.13, we know that for $p=1$, the term

$\mathbb{E}\left(\int_{0}^{\bar{\tau}_{\kappa}\wedge
T}u_{\kappa}^{2}(s)v_{\kappa}^{2}(s)\,ds\right)^{l}\leq\mathbb{E}|u_{0}|_{L^{2}}^{l},$

which can be uniformly bounded by $R_{0}^{l}$. Hence, we conclude by (3.26)
that $\mathbb{E}\|\bar{v}_{\kappa}\|_{\mathbb{Z}}^{m_{0}}\leq R_{1}$ and the
choice of $R_{2}$ that
$\mathbb{E}\|\bar{v}_{\kappa}\|_{\mathbb{H_{\rho}}}^{m_{0}}\leq R_{2}$.
Finally, by Proposition A.4 we know for
$\frac{d}{2}-\rho\leq\frac{2}{m}+\frac{d}{m_{0}}$ that
$\mathbb{H_{\rho}}\hookrightarrow\mathbb{Z}$. Due to the choice of $m$ and
$m_{0}$, we know that the inequality is satisfied. Hence, setting

(3.28) $\displaystyle R_{1}$ $\displaystyle\geq$ $\displaystyle R_{2},$

we know that $\mathbb{E}\|\bar{v}_{\kappa}\|_{\mathbb{Z}}^{m_{0}}\leq R_{1}$.
In particular, there exists $R_{0},R_{1}$ and $R_{2}$ such that for all
$\kappa\in\mathbb{N}$ we have

(3.29) $\mathbb{E}\left[\sup_{0\leq s\leq
T}|\bar{u}_{\kappa}(s)|_{L^{p^{\ast}}}^{p^{\ast}_{0}}\right]\leq
R_{0}^{p_{0}^{\ast}},$

$\mathbb{E}\|\bar{v}_{\kappa}\|_{\mathbb{Z}}^{m_{0}}\leq R_{1}$, and
$\mathbb{E}\|\bar{v}_{\kappa}\|_{\mathbb{H_{\rho}}}^{m_{0}}\leq R_{2}$.

  * Step (V)

Passing on to the limit.

In the final step, we will show that ${\mathbb{P}}$-a.s. a martingale solution
to (3.6)-(3.7) exists.

######  Claim 3.15.

There exists a measurable set $\tilde{\Omega}\subset\Omega$ with
${\mathbb{P}}(\tilde{\Omega})=1$ such that a martingale solution $(u,v)$ to
the system (3.6)-(3.7) exists on $\tilde{\Omega}$.

###### Proof.

For any $\kappa\in\mathbb{N}$, let us define the set

$A_{\kappa}:=\left\\{\omega\in\Omega\;\colon\;\bar{\tau}_{\kappa}(\omega)\geq
T\right\\}.$

It can be clearly observed that there exists a progressively measurable
process $(u,v)$ over $\mathfrak{A}$ such that $(u,v)$ solves
${\mathbb{P}}$–a.s. the integral equation given by (3.6)-(3.7) up to time $T$.
In particular, we have for the conditional probability

${\mathbb{P}}\left(\\{\mbox{there exists a solution $(u,v)$ to
\eqref{eq:uIto}-\eqref{eq:vIto}}\\}\mid A_{\kappa}\right)=1.$

Set $\tilde{\Omega}=\bigcup_{\kappa=1}^{\infty}A_{\kappa}$. Then, it is
elementary to verify that

$\displaystyle\qquad{\mathbb{P}}\left(\left\\{\mbox{there exists a solution
$(u,v)$ to \eqref{eq:uIto}-\eqref{eq:vIto}}\right\\}\cap\tilde{\Omega}\right)$
$\displaystyle=$
$\displaystyle\lim_{\kappa\to\infty}{\mathbb{P}}\left(\\{\mbox{there exists a
solution $(u,v)$ to \eqref{eq:uIto}-\eqref{eq:vIto}}\\}\cap A_{\kappa}\right)$
$\displaystyle=$
$\displaystyle\lim_{\kappa\to\infty}\left[{\mathbb{P}}\left(\\{\mbox{there
exists a solution $(u,v)$ to \eqref{eq:uIto}-\eqref{eq:vIto}}\\}\mid
A_{\kappa}\right){\mathbb{P}}\left(A_{\kappa}\right)\right].$

Since ${\mathbb{P}}\left(\\{\mbox{a solution $(u,v)$ to
\eqref{eq:uIto}-\eqref{eq:vIto} exists}\\}\mid A_{\kappa}\right)=1$, it
remains to show that ${\lim_{\kappa\to\infty}}$ ${\mathbb{P}}(A_{\kappa})=1$.
Then, as $A_{\kappa}\subset A_{\kappa+1}$ for $\kappa\in\mathbb{N}$, it
follows that

${\mathbb{P}}\left(\left\\{\mbox{there exists a solution $(u,v)$ to
\eqref{eq:uIto}-\eqref{eq:vIto}}\right\\}\cap\tilde{\Omega}\right)=1.$

However, due to Step IV, there exists a constant $C(T)>0$ such that

$\mathbb{E}\left[h(\bar{u}_{\kappa},\bar{v}_{\kappa},t)\right]\leq C(T),\quad
t\in[0,T],\;\kappa\in\mathbb{N},$

thus by the Markov inequality,

${\mathbb{P}}\left(\Omega\setminus A_{\kappa}\right)\leq\frac{C(T)}{\kappa}\to
0.$

Thus the solution process is well defined on
$\tilde{\Omega}=\bigcup_{\kappa=1}^{\infty}A_{\kappa}$ with
${\mathbb{P}}(\tilde{\Omega})=1$. ∎

The proof of Theorem 3.6 is complete.

∎

##  4\. Proof of the stochastic Schauder-Tychonoff theorem

###### Proof of Theorem 2.1.

Fix ${\mathfrak{A}}$ and $W$ and fix any
$v^{(0)}\in{\mathcal{X}}({\mathfrak{A}})$.

  * Step (I)

For $k\in\mathbb{N}$, define
$v^{(k)}:={\mathcal{V}}_{{\mathfrak{A}},W}(v^{(k-1)})$ recursively for some
initial datum $v^{(k)}(0)=v_{0}\in
L^{m}(\Omega,{\mathcal{F}}_{0},\mathbb{P};E)$, $k\in\mathbb{N}$. We claim that
the laws of the sequence $\\{v^{(k)}\\}_{k\in\mathbb{N}}$, are tight on
$\mathbb{X}$. By compactness of the embedding
$\mathbb{X}^{\prime}\hookrightarrow\mathbb{X}$, and by reflexivity of
$\mathbb{X}^{\prime}$, it follows by standard arguments that

$K_{R}:=\\{\xi\in\mathbb{X}\;\colon\;|\xi|_{\mathbb{X}^{\prime}}^{m_{0}}\leq
R\\}$

is compact in $\mathbb{X}$. By assumption (i),
$v^{(k)}\in{\mathcal{X}}({\mathfrak{A}})$ for every $k\in\mathbb{N}$, and thus
it follows by assumption (iii) that $v^{(k)}\in K_{R}$ for any
$k\in\mathbb{N}$. Therefore, the probability laws
$\rho_{k}:=\text{Law}(v^{(k)})$, $k\in\mathbb{N}$, of the sequence
$\\{v^{(k)}\\}_{k\in\mathbb{N}}$ are tight on $\mathbb{X}$.

  * Step (II)

By Step I and Prokhorov’s theorem, we know that there exists a (nonrelabeled)
subsequence $\\{\rho_{k}\\}_{k\in\mathbb{N}}$ and a Borel probability law
$\rho^{\ast}$ on $\mathbb{X}$ such that

$\rho_{k}\rightharpoonup\rho^{\ast}$

as $k\to\infty$ weakly in the sense of probability measures. Next, by the
Skorokhod lemma [34, Theorem 4.30], there exists a probability space
$(\tilde{\Omega},\tilde{{\mathcal{F}}},\tilde{\mathbb{P}})$, a sequence of
$\mathbb{X}$-valued random variables $\\{\tilde{v}^{(k)}\\}_{k\in\mathbb{N}}$
and $\tilde{v}^{\ast}$ such that

(4.1) $\displaystyle\text{Law}(\tilde{v}^{(k)})=\rho_{k},\quad
k\in\mathbb{N},\quad\text{Law}(\tilde{v}^{\ast})=\rho^{\ast},$

and such that

$\tilde{v}^{(k)}\to\tilde{v}^{\ast}\quad\tilde{\mathbb{P}}\text{-a.s.}$

on $\mathbb{X}$ as $k\to\infty$.

  * Step (III)

For each $k\in\mathbb{N}$, let $\tilde{M}^{(k)}$ be the process given by

$\displaystyle\tilde{M}^{(k)}(t)$ $\displaystyle:=$
$\displaystyle\tilde{v}^{(k)}(t)-\tilde{v}_{0}^{(k)}-\int_{0}^{t}\left[A\tilde{v}^{(k)}(s)+F(\tilde{v}^{(k-1)}(s),s)\right]\,ds,$

and let

$\tilde{{\mathcal{G}}}_{t}^{(k)}:=\sigma\left(\left\\{\,\left(\tilde{v}^{(k)}(s),\tilde{v}^{(k-1)}(s)\right)\;\colon\;0\leq
s\leq t\,\right\\}\right),\quad k\in\mathbb{N},\quad t\in[0,T],$

where $\tilde{v}_{0}^{(k)}$ is a $\tilde{{\mathcal{G}}}^{(k)}_{0}$ measurable
version of $v_{0}$. Then, $\tilde{M}^{(k)}$ is a local martingale over
$(\tilde{\Omega},\tilde{{\mathcal{F}}},\tilde{\mathbb{P}})$ with respect to
$(\tilde{{\mathcal{G}}}_{t}^{(k)})_{t\in[0,T]}$ that has the quadratic
variation

(4.3)
$\langle\\!\langle\tilde{M}^{(k)}\rangle\\!\rangle(t):=\langle\\!\langle\tilde{M}^{(k)}(t),\tilde{M}^{(k)}(t)\rangle\\!\rangle=\int_{0}^{t}\sum_{i\in\mathbb{I}}|\Sigma(\tilde{v}^{(k)}(s))\psi_{i}|_{E}^{2}\,ds,\quad
t\in[0,T].$

The representation (4.3) follows from (4.1) and the fact that
$v^{(k)}=\mathcal{V}_{\mathfrak{A},W}(v^{(k-1)})$, see (2.2). To show that
$\langle\\!\langle\tilde{M}^{(k)}\rangle\\!\rangle$ is adapted to the
filtration $(\tilde{{\mathcal{G}}}_{t}^{(k)})_{t\in[0,T]}$, we use the fact
that by (4.1) for all measurable maps $\varphi:\mathbb{X}\to[0,+\infty]$ and
all $0\leq s\leq t\leq T$ we have

$\mathbb{E}\left[\left(\tilde{M}^{(k)}(t)-\tilde{M}^{(k)}(s)\right)\varphi(\mathbbm{1}_{[0,s)}\tilde{v}^{(k)})\right]=0.$

  * Step (IV)

Next, we verify the following statements with the goal to pass on to the
limit:

    * •

there exists a constant $C>0$ such that
$\sup_{k\in{\mathbb{N}}}\tilde{\mathbb{E}}\left[|\tilde{v}^{(k)}|^{m_{0}}_{\mathbb{X}}\right]\leq
C$ and

    * •

for any $r\in(1,m_{0})$ we have

$\lim_{k\to\infty}\tilde{{\mathbb{E}}}\left[|\tilde{v}^{(k)}-\tilde{v}^{\ast}|_{\mathbb{X}}^{r}\right]=0.$

Clearly, since
$\\{v^{(k)}\\}_{k\in\mathbb{N}}\subset\mathcal{X}(\tilde{\mathfrak{A}})$, and
$\mathcal{X}(\tilde{\mathfrak{A}})$ is bounded in $\mathbb{X}$, we can
conclude from the application of the Skorokhod lemma in Step II that

$\mathbb{E}|v^{(k)}|_{\mathbb{X}}^{r}=\tilde{\mathbb{E}}|\tilde{v}^{(k)}|_{\mathbb{X}}^{r},$

for any $r\in[1,m_{0}]$, so that we get by (iii) that

$\sup_{k}\tilde{\mathbb{E}}|\tilde{v}^{(k)}|_{\mathbb{X}}^{m_{0}}\leq CR,$

where $C>0$ is a constant such that $|\cdot|_{\mathbb{X}}\leq
C|\cdot|_{\mathbb{X}^{\prime}}$.

Hence, we know that $\\{|\tilde{v}^{(k)}|_{\mathbb{X}}^{r}\\}$ is uniformly
integrable for any $r\in(1,m_{0}]$ w.r.t. the probability measure
$\tilde{\mathbb{P}}$. By Step II, $\tilde{v}^{(k)}\to\tilde{v}^{\ast}$
$\tilde{\mathbb{P}}$-a.s., so we get by the Vitali convergence theorem that

(4.4)
$\lim_{k\to\infty}\tilde{\mathbb{E}}\left|\tilde{v}^{(k)}-\tilde{v}^{\ast}\right|_{\mathbb{X}}^{r}=0$

for any $r\in(1,m_{0})$.

  * Step (V)

Since $|\tilde{v}^{(k)}(s)|_{\mathbb{X}}^{r}$ is uniformly integrable with
respect to the probability measure $\tilde{\mathbb{P}}$ for any $r<m_{0}$, it
is integrable for $r=m_{0}$. Moreover,

$\int_{0}^{t}\sum_{i\in\mathbb{I}}\left|\Sigma\left(\tilde{v}^{(k)}(s)\right)\psi_{i}\right|_{E}^{2}\,ds\leq
C\,\int_{0}^{t}|\tilde{v}^{(k)}(s)|_{\mathbb{X}}^{2}\,ds,\quad t\in[0,T].$

Hence we invoke the Vitali convergence theorem to take the mean square limit
in (LABEL:qv1) and set
$v^{\ast}_{0}:=\lim_{k\to\infty}\tilde{v}_{0}^{(k)}=\lim_{k\to\infty}\tilde{v}^{(k)}(0)$
as well as

$\tilde{M}^{\ast}:=\lim_{k\to\infty}\tilde{M}^{(k)}.$

Next, let $\tilde{\mathbb{G}}=(\tilde{\mathcal{G}}_{t})_{t\in[0,T]}$ be the
filtration defined by

$\tilde{\mathcal{G}}_{t}=\sigma\left(\sigma(\\{\tilde{v}^{\ast}(s)\;\colon\;0\leq
s\leq t\\})\cup\mathcal{N}\right),\quad t\in[0,T],$

where $\mathcal{N}$ denotes the collection of $\tilde{{\mathbb{P}}}$-null sets
of $\tilde{\mathcal{F}}$. Since $v^{(k)}\in\mathbb{X}$ for every
$k\in\mathbb{N}$, the limit is well defined and for $0\leq s\leq t$, we have
all measurable maps $\varphi:\mathbb{X}\to[0,+\infty]$ and for all $0\leq
s\leq t\leq T$,

$\tilde{\mathbb{E}}\left[\left(\tilde{M}^{\ast}(t)-\tilde{M}^{\ast}(s)\right)\varphi(\mathbbm{1}_{[0,s)}\tilde{v}^{\ast})\right]=0.$

We can easily conclude that $\tilde{M}^{\ast}$ is a local martingale with
respect to $\tilde{\mathbb{G}}$ such that

(4.5)
$\langle\\!\langle\tilde{M}^{\ast}\rangle\\!\rangle(t)=\int_{0}^{t}\sum_{i\in\mathbb{I}}|\Sigma(\tilde{v}^{\ast}(s))\psi_{i}|_{E}^{2}\,ds,\quad
t\in[0,T].$

By the representation theorem [15, Theorem 8.2], and more specifically, for
the case of Banach space valued noise, [18] and [8, Proof of Theorem 4.5],
there exists a filtered probability space
$(\tilde{\tilde{\Omega}},\tilde{\tilde{{{\mathcal{F}}}}},\tilde{\tilde{{\mathbb{P}}}})$
with filtration $(\tilde{\tilde{{{\mathcal{F}}}}}_{t})_{t\in[0,T]}$, a Wiener
processes $\tilde{W}$ on $H$ with covariance $Q$ defined on
$\tilde{\mathfrak{A}}:=(\tilde{\Omega}\times\tilde{\tilde{\Omega}},\tilde{{\mathcal{F}}}\otimes\tilde{\tilde{{{\mathcal{F}}}}},\tilde{\mathbb{P}}\otimes\tilde{\tilde{{\mathbb{P}}}})$
and adapted to
$(\tilde{{\mathcal{G}}}_{t}\otimes\tilde{\tilde{{{\mathcal{F}}}}}_{t})_{t\in[0,T]}$
such that

$\tilde{M}^{\ast}(t,\omega,\tilde{\omega}):=\int_{0}^{t}v^{\ast}(s,\tilde{\omega})\,d\tilde{W}(s,\tilde{\omega},\tilde{\tilde{\omega}}),\quad
t\in[0,T],\,(\tilde{\omega},\tilde{\tilde{\omega}})\in\tilde{\Omega}\times\tilde{\tilde{\Omega}},$

In this way, we obtain a pair $v^{\ast}\in C([0,t];U)$ over
$\tilde{\mathfrak{A}}$ and the processes $\tilde{W}$ on $H$ over
$\tilde{\mathfrak{A}}$.

  * Step (VI)

In this step we show that $v^{\ast}$ over $\tilde{\mathfrak{A}}$ together with
$\tilde{W}$ is indeed a martingale solution to (2.2). Let us consider the
Banach space ${{{\mathcal{M}}}}^{m}_{\tilde{\mathfrak{A}}}(\mathbb{X})$ over
$\tilde{\mathfrak{A}}$ and the convex subset
$\mathcal{X}(\tilde{\mathfrak{A}})$. The operator
$\tilde{{\mathcal{V}}}_{{\mathfrak{A}},W}$ acts now on the space
${\mathcal{X}}(\tilde{\mathfrak{A}})$ and is defined by system (2.2).

Fix $\varepsilon>0$. Set, for simplicity,
$\tilde{{\mathcal{V}}}:={\mathcal{V}}_{\tilde{{\mathfrak{A}}},\tilde{W}}$. We
claim that

$\tilde{\mathbb{E}}\left|\tilde{{\mathcal{V}}}(\tilde{v}^{\ast})-\tilde{v}^{\ast}\right|_{\mathbb{X}}<\varepsilon.$

By Step IV and the Banach-Alaoglu theorem, we get that
$\tilde{v}^{(k)}\rightharpoonup\tilde{v}^{\ast}$ weakly∗ for $k\to\infty$ in
the Banach space ${\mathcal{M}}_{\tilde{{\mathfrak{A}}}}^{m}$. By the weak∗
closedness assumption, we get that
$\tilde{v}^{\ast}\in{\mathcal{X}}(\tilde{{\mathfrak{A}}})$.

By assumption (ii), $\tilde{{\mathcal{V}}}$ is continuous on
${\mathcal{X}}(\tilde{{\mathfrak{A}}})$ and hence there exists
$\delta=\delta(\varepsilon)>0$ such that

$\tilde{\mathbb{E}}\left|\tilde{{\mathcal{V}}}(\tilde{v}^{\ast})-\tilde{{\mathcal{V}}}(\tilde{v}^{(k-1)})\right|_{\mathbb{X}}^{m}<\frac{\varepsilon}{2},$

whenever
$\tilde{\mathbb{E}}|\tilde{v}^{\ast}-\tilde{v}^{(k-1)}|_{\mathbb{X}}^{m}<\delta$,
which is the case by Step IV (and the assumption $m_{0}\geq m$) for some large
$k$. Also, by construction,

$\tilde{\mathbb{E}}\left|\tilde{{\mathcal{V}}}(\tilde{v}^{(k-1)})-\tilde{v}^{(k)}\right|_{\mathbb{X}}=\mathbb{E}\left|{\mathcal{V}}(v^{(k-1)})-v^{(k)}\right|_{\mathbb{X}}=0.$

Furthermore, by (4.4),

$\tilde{\mathbb{E}}\left|\tilde{v}^{(k)}-\tilde{v}^{\ast}\right|_{\mathbb{X}}<\frac{\varepsilon}{2}$

for some large $k$. Altogether, writing

$\tilde{{\mathcal{V}}}(\tilde{v}^{\ast})-\tilde{v}^{\ast}=\tilde{{\mathcal{V}}}(\tilde{v}^{\ast})-\tilde{{\mathcal{V}}}(\tilde{v}^{(k-1)})+\tilde{{\mathcal{V}}}(\tilde{v}^{(k-1)})-\tilde{v}^{(k)}+\tilde{v}^{(k)}-\tilde{v}^{\ast}.$

we can easily complete the proof of the claim. As a consequence,

$\tilde{{\mathcal{V}}}(\tilde{v}^{\ast})=\tilde{v}^{\ast},\quad\tilde{\mathbb{P}}\text{-a.s.}$

As seen above, $\tilde{v}^{\ast}\in{\mathcal{X}}(\tilde{{\mathfrak{A}}})$, so
that by (iv), $\tilde{{\mathcal{V}}}(\tilde{v}^{\ast})\in\mathbb{D}([0,T];U)$,
and therefore $\tilde{v}^{\ast}\in\mathbb{D}([0,T];U)$
$\tilde{\mathbb{P}}$-a.s. Hence for all $t\in[0,T]$, $\tilde{\mathbb{P}}$-a.s.

$\tilde{{\mathcal{V}}}(\tilde{v}^{\ast})(t)=\tilde{v}^{\ast}(t)$

and the proof is complete. By construction and Step V, we see that $v^{\ast}$
solves

(4.6) $\displaystyle dv^{\ast}(t)$ $\displaystyle=$
$\displaystyle\left(Av^{\ast}(t)+F(v^{\ast}(t),t)\right)\,dt+\Sigma(v^{\ast}(t))\,d\tilde{W}(t),\quad
v^{\ast}(0)=v^{\ast}_{0}.$

on $\tilde{\mathfrak{A}}$, where $v_{0}^{\ast}$ is a
$\tilde{{\mathcal{G}}}_{0}\otimes{\tilde{\tilde{{\mathcal{F}}}}}_{0}$-measurable
version of $v_{0}$.

∎

##  5\. Results on regularity and technical propositions

This section contains the remaining results which are used in the proof of the
main result Theorem 3.6.

###  5.1. Assumptions on the noise and consequences

Let us recall, we denoted by $\\{\psi_{k}:k\in\mathbb{N}\\}$ the
eigenfunctions of the Laplace operator $-\Delta$ in $L^{2}({{\mathcal{O}}})$
and by $\\{\nu_{k}:k\in\mathbb{N}\\}$ the corresponding eigenvalues, where the
enumeration is chosen in increasing order counting the multiplicity.

Let us characterize the asymptotic behavior of $\\{\nu_{k}:k\in\mathbb{N}\\}$
and $\\{\psi_{k}:k\in\mathbb{N}\\}$ for an arbitrary domain ${{\mathcal{O}}}$
with $C^{2}$–boundary. Here, we know by Weyl’s law [70, 71] that there exist
constants $c,C>0$ such that,

(5.1) $\displaystyle\\#\\{j\;\colon\;\nu_{j}\leq\lambda\\}\leq
C\lambda^{\frac{d}{2}},$

compare with [40, 32] and [33, Corollary 17.8.5], and there exists a constant
$C>0$ such that,

(5.2) $\displaystyle\sup_{x\in{{\mathcal{O}}}}|\psi_{k}(x)|$
$\displaystyle\leq$ $\displaystyle C\nu_{k}^{\frac{d-1}{2}},\quad
k\in\mathbb{N},$

compare with [30].

######  Remark 5.1.

If ${{\mathcal{O}}}=[0,1]^{d}$ is a rectangular domain, then a complete
orthonormal system $\\{\psi_{k}:k\in\mathbb{Z}\\}$ of the underlying Lebesgue
space $L^{2}({{\mathcal{O}}})$ is given by the trigonometric functions, see
[5, p. 352] (now, $\mathbb{Z}$ obviously denotes the integer numbers). Let us
define firstly the eigenvalues for the Laplace on $L^{2}([0,1])$

(5.3)
$e_{k}(x)=\begin{cases}{\sqrt{2}}\,\sin\big{(}2\pi{k}x\big{)}&\\!\\!\text{if
}k\geq 1,\,x\in{{\mathcal{O}}},\\\ {1}&\\!\\!\text{if
}k=0,\,x\in{{\mathcal{O}}}\,,\\\
{\sqrt{2}}\,\cos\big{(}2\pi{k}x\big{)}&\\!\\!\text{if
}k\leq-1,x\in{{\mathcal{O}}}.\end{cases}$

For instance, the eigenfunctions for $L^{2}([0,1]^{2})$ for the multi-index
$k=(k_{1},k_{2})\in\mathbb{Z}^{2}$ are given by the tensor product

$\psi_{k}(x_{1},x_{2})=e_{k_{1}}(x_{1})e_{k_{2}}(x_{2}),\quad x\in[0,1]^{2},$

the corresponding eigenvalues are given by $\nu_{k}=\pi^{2}|k|^{2}$, where
$|k|=k_{1}+k_{2}$. The case $d=3$ works analogously. In this special case the
conditions on $\delta_{1}$ and $\delta_{2}$ in Hypothesis 3.5 can be relaxed
to

$\lambda_{k}^{(1)}\leq C\nu_{k}^{\delta_{1}},\quad\lambda^{(2)}_{k}\leq
C\nu_{k}^{\delta_{2}},\quad k=(k_{1},k_{2})$

to $\delta_{1},\delta_{2}>\frac{1}{2}$. See also the discussion in [2,
Examples 1.2.1 and 2.1.2].

We begin with a remark on the noise coefficients. We partly work in the Banach
spaces $L^{m}({{\mathcal{O}}})$ and $H^{m}_{\delta}({{\mathcal{O}}})$ for
$\delta$ being arbitrary small but positive.

Given a Wiener process $W$ on $H=L^{2}({{\mathcal{O}}})$ over
$\mathfrak{A}=(\Omega,{\mathcal{F}},\mathbb{F},{\mathbb{P}})$, and a
progressively measurable process
$\xi\in\mathcal{M}^{2}_{\mathfrak{A}}(L^{2}({{\mathcal{O}}}))$,
$\rho\in[0,\frac{1}{2}]$, let us define $\\{Y(t):t\in[0,T]\\}$ by

$Y(t):=\int_{0}^{t}\xi(s)\,d{W}(s),\quad t\in[0,T].$

Here, for each $t\in[0,T]$, $\xi(t)$ is interpreted as a multiplication
operator acting on the elements of $H$, namely,
$\xi:H\ni\psi\mapsto\xi\psi\in{{\mathcal{S}}}^{\prime}({\mathcal{O}})$, where
${{\mathcal{S}}}^{\prime}({\mathcal{O}})$ denotes the space of _tempered
distributions_ on ${\mathcal{O}}$. Let $E$ be a Banach space. An inequality
needed in several places within the proof is the Burkholder-Davis-Gundy
inequality given for $p\geq 1$ as follows

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y(t)|^{p}_{E}\right]\leq
C(p)\,\mathbb{E}\,\left[\int_{0}^{T}|\xi(t)|^{2}_{\gamma(H,E)}\,dt\right]^{\frac{p}{2}},$

where $\gamma(H,E)$ denotes the space of $\gamma$-radonifying operators and
$|\cdot|_{\gamma(H,E)}$ the corresponding norm, cf. [6, 67]. In case $E$ is a
Hilbert space, the $\gamma$-norm coincides with the Hilbert-Schmidt norm
$|\cdot|_{L_{\textup{HS}}(H,E)}$. See Appendix C for further details.

By $|\xi\psi_{k}|_{L^{m}}\leq|\xi|_{L^{m}}\,|\psi_{k}|_{L^{\infty}}$ and (5.2)

we know that

$\displaystyle|\xi|^{2}_{\gamma(H,L^{m})}\leq\sum_{k\in\mathbb{N}}\lambda^{k}_{k}|\xi\psi_{k}|_{L^{m}}^{2}\leq|\xi|_{L^{m}}\sum_{k\in\mathbb{N}}\lambda^{\delta}_{k}|\psi_{k}|_{L^{\infty}}^{2}\leq
C\sum_{k\in\mathbb{N}}\nu_{k}^{2(1-\delta)},$

which is finite if $\delta>\frac{1}{2}$.

  * •

Firstly, let $E=H^{-1}_{2}({{\mathcal{O}}})$. Since

$|\xi\psi|_{H^{-1}_{2}}\leq|\xi|_{H^{-1}_{2}}|\psi|_{H^{\delta}_{2}}\leq|\xi|_{H^{\rho}_{2}}|\psi|_{L^{\infty}},$

we know that $\delta>\frac{1}{2}$, then

$|\sigma_{1}(\xi)|_{\gamma(H,H^{\rho}_{2})}\leq C\,|\xi|_{H^{\rho}_{2}}.$

  * •

Secondly, let $E=H^{\rho}_{2}({{\mathcal{O}}})$, where $\rho$ is given in
Hypothesis 3.4. Since

$|\xi\psi|_{H^{\rho}_{2}}\leq|\xi|_{H^{\rho}_{2}}|\psi|_{H^{2|\rho|}_{2}}\leq|\xi|_{H^{\rho}_{2}}|\psi|_{L^{\infty}},$

we know that $\delta>\frac{1}{2}$, then

$|\sigma_{2}(\xi)|_{\gamma(H,H^{\rho}_{2})}\leq C\,|\xi|_{H^{\rho}_{2}}.$

######  Remark 5.2.

From Hypotheses 3.5, one can infer that there exists a constant $C>0$ such
that

$\sum_{k=1}^{\infty}[(\sigma_{j}(\xi)f_{k})(x)]^{2}\leq
C|\xi(x)|_{L^{2}}^{2},\quad\forall\xi\in
L^{2}({{\mathcal{O}}}),\;x\in{\mathcal{O}},\mbox{ and }j=1,2.$

Here $\\{f_{k}\\}$ is an orthonormal basis in $H^{-1}_{2}({{\mathcal{O}}})$,
compare with [2, Hypothesis 3, p. 42]. In addition, note, that

  * •

Firstly, $\sigma_{j}:H^{-1}_{2}({{\mathcal{O}}})\to
L_{\textup{HS}}(H_{j},H^{-1}_{2}({{\mathcal{O}}}))$ is of linear growth and
Lipschitz continuous. In particular, there exists constants $C_{1},L_{1}>0$
such that

$\displaystyle|\sigma_{j}(\xi)|_{L_{\textup{HS}}(H_{j},H^{-1}_{2})}$
$\displaystyle\leq$ $\displaystyle C_{1}(1+|\xi|_{H^{-1}_{2}}),\quad\xi\in
H^{-1}_{2}({{\mathcal{O}}});$
$\displaystyle|\sigma_{j}(\xi)-\sigma_{j}(\eta)|_{L_{\textup{HS}}(H_{j},H^{-1}_{2})}$
$\displaystyle\leq$ $\displaystyle
L_{1}|\xi-\eta|_{H^{-1}_{2}},\quad\xi,\eta\in H^{-1}_{2}({{\mathcal{O}}});.$

  * •

Secondly, $\sigma_{j}:L^{2}({{\mathcal{O}}})\to
L_{\textup{HS}}(H_{j},L^{2}({{\mathcal{O}}}))$ is of linear growth and
Lipschitz continuous. In particular, there exists constants $C_{2},L_{2}>0$
such that

$\displaystyle|\sigma_{j}(\xi)|_{L_{\textup{HS}}(H_{j},L^{2})}$
$\displaystyle\leq$ $\displaystyle C_{2}(1+|\eta|_{L^{2}}),\quad\eta\in
L^{2}({{\mathcal{O}}});$
$\displaystyle|\sigma_{j}(\xi)-\sigma_{j}(\eta)|_{L_{\textup{HS}}(H_{j},L^{2})}$
$\displaystyle\leq$ $\displaystyle L_{2}|\xi-\eta|_{L^{2}},\quad\xi,\eta\in
L^{2}({{\mathcal{O}}}).$

  * •

Similarly, straightforward computations and using the fact that
$|f_{k}|_{L^{\infty}}\leq\nu_{k}^{\frac{d-1}{2}}$, see (5.2) and [2, p. 46],
we get that

$\displaystyle|\sigma_{2}(\xi)|_{\gamma(H_{2},L^{m})}$ $\displaystyle\leq$
$\displaystyle C_{1}(1+|\xi|_{L^{m}}),\quad\xi\in L^{m}({{\mathcal{O}}});$
$\displaystyle|\sigma_{2}(\xi)-\sigma_{2}(\eta)|_{{\gamma}(H_{2},L^{m})}$
$\displaystyle\leq$ $\displaystyle L_{1}|\xi-\eta|_{L^{m}},\quad\xi,\eta\in
L^{m}({{\mathcal{O}}}),$

where $\gamma(H_{2},L^{m})$ denotes the space of $\gamma$-radonifying
operators, cf. [6, 67].

###  5.2. Properties of the subsystem (3.13)

In this subsection, we are analyzing equation (3.13), where
$\kappa\in\mathbb{N}$ and the couple
$(\xi,\eta)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$
are given.

First, we will show that a unique solution $u_{\kappa}$ to the system (3.13)
exists and is nonnegative. Second, we will show by variational methods that
this solution satisfies some integrability properties, given $u_{0}\in
L^{p+1}({\mathcal{O}})$, $p\geq 1$.

######  Theorem 5.3.

Assume that the Hypotheses of Theorem 3.6 hold. For any
$(\xi,\eta)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$,
and for any $u_{0}\in
L^{2}(\Omega;{\mathcal{F}}_{0},\mathbb{P};H^{-1}_{2}({{\mathcal{O}}}))$ the
system

(5.4) $du_{\kappa}(t)=r_{u}\Delta
u_{\kappa}^{[\gamma]}(t)\,dt-\chi\phi_{\kappa}(h(\eta,\xi,t))\,\eta(t)\xi^{2}(t)\,dt+\sigma_{1}u_{\kappa}(t)\,dW_{1}(t),$

with $u_{\kappa}(0)=u_{0}$ has a unique solution $u_{\kappa}$ on the time
interval $[0,T]$ which is $\mathbb{P}$-a.s. continuous in $H_{2}^{-1}$ and
satisfies

$\mathbb{E}\left[\sup_{0\leq s\leq
T}|u_{\kappa}(s)|^{2}_{H^{-1}_{2}}\right]+\mathbb{E}\int_{0}^{T}|u_{\kappa}(s)|_{L^{\gamma+1}}^{\gamma+1}\,ds<\infty.$

In particular, there exists a constant $C(T,\kappa)>0$ such that

$\mathbb{E}\|u_{\kappa}\|^{2}_{\mathbb{Y}}\leq C(T,\kappa).$

###### Proof.

Before starting with the proof, we introduce the setting used by the book by
Barbu, Da Prato and Röckner [2], and the book by Liu and Röckner [42],
respectively. Let ${{\mathcal{H}}}:=H_{2}^{-1}({\mathcal{O}})$, the dual space
of ${{\mathcal{H}}}^{\ast}={H}_{2}^{1}({\mathcal{O}})$ (corresponding to
Neumann boundary conditions). Let $V:=L^{\gamma+1}({\mathcal{O}})$. By the
Sobolev embedding theorem, ${\mathcal{H}}^{\ast}\hookrightarrow
L^{(\gamma+1)/\gamma}({\mathcal{O}})$ densely and compactly. Thus, upon
identifying ${\mathcal{H}}$ with its own dual space via the Riesz-map
$(-\Delta)^{-1}$ of ${\mathcal{H}}$, we have a Gelfand triple

$V\subset{{\mathcal{H}}}\equiv{{\mathcal{H}}}^{\ast}\subset V^{\ast}.$

We set

$A(t,u,\omega):=r_{u}\Delta
u^{[\gamma]}-\chi\phi_{\kappa}(h(\eta(\omega),\xi(\omega),t))\xi^{2}(t,\omega)\eta(t,\omega).$

Note, due to the assumption
$(\xi,\eta)\in\mathcal{X}_{\mathfrak{A}}(R_{0},R_{1},R_{2})\subset{{\mathcal{M}}}_{\mathfrak{A}}(\mathbb{Y},\mathbb{Z})$
and by (3.10), $\eta$ and $\xi$ are adapted such that there exists a constant
$C>0$ such that

(5.5) $\mathbb{E}\

authors: Erika Hausenblas and Jonas M. T\"olle

submitter: Jonas M. T\"olle Dr. math.