text: 28 NICOLAS CHOPIN AND JAMES RIDGWAY Statist., 20(3):616–635. Jasra, A., Stephens, D., A. Doucet, A., and Tsagaris, T. (2011). Inference for Lévy driven stochastic volatility models via Sequential Monte Carlo. Scand. J. of Statist., 38(1). Kabán, A. (2007). On Bayesian classiﬁcation with Laplace priors. Pattern Recog- nition Letters, 28(10):1271–1282. Kong, A., Liu, J. S., and Wong, W. H. (1994). Sequential imputation and Bayesian missing data problems. J. Am. Statist. Assoc., 89:278–288. Lamnisos, D., Griﬃn, J. E., and Steel, M. F. J. (2013). Adaptive Monte Carlo for Bayesian variable selection in regression models. J. Comput. Graph. Statist., 22(3):729–748. Lee, A., Yau, C., Giles, M. B., Doucet, A., and Holmes, C. C. (2010). On the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. J. Comput. Graph. Statist., 19(4):769–789. Lemieux, C. (2009). Monte Carlo and Quasi-Monte Carlo Sampling (Springer Series in Statistics). Springer. Minka, T. (2001). Expectation Propagation for approximate Bayesian inference. Proceedings of Uncertainty in Artiﬁcial Intelligence, 17:362–369. Neal, R. M. (2001). Annealed importance sampling. Statist. Comput., 11:125–139. Neal, R. M. (2010). MCMC using Hamiltonian dynamics. In Brooks, S., Gelman, A., Jones, G. L., and Meng, X.-L., editors, Handbook of Markov Chain Monte Carlo, pages 113–162. Chapman & Hall / CRC Press. Nickisch, H. and Rasmussen, C. (2008). Approximations for Binary Gaussian Pro- cess Classiﬁcation. J. Machine Learning Research, 9(10):2035–2078. Polson, N. G., Scott, J. G., and Windle, J. (2013). Bayesian inference for logistic models using pólya˘gamma latent variables. Journal of the American Statistical Association, 108(504):1339–1349. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (2007). Nu- merical Recipes: The Art of Scientiﬁc Computing. Cambridge University Press. Ridgway, J. (2014). Computation of Gaussian orthant probabilities in high dimen- sion. arXiv preprint arXiv:1411.1314. Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer-Verlag, New York. Roberts, G. O. and Rosenthal, J. S. (2001). Optimal scaling for various Metropolis- Hastings algorithms. Statist. Science, 16(4):351–367. Rue, H., Martino, S., and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Statist. Soc. B, 71(2):319–392. Schäfer, C. (2012). Monte Carlo methods for sampling high-dimensional binary vectors. PhD thesis, Université Paris Dauphine. Schäfer, C. and Chopin, N. (2011). Sequential monte carlo on large binary sampling spaces. Statistics and Computing, pages 1–22. Scott, S. L., Blocker, A. W., and Bonassi, F. V. (2013). Bayes and big data: The consensus monte carlo algorithm. In Bayes 250. Seeger, M. (2005). Expectation Propagation for Exponential Families. Technical report, Univ. California Berkeley. Shahbaba, B., Lan, S., Johnson, W. O., and Neal, R. M. (2011). Split Hamiltonian Monte Carlo. Statist. Comput., pages 1–11. Skilling, J. (2006). Nested sampling for general Bayesian computation. Bayesian Analysis, 1(4):833–860. Suchard, M. A., Wang, Q., Chan, C., Frelinger, J., Cron, A., and West, M. (2010). Understanding GPU programming for statistical computation: Studies in

text: NICOLAS

text: CHOPIN

text: AND

text: JAMES

text: RIDGWAY

text: Statist.,

label: reference

text: Jasra,

text: A.,

text: Stephens,

text: D.,

text: A.

text: Doucet,

text: A.,

text: and

text: Tsagaris,

text: T.

text: Inference

text: for

text: Lévy

text: driven

text: stochastic

text: volatility

text: models

text: via

text: Sequential

text: Monte

text: Carlo.

text: Scand.

text: J.

text: of

text: Statist.,

text: Kabán,

text: A.

text: On

text: Bayesian

text: classiﬁcation

text: with

text: Laplace

text: priors.

text: Pattern

text: Recog-

text: nition

text: Letters,

text: Kong,

text: A.,

text: Liu,

text: J.

text: S.,

text: and

text: Wong,

text: W.

text: H.

text: Sequential

text: imputation

text: and

text: Bayesian

text: missing

text: data

text: problems.

text: J.

text: Am.

text: Statist.

text: Assoc.,

text: Lamnisos,

text: D.,

text: Griﬃn,

text: J.

text: E.,

text: and

text: Steel,

text: M.

text: F.

text: J.

text: Adaptive

text: Monte

text: Carlo

text: for

text: Bayesian

text: variable

text: selection

text: in

text: regression

text: models.

text: J.

text: Comput.

text: Graph.

text: Statist.,

text: Lee,

text: A.,

text: Yau,

text: C.,

text: Giles,

text: M.

text: B.,

text: Doucet,

text: A.,

text: and

text: Holmes,

text: C.

text: C.

text: On

text: the

text: utility

text: of

text: graphics

text: cards

text: to

text: perform

text: massively

text: parallel

text: simulation

text: of

text: advanced

text: Monte

text: Carlo

text: methods.

text: J.

text: Comput.

text: Graph.

text: Statist.,

text: Lemieux,

text: C.

text: Monte

text: Carlo

text: and

text: Quasi-Monte

text: Carlo

text: Sampling

text: (Springer

text: Series

text: in

text: Statistics).

text: Springer.

text: Minka,

text: T.

text: Expectation

text: Propagation

text: for

text: approximate

text: Bayesian

text: inference.

text: Proceedings

text: of

text: Uncertainty

text: in

text: Artiﬁcial

text: Intelligence,

text: Neal,

text: R.

text: M.

text: Annealed

text: importance

text: sampling.

text: Statist.

text: Comput.,

text: Neal,

text: R.

text: M.

text: MCMC

text: using

text: Hamiltonian

text: dynamics.

text: In

text: Brooks,

text: S.,

text: Gelman,

text: A.,

text: Jones,

text: G.

text: L.,

text: and

text: Meng,

text: X.-L.,

text: editors,

text: Handbook

text: of

text: Markov

text: Chain

text: Monte

text: Carlo,

text: pages

text: Chapman

text: Hall

text: CRC

text: Press.

text: Nickisch,

text: H.

text: and

text: Rasmussen,

text: C.

text: Approximations

text: for

text: Binary

text: Gaussian

text: Pro-

text: cess

text: Classiﬁcation.

text: J.

text: Machine

text: Learning

text: Research,

text: Polson,

text: N.

text: G.,

text: Scott,

text: J.

text: G.,

text: and

text: Windle,

text: J.

text: Bayesian

text: inference

text: for

text: logistic

text: models

text: using

text: pólya˘gamma

text: latent

text: variables.

text: Journal

text: of

text: the

text: American

text: Statistical

text: Association,

text: Press,

text: W.

text: H.,

text: Teukolsky,

text: S.

text: A.,

text: Vetterling,

text: W.

text: T.,

text: and

text: Flannery,

text: B.

text: P.

text: Nu-

text: merical

text: Recipes:

text: The

text: Art

text: of

text: Scientiﬁc

text: Computing.

text: Cambridge

text: University

text: Press.

text: Ridgway,

text: J.

text: Computation

text: of

text: Gaussian

text: orthant

text: probabilities

text: in

text: high

text: dimen-

text: sion.

text: arXiv

text: preprint

text: Robert,

text: C.

text: P.

text: and

text: Casella,

text: G.

text: Monte

text: Carlo

text: Statistical

text: Methods,

text: ed.

text: Springer-Verlag,

text: New

text: York.

text: Roberts,

text: G.

text: O.

text: and

text: Rosenthal,

text: J.

text: S.

text: Optimal

text: scaling

text: for

text: various

text: Metropolis-

text: Hastings

text: algorithms.

text: Statist.

text: Science,

text: Rue,

text: H.,

text: Martino,

text: S.,

text: and

text: Chopin,

text: N.

text: Approximate

text: Bayesian

text: inference

text: for

text: latent

text: Gaussian

text: models

text: by

text: using

text: integrated

text: nested

text: Laplace

text: approximations.

text: J.

text: R.

text: Statist.

text: Soc.

text: B,

text: Schäfer,

text: C.

text: Monte

text: Carlo

text: methods

text: for

text: sampling

text: high-dimensional

text: binary

text: vectors.

text: PhD

text: thesis,

text: Université

text: Paris

text: Dauphine.

text: Schäfer,

text: C.

text: and

text: Chopin,

text: N.

text: Sequential

text: monte

text: carlo

text: on

text: large

text: binary

text: sampling

text: spaces.

text: Statistics

text: and

text: Computing,

text: pages

text: Scott,

text: S.

text: L.,

text: Blocker,

text: A.

text: W.,

text: and

text: Bonassi,

text: F.

text: V.

text: Bayes

text: and

text: big

text: data:

text: The

text: consensus

text: monte

text: carlo

text: algorithm.

text: In

text: Bayes

text: Seeger,

text: M.

text: Expectation

text: Propagation

text: for

text: Exponential

text: Families.

text: Technical

text: report,

text: Univ.

text: California

text: Berkeley.

text: Shahbaba,

text: B.,

text: Lan,

text: S.,

text: Johnson,

text: W.

text: O.,

text: and

text: Neal,

text: R.

text: M.

text: Split

text: Hamiltonian

text: Monte

text: Carlo.

text: Statist.

text: Comput.,

text: pages

text: Skilling,

text: J.

text: Nested

text: sampling

text: for

text: general

text: Bayesian

text: computation.

text: Bayesian

text: Analysis,

text: Suchard,

text: M.

text: A.,

text: Wang,

text: Q.,

text: Chan,

text: C.,

text: Frelinger,

text: J.,

text: Cron,

text: A.,

text: and

text: West,

text: M.

text: Understanding

text: GPU

text: programming

text: for

text: statistical

text: computation:

text: Studies

text: in


text: 4 NICOLAS CHOPIN AND JAMES RIDGWAY (cid:26) (cid:27) − − ∂ |D (2.2) β = β H 1 log p(β ) (new) (old) ∂β (old) until convergence is reached; here H is Hessian of the log posterior at β = β , (old) as computed above. The iteration above corresponds to ﬁnding the zero of a local, quadratic approximation of the log-posterior. Newton-Raphson typically works very well (converges in a small number of iterations) when the function to maximise is concave. A variant of this approach is We note two points in passing. First, one may obtain the MLE (maximum likelihood estimator) by simply taking p(β) = 1 above (i.e. a Gaussian with in- ﬁnite variance). But the MLE is not properly deﬁned when complete separation occurs, that is, there exists a hyperplane that separates perfectly the two outcomes: ≥ ∈ y βT x 0 for some β and all i 1 : N. This remark gives an extra incentive i CS i CS for performing Bayesian inference, or at least MAP estimation, in cases where com- plete separation may occur, in particular when the number of covariates is large (cid:110) (cid:111) (Firth, 1993; Gelman et al., 2008). Variants of Newton-Raphson may be obtained by adapting automatically the − − |D step size (e.g. update is β = β λH 1 ∂ log p(β ) , and step size (new) (old) ∂β (old) λ is determined by line search) or replacing the Hessian H by some approximation. Some of these algorithms such as IRLS (iterated reweighted least squares) have a nice statistical interpretation. For our purposes however, these variants seem to show roughly similar performance, so we will stick to the standard version of Newton-Raphson. (cid:88) (cid:88) The log-density of the Cauchy prior is not concave: p p − − log p(β) = log (πσ ) log(1 + β2/σ2) j j j j=1 j=1 for scales σ chosen as explained in Section ??. Hence, the corresponding log- j posterior is no longer guaranteed to be concave, which in turn means that Newton- Raphson might fail to converge. However, we shall observe that, for most of the datasets considered in this paper, Newton-Raphson does converge quickly even for our Cauchy prior. In each case, we used as starting point for the Newton-Raphson iterations the OLS (ordinary least square) estimate. We suspect what happens is that, for most standard datasets, the posterior derived from a Cauchy prior remains log-concave, at least in a region that encloses the MAP estimator and our starting point. 3. Fast approximation methods This section discusses fast approximation methods, that is methods that are deterministic, fast (compared to sampling-based methods), but which comes with an approximation error which is diﬃcult to assess. These methods include the Laplace approximation, which was popular in Statistics before the advent of MCMC methods, but also recent Machine Learning methods, such as EP (Expectation Propagation, Minka, 2001), and VB (Variational Bayes, e.g. Bishop, 2006, Chap. 10). We will focus on Laplace and EP; for VB, see Consonni and Marin (2007) for a discussion of why VB (or at least a certain standard version of VB, known as mean ﬁeld VB) may not work so well for probit models. ##LTLine## ##LTLine##

label: paragraph

text: NICOLAS

text: CHOPIN

text: AND

text: JAMES

text: RIDGWAY

label: equation

text: |D

text: β

text: β

text: H

text: log

text: p(β

text: (new)

text: (old)

text: ∂β

text: (old)

text: until

text: convergence

text: is

text: reached;

text: here

text: H

text: is

text: Hessian

text: of

text: the

text: log

text: posterior

text: at

text: β

text: β

text: (old)

text: as

text: computed

text: above.

text: The

text: iteration

text: above

text: corresponds

text: to

text: ﬁnding

text: the

text: zero

text: of

text: a

text: local,

text: quadratic

text: approximation

text: of

text: the

text: log-posterior.

text: Newton-Raphson

text: typically

text: works

text: very

text: well

text: (converges

text: in

text: a

text: small

text: number

text: of

text: iterations)

text: when

text: the

text: function

text: to

text: maximise

text: is

text: concave.

text: A

text: variant

text: of

text: this

text: approach

text: is

text: We

text: note

text: two

text: points

text: in

text: passing.

text: First,

text: one

text: may

text: obtain

text: the

text: MLE

text: (maximum

text: likelihood

text: estimator)

text: by

text: simply

text: taking

text: p(β)

text: above

text: (i.e.

text: a

text: Gaussian

text: with

text: in-

text: ﬁnite

text: variance).

text: But

text: the

text: MLE

text: is

text: not

text: properly

text: deﬁned

text: when

text: complete

text: separation

text: occurs,

text: that

text: is,

text: there

text: exists

text: a

text: hyperplane

text: that

text: separates

text: perfectly

text: the

text: two

text: outcomes:

text: y

text: βT

text: x

text: for

text: some

text: β

text: and

text: all

text: i

text: N.

text: This

text: remark

text: gives

text: an

text: extra

text: incentive

text: i

text: CS

text: i

text: CS

text: for

text: performing

text: Bayesian

text: inference,

text: or

text: at

text: least

text: MAP

text: estimation,

text: in

text: cases

text: where

text: com-

text: plete

text: separation

text: may

text: occur,

text: in

text: particular

text: when

text: the

text: number

text: of

text: covariates

text: is

text: large

text: (Firth,

text: Gelman

text: et

text: al.,

text: Variants

text: of

text: Newton-Raphson

text: may

text: be

text: obtained

text: by

text: adapting

text: automatically

text: the

text: |D

text: step

text: size

text: (e.g.

text: update

text: is

text: β

text: β

text: λH

text: log

text: p(β

text: and

text: step

text: size

text: (new)

text: (old)

text: ∂β

text: (old)

text: λ

text: is

text: determined

text: by

text: line

text: search)

text: or

text: replacing

text: the

text: Hessian

text: H

text: by

text: some

text: approximation.

text: Some

text: of

text: these

text: algorithms

text: such

text: as

text: IRLS

text: (iterated

text: reweighted

text: least

text: squares)

text: have

text: a

text: nice

text: statistical

text: interpretation.

text: For

text: our

text: purposes

text: however,

text: these

text: variants

text: seem

text: to

text: show

text: roughly

text: similar

text: performance,

text: so

text: we

text: will

text: stick

text: to

text: the

text: standard

text: version

text: of

text: Newton-Raphson.

text: The

text: log-density

text: of

text: the

text: Cauchy

text: prior

text: is

text: not

text: concave:

text: p

text: p

text: log

text: p(β)

text: log

text: (πσ

text: j

text: j

text: j

text: for

text: scales

text: σ

text: chosen

text: as

text: explained

text: in

text: Section

text: Hence,

text: the

text: corresponding

text: log-

text: j

text: posterior

text: is

text: no

text: longer

text: guaranteed

text: to

text: be

text: concave,

text: which

text: in

text: turn

text: means

text: that

text: Newton-

text: Raphson

text: might

text: fail

text: to

text: converge.

text: However,

text: we

text: shall

text: observe

text: that,

text: for

text: most

text: of

text: the

text: datasets

text: considered

text: in

text: this

text: paper,

text: Newton-Raphson

text: does

text: converge

text: quickly

text: even

text: for

text: our

text: Cauchy

text: prior.

text: In

text: each

text: case,

text: we

text: used

text: as

text: starting

text: point

text: for

text: the

text: Newton-Raphson

text: iterations

text: the

text: OLS

text: (ordinary

text: least

text: square)

text: estimate.

text: We

text: suspect

text: what

text: happens

text: is

text: that,

text: for

text: most

text: standard

text: datasets,

text: the

text: posterior

text: derived

text: from

text: a

text: Cauchy

text: prior

text: remains

text: log-concave,

text: at

text: least

text: in

text: a

text: region

text: that

text: encloses

text: the

text: MAP

text: estimator

text: and

text: our

text: starting

text: point.

text: Fast

text: approximation

text: methods

text: This

text: section

text: discusses

text: fast

text: approximation

text: methods,

text: that

text: is

text: methods

text: that

text: are

text: deterministic,

text: fast

text: (compared

text: to

text: sampling-based

text: methods),

text: but

text: which

text: comes

text: with

text: an

text: approximation

text: error

text: which

text: is

text: diﬃcult

text: to

text: assess.

text: These

text: methods

text: include

text: the

text: Laplace

text: approximation,

text: which

text: was

text: popular

text: in

text: Statistics

text: before

text: the

text: advent

text: of

text: MCMC

text: methods,

text: but

text: also

text: recent

text: Machine

text: Learning

text: methods,

text: such

text: as

text: EP

text: (Expectation

text: Propagation,

text: Minka,

text: and

text: VB

text: (Variational

text: Bayes,

text: e.g.

text: Bishop,

text: Chap.

text: We

text: will

text: focus

text: on

text: Laplace

text: and

text: EP;

text: for

text: VB,

text: see

text: Consonni

text: and

text: Marin

text: for

text: a

text: discussion

text: of

text: why

text: VB

text: (or

text: at

text: least

text: a

text: certain

text: standard

text: version

text: of

text: VB,

text: known

text: as

text: mean

text: ﬁeld

text: VB)

text: may

text: not

text: work

text: so

text: well

text: for

text: probit

text: models.

text: ##LTLine##

text: ##LTLine##


text: [35] V. Rosato, S. Bologna, and F. Tiriticco. Topological properties of high-voltage electrical transmission networks. Electric Power Systems Research, 77(2):99 – 105, 2007. [36] J. Salmeron, K. Wood, and R. Baldick. Analysis of electric grid security under terrorist threat. Power Systems, IEEE Transactions on, 19(2):905–912, 2004. [37] J. Schneider, A. Gaul, C. Neumann, J. Hografer, W. Wellsow, M. Schwan, and A. Schnettler. Asset management techniques. International Journal of Electrical Power and Energy Systems, 28(9):643 – 654, 2006. [38] M. Schwan, K. Schilling, and A. Arssuﬁ De Melo. Reliability centered asset management in distribution networks -process and application examples. In Elec- tricity Distribution , 2007. The 19th International Conference on, 2007. [39] J. Setreus. Identifying critical components for system reliability in power trans- mission systems. PhD thesis, KTH Royal Institute of Technology, 2011. [40] J. Setreus, P. Hilber, S. Arnborg, and N. Taylor. Identifying critical compo- nents for transmission system reliability. Power Systems, IEEE Transactions on, 27(4):2106–2115, Nov 2012. [41] R. V. Sole, M. Rosas-Casals, B. Corominas Murtra, and S. Valverde. Robustness of the european power grids under intentional attack. Phys. Rev. E, 77:026102, 2008. [42] J. P. Sterbenz, E. K. Cetinkaya, M. A. Hameed, A. Jabbar, S. Qian, and J. Rohrer. Evaluation of network resilience, survivability, and disruption tolerance: analy- sis, topology generation, simulation, and experimentation. Telecommunication Systems, 52(2):705–736, 2013. [43] M. Van Eeten, A. Nieuwenhuijs, E. Luiijf, M. Klaver, and E. Cruz. The state and the threat of cascading failure across critical infrastructures. Public Administra- tion, 89(2), 2011. [44] P. Van Mieghem. Graph Spectra for Complex Networks. Cambridge University Press, Cambridge, UK, 2011. [45] Z. Wang, A. Scaglione, and R. Thomas. Generating statistically correct random topologies for testing smart grid communication and control networks. Smart Grid, IEEE Transactions on, 1(1):28–39, 2010. [46] D. Watts and S. Strogatz. Collective dynamics of ’small-world’ networks. Nature, (393):440–442, 1998. [47] F. Yang. A Comprehensive Approach for Bulk Power System Reliability Assess- ment. PhD thesis, KTH Royal Institute of Technology, 2007. [48] A. Yazdani and P. Jeffrey. Applying network theory to quantify the redundancy and structural robustness of water distribution systems. Journal of Water Re- sources Planning and Management, 138(2):153–161, 2012. 19

text: V.

text: Rosato,

text: S.

text: Bologna,

text: and

text: F.

text: Tiriticco.

text: Topological

text: properties

text: of

text: high-voltage

text: electrical

text: transmission

text: networks.

text: Electric

text: Power

text: Systems

text: Research,

text: J.

text: Salmeron,

text: K.

text: Wood,

text: and

text: R.

text: Baldick.

text: Analysis

text: of

text: electric

text: grid

text: security

text: under

text: terrorist

text: threat.

text: Power

text: Systems,

text: IEEE

text: Transactions

text: on,

text: J.

text: Schneider,

text: A.

text: Gaul,

text: C.

text: Neumann,

text: J.

text: Hografer,

text: W.

text: Wellsow,

text: M.

text: Schwan,

text: and

text: A.

text: Schnettler.

text: Asset

text: management

text: techniques.

text: International

text: Journal

text: of

text: Electrical

text: Power

text: and

text: Energy

text: Systems,

text: M.

text: Schwan,

text: K.

text: Schilling,

text: and

text: A.

text: Arssuﬁ

text: De

text: Melo.

text: Reliability

text: centered

text: asset

text: management

text: in

text: distribution

text: networks

text: -process

text: and

text: application

text: examples.

text: In

text: Elec-

text: tricity

text: Distribution

text: The

text: International

text: Conference

text: on,

text: J.

text: Setreus.

text: Identifying

text: critical

text: components

text: for

text: system

text: reliability

text: in

text: power

text: trans-

text: mission

text: systems.

text: PhD

text: thesis,

text: KTH

text: Royal

text: Institute

text: of

text: Technology,

text: J.

text: Setreus,

text: P.

text: Hilber,

text: S.

text: Arnborg,

text: and

text: N.

text: Taylor.

text: Identifying

text: critical

text: compo-

text: nents

text: for

text: transmission

text: system

text: reliability.

text: Power

text: Systems,

text: IEEE

text: Transactions

text: on,

text: Nov

text: R.

text: V.

text: Sole,

text: M.

text: Rosas-Casals,

text: B.

text: Corominas

text: Murtra,

text: and

text: S.

text: Valverde.

text: Robustness

text: of

text: the

text: european

text: power

text: grids

text: under

text: intentional

text: attack.

text: Phys.

text: Rev.

text: E,

text: J.

text: P.

text: Sterbenz,

text: E.

text: K.

text: Cetinkaya,

text: M.

text: A.

text: Hameed,

text: A.

text: Jabbar,

text: S.

text: Qian,

text: and

text: J.

text: Rohrer.

text: Evaluation

text: of

text: network

text: resilience,

text: survivability,

text: and

text: disruption

text: tolerance:

text: analy-

text: sis,

text: topology

text: generation,

text: simulation,

text: and

text: experimentation.

text: Telecommunication

text: Systems,

text: M.

text: Van

text: Eeten,

text: A.

text: Nieuwenhuijs,

text: E.

text: Luiijf,

text: M.

text: Klaver,

text: and

text: E.

text: Cruz.

text: The

text: state

text: and

text: the

text: threat

text: of

text: cascading

text: failure

text: across

text: critical

text: infrastructures.

text: Public

text: Administra-

text: tion,

text: P.

text: Van

text: Mieghem.

text: Graph

text: Spectra

text: for

text: Complex

text: Networks.

text: Cambridge

text: University

text: Press,

text: Cambridge,

text: UK,

text: Z.

text: Wang,

text: A.

text: Scaglione,

text: and

text: R.

text: Thomas.

text: Generating

text: statistically

text: correct

text: random

text: topologies

text: for

text: testing

text: smart

text: grid

text: communication

text: and

text: control

text: networks.

text: Smart

text: Grid,

text: IEEE

text: Transactions

text: on,

text: D.

text: Watts

text: and

text: S.

text: Strogatz.

text: Collective

text: dynamics

text: of

text: ’small-world’

text: networks.

text: Nature,

text: F.

text: Yang.

text: A

text: Comprehensive

text: Approach

text: for

text: Bulk

text: Power

text: System

text: Reliability

text: Assess-

text: ment.

text: PhD

text: thesis,

text: KTH

text: Royal

text: Institute

text: of

text: Technology,

text: A.

text: Yazdani

text: and

text: P.

text: Jeffrey.

text: Applying

text: network

text: theory

text: to

text: quantify

text: the

text: redundancy

text: and

text: structural

text: robustness

text: of

text: water

text: distribution

text: systems.

text: Journal

text: of

text: Water

text: Re-

text: sources

text: Planning

text: and

text: Management,


text: (cid:88) • 3 a (t, x, v) µ(t, β , v0) = µ (t, β , v0), where µ (t, x, v) := k , t t k t t k γ(x, v)k k=1 ∈ R ∈ ∈ { } R × R × → R for x , v H and k 1, 2, 3 , where a : + H are functions k given by (cid:104) ∗ (cid:48) (cid:105) (cid:104) − (cid:48) (cid:105) a (t, x, v) := v, A ϕ + f(v + ϕ ) f(ϕ ), ϕ 1 (cid:104) x ∗ (cid:48) (cid:48)(cid:48)(cid:105)0 0 x a (t, x, v) := ε2 B(t)QB (t)ϕ , ϕ 2 x x (5.8) ε2 (cid:104) − (cid:48)(cid:48)(cid:48)(cid:105)(cid:104) ∗ (cid:48) (cid:48) (cid:105) a (t, x, v) := v + ϕ ϕ , ϕ B(t)QB (t)ϕ , ϕ . 3 2 0 x x x x Formally, the SDE (5.6) can be obtained by Itô’s formula and a comparison of coeﬃcients: if one assumes that (β ) ≥ satisﬁes an SDE driven by (W Q) ≥ with t t 0 t t 0 drift and diﬀusion coeﬃcients to be determined, then by formally applying Itô’s (cid:104) − (cid:48) (cid:105) formula, one can write down an SDE for ( u ϕ , ϕ ) ≥ . Setting the result to zero (since we want a solution to (5.4)) and ctompaβrtingβtcogeﬃ0cients leads to (5.6). However, since we cannot ﬁnd any (inﬁnite-dimensional) Itô-type lemma that directly applies to our situation, we take care in Proposition 5.3 below to rigorously prove the result. In any case, we start with the following existence and uniqueness result. {F } Proposition 5.2. Let τ be a stopping time with respect to the ﬁltration ≥ . For F t t 0 any -measurable random variable β such that γ(β , v0) > 0 almost surely (where τ E ∞τ τ τ γ is deﬁned in (5.5)) and (β2) < , there exists a unique continuous solution τ (β ) ∈ to the SDE (5.6), with initial condition β at τ, up until the stopping t t [τ,τ∞) τ time τ∞ = lim →∞ τ > τ, where n n { ≥ } τ := inf t τ : γ(β , v0) = 1/n . n t t (cid:90) (cid:90) (cid:88) In other words ∧ ∧ 3 t τn t τn ≥ ≥ β ∧ = β + µ (s, β , v0)ds + σ(s, β , v0)dW Q, t τ, n 1. t τn τ k s s s s s τ τ k=1 Proof. The proof follows the fairly standard proof of existence and uniqueness of solutions to SDEs with locally Lipschitz coeﬃcients up until an explosion time (see for example [19, Theorem 1.18]). We however recall the key arguments here, since we are in a slightly non-standard set-up. We also suppose that (the general τ = 0 case is the same). (cid:40) ≥ ∈ R ∈ (cid:107) (cid:107) ≤ ≥ Step 1: Existence. Deﬁne for , , such that and , t 0 x v H v R n 1 ≥ b(t,x) if x is s.t γ(x, v) 1 σn(t, x, v) = γ(x,v) n , nb(t, x) otherwise 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: a

text: (t,

text: x,

text: v)

text: µ(t,

text: β

text: µ

text: (t,

text: β

text: where

text: µ

text: (t,

text: x,

text: v)

text: k

text: t

text: t

text: k

text: t

text: t

text: k

text: γ(x,

text: v)k

text: R

text: R

text: R

text: R

text: for

text: x

text: v

text: H

text: and

text: k

text: where

text: a

text: H

text: are

text: functions

text: k

text: given

text: by

text: a

text: (t,

text: x,

text: v)

text: v,

text: A

text: ϕ

text: f(v

text: ϕ

text: f(ϕ

text: ϕ

text: x

text: x

text: a

text: (t,

text: x,

text: v)

text: ε2

text: B(t)QB

text: (t)ϕ

text: ϕ

text: x

text: x

text: ε2

text: a

text: (t,

text: x,

text: v)

text: v

text: ϕ

text: ϕ

text: ϕ

text: B(t)QB

text: (t)ϕ

text: ϕ

text: x

text: x

text: x

text: x

text: Formally,

text: the

text: SDE

text: can

text: be

text: obtained

text: by

text: Itô’s

text: formula

text: and

text: a

text: comparison

text: of

text: coeﬃcients:

text: if

text: one

text: assumes

text: that

text: (β

text: satisﬁes

text: an

text: SDE

text: driven

text: by

text: (W

text: Q)

text: with

text: t

text: t

text: t

text: t

text: drift

text: and

text: diﬀusion

text: coeﬃcients

text: to

text: be

text: determined,

text: then

text: by

text: formally

text: applying

text: Itô’s

text: formula,

text: one

text: can

text: write

text: down

text: an

text: SDE

text: for

text: u

text: ϕ

text: ϕ

text: Setting

text: the

text: result

text: to

text: zero

text: (since

text: we

text: want

text: a

text: solution

text: to

text: and

text: leads

text: to

text: However,

text: since

text: we

text: cannot

text: ﬁnd

text: any

text: (inﬁnite-dimensional)

text: Itô-type

text: lemma

text: that

text: directly

text: applies

text: to

text: our

text: situation,

text: we

text: take

text: care

text: in

text: Proposition

text: below

text: to

text: rigorously

text: prove

text: the

text: result.

text: In

text: any

text: case,

text: we

text: start

text: with

text: the

text: following

text: existence

text: and

text: uniqueness

text: result.

text: {F

text: Proposition

text: Let

text: τ

text: be

text: a

text: stopping

text: time

text: with

text: respect

text: to

text: the

text: ﬁltration

text: For

text: F

text: t

text: t

text: any

text: -measurable

text: random

text: variable

text: β

text: such

text: that

text: γ(β

text: almost

text: surely

text: (where

text: τ

text: E

text: ∞τ

text: τ

text: τ

text: γ

text: is

text: deﬁned

text: in

text: and

text: there

text: exists

text: a

text: unique

text: continuous

text: solution

text: τ

text: (β

text: to

text: the

text: SDE

text: with

text: initial

text: condition

text: β

text: at

text: τ,

text: up

text: until

text: the

text: stopping

text: t

text: t

text: [τ,τ∞)

text: τ

text: time

text: τ∞

text: lim

text: τ

text: τ,

text: where

text: n

text: n

text: τ

text: inf

text: t

text: τ

text: γ(β

text: n

text: t

text: t

text: In

text: other

text: words

text: t

text: τn

text: t

text: τn

text: β

text: β

text: µ

text: (s,

text: β

text: σ(s,

text: β

text: Q,

text: t

text: τ,

text: n

text: t

text: τn

text: τ

text: k

text: s

text: s

text: s

text: s

text: s

text: τ

text: τ

text: Proof.

text: The

text: proof

text: follows

text: the

text: fairly

text: standard

text: proof

text: of

text: existence

text: and

text: uniqueness

text: of

text: solutions

text: to

text: SDEs

text: with

text: locally

text: Lipschitz

text: coeﬃcients

text: up

text: until

text: an

text: explosion

text: time

text: (see

text: for

text: example

text: Theorem

text: We

text: however

text: recall

text: the

text: key

text: arguments

text: here,

text: since

text: we

text: are

text: in

text: a

text: slightly

text: non-standard

text: set-up.

text: We

text: also

text: suppose

text: that

text: (the

text: general

text: τ

text: case

text: is

text: the

text: same).

text: R

text: Step

text: Existence.

text: Deﬁne

text: for

text: such

text: that

text: and

text: t

text: x

text: v

text: H

text: v

text: R

text: n

text: b(t,x)

text: if

text: x

text: is

text: s.t

text: γ(x,

text: v)

text: σn(t,

text: x,

text: v)

text: γ(x,v)

text: n

text: nb(t,

text: x)

text: otherwise

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:107) (cid:48) (cid:107) for a new constant depending on but independent of and . Returning C ϕ ε M 0 now to (5.14), we see that agin thanks to our choice of ξk, (cid:104) (cid:105) ≥ − (cid:107) (cid:107) − ε (cid:107) (cid:48) (cid:107) ≥ v0 , e κ v0 εCMeCM ϕ κ/2 τ∞ k τ∞ k2 0 (cid:107) (cid:107) (cid:107) (cid:48) (cid:107) ≤ for ε small enough so (cid:13)that(cid:13)ε( v0(cid:88)CMeCM + ε ϕ (cid:88)) κ/2. Thus (cid:13) (cid:13) τ∞ 0 (cid:104) (cid:105) ≥ M v0 , e 2 ≥ κ2 M 1 v0 2 (cid:107)τ∞ (cid:107)k (cid:107) (cid:107) . τ∞ e 2 4 e 2 (cid:80) k k k=1 k=1 (cid:107) (cid:107) → (cid:107) (cid:107) This is clearly a contradiction if we take large enough since as M e ϕ → ∞ (cid:107) (cid:107)− → ∞ → ∞ k 0 k (which implies that M e 2 as M ). k=1 k We now prove that lim → β exists almost surely under assumption (5.13). Fix t τ∞ t . Deﬁne α < α¯ { ∈ ≥ } ξ0 := inf t [0, τ∞) : β α¯ , t and { ∈ ≤ } { ∈ ≥ } ξ2k+1 := inf t (ξ2k, τ∞) : β α , ξ2k+2 := inf t (ξ2k+1, τ∞) : β α¯ , t t ≥ {∅} for all k 0, with inf = τ∞ by convention. Suppose for a contradiction that ξn < τ∞ for all n. Let |(cid:104) − (cid:48) (cid:105)| s = sup ( u ϕ , ϕ ) , n ∈ ξn θ θ θ [α,α¯] ∗ ∈ and let be such that this supremum is attained (this exists by continuity). θ [α, α¯] n (cid:54)→ → ∞ Suppose for contradiction that as i.e. that there exists a s 0 n ∞ n ≥ ≥ subsequence such that for some , for all . We know that (s ) δ > 0 s δ r 1 ≥ nr r=1 ∈ nr (cid:104) − (cid:48) (cid:105) for all r 1 there exists some t [ξnr, ξnr+1], such that u ϕ ∗ , ϕ ∗ = 0. This is because at time ξnr, α¯ (or α) is a local minimum of the futnctionθnrα∗ (cid:55)→∈θnr(cid:107)uξnr −ϕα(cid:107)2 while at time ξnr+1 α (or α¯) is. Therefore, by continuity, since θ [α, α¯], there must exist t ∈ [ξnr, ξnr+1] such that θ∗ is a local minimum of thisnfrunction. In this case nr (cid:107) − (cid:107) ≥ (cid:107) − (cid:107) ≥ 1 (cid:104) − (cid:48) (cid:105) r,s∈[ξsnur,pξnr+1] ur us 2 ut uξnr 2 (cid:107)ϕ(cid:48)0(cid:107)2 ut uξnr , ϕθn∗r 2 1 (cid:104) − − (cid:48) (cid:105) = (cid:107)ϕ(cid:48) (cid:107)2 ut ϕθn∗r + ϕθn∗r uξnr , ϕθn∗r 2 0 1 (cid:104) − (cid:48) (cid:105) s2 ≥ δ2 = (cid:107)ϕ(cid:48) (cid:107)2 uξnr ϕθn∗r , ϕθn∗r 2 = (cid:107)ϕn(cid:48) r(cid:107)2 (cid:107)ϕ(cid:48) (cid:107)2 . 0 0 0 This is a contradiction since we are assuming that ξn < τ∞ for all n. Indeed it (cid:107) (cid:107) implies inﬁnite oscillations (of a nontrivial magnitude) of over a compact time u → → ∞ t interval. We can therefore conclude that as . s 0 n n 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: a

text: new

text: constant

text: depending

text: on

text: but

text: independent

text: of

text: and

text: Returning

text: C

text: ϕ

text: ε

text: M

text: now

text: to

text: we

text: see

text: that

text: agin

text: thanks

text: to

text: our

text: choice

text: of

text: ξk,

text: ε

text: e

text: κ

text: εCMeCM

text: ϕ

text: τ∞

text: k

text: τ∞

text: for

text: ε

text: small

text: enough

text: so

text: ε

text: ϕ

text: Thus

text: τ∞

text: M

text: e

text: κ2

text: M

text: τ∞

text: e

text: e

text: k

text: k

text: This

text: is

text: clearly

text: a

text: contradiction

text: if

text: we

text: take

text: large

text: enough

text: since

text: as

text: M

text: e

text: ϕ

text: k

text: k

text: (which

text: implies

text: that

text: M

text: e

text: as

text: M

text: k

text: We

text: now

text: prove

text: that

text: lim

text: β

text: exists

text: almost

text: surely

text: under

text: assumption

text: Fix

text: t

text: τ∞

text: t

text: Deﬁne

text: α

text: α¯

text: ξ0

text: inf

text: t

text: τ∞)

text: β

text: α¯

text: t

text: and

text: inf

text: t

text: τ∞)

text: β

text: α

text: inf

text: t

text: τ∞)

text: β

text: α¯

text: t

text: t

text: for

text: all

text: k

text: with

text: inf

text: τ∞

text: by

text: convention.

text: Suppose

text: for

text: a

text: contradiction

text: that

text: ξn

text: τ∞

text: for

text: all

text: n.

text: Let

text: s

text: sup

text: u

text: ϕ

text: ϕ

text: n

text: ξn

text: θ

text: θ

text: θ

text: [α,α¯]

text: and

text: let

text: be

text: such

text: that

text: this

text: supremum

text: is

text: attained

text: (this

text: exists

text: by

text: continuity).

text: θ

text: [α,

text: α¯]

text: n

text: Suppose

text: for

text: contradiction

text: that

text: as

text: i.e.

text: that

text: there

text: exists

text: a

text: s

text: n

text: n

text: subsequence

text: such

text: that

text: for

text: some

text: for

text: all

text: We

text: know

text: that

text: (s

text: δ

text: s

text: δ

text: r

text: nr

text: nr

text: for

text: all

text: r

text: there

text: exists

text: some

text: t

text: [ξnr,

text: such

text: that

text: u

text: ϕ

text: ϕ

text: This

text: is

text: because

text: at

text: time

text: ξnr,

text: α¯

text: (or

text: α)

text: is

text: a

text: local

text: minimum

text: of

text: the

text: futnctionθnrα∗

text: while

text: at

text: time

text: α

text: (or

text: α¯)

text: is.

text: Therefore,

text: by

text: continuity,

text: since

text: θ

text: [α,

text: α¯],

text: there

text: must

text: exist

text: t

text: [ξnr,

text: such

text: that

text: θ∗

text: is

text: a

text: local

text: minimum

text: of

text: thisnfrunction.

text: In

text: this

text: case

text: nr

text: ur

text: us

text: ut

text: uξnr

text: ut

text: uξnr

text: ϕθn∗r

text: ut

text: ϕθn∗r

text: ϕθn∗r

text: uξnr

text: ϕθn∗r

text: δ2

text: uξnr

text: ϕθn∗r

text: ϕθn∗r

text: This

text: is

text: a

text: contradiction

text: since

text: we

text: are

text: assuming

text: that

text: ξn

text: τ∞

text: for

text: all

text: n.

text: Indeed

text: it

text: implies

text: inﬁnite

text: oscillations

text: (of

text: a

text: nontrivial

text: magnitude)

text: of

text: over

text: a

text: compact

text: time

text: u

text: t

text: interval.

text: We

text: can

text: therefore

text: conclude

text: that

text: as

text: s

text: n

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6.1 Proofs In order to prove the results of Section , we will need the following lemmas. K ∈ R ≥ Lemma 6.7. There exists a constant such that for all α , α and h 0, 1 2 (cid:107) − (cid:107) ≤ K U (h) U (h) α h, α1+α2 α2 1 L (recall that (U (t)) ≥ is the semigroup generated by given by (6.4)). α t 0 α (cid:107) − (cid:107) ≤ (cid:107) − (cid:107) (cid:107) (cid:107) ≤ Proof. Note that , since for U (h) U (h) U (h) U (h) U (t) 1 ∈ R α1+α2 L − Lα2 (cid:48) α1− (cid:48) 0 α all . The operator is bounded over its domain α ( ) = f (ϕ ) f (ϕ ) α1 0 α1 0 by assumption. We may therefore use the variation of parameters formula [11, E (cid:90) ∈ Page 161] to write for any v H − h − (cid:48) − (cid:48) (U (h) U (h))v = U (h r) (f (ϕ ) f (ϕ )) U (r)vdr. α1 0 0 α1 0 α1 0 (cid:48) (cid:55)→ The result now follows from the Lipschitz property of and , as well as the f α ϕ (cid:107) (cid:107) ≤ ∈ R α fact that for all . U (t) 1 α α Lemma 6.8. For G deﬁned by (6.3), it holds that (cid:107) (cid:107) ≤ c (cid:107) (cid:107) ∀ ∈ ∈ R G(z, β, β) z 2, z H, β , 2 (cid:48) where is the Lipschitz constant of (which is independent of and ). c f z β (cid:90) Proof. We ﬁrst note (by Assumption 2.2 (ii) on ) that we may write f − 1 (cid:48) f(z + ϕ ) f(ϕ ) = f (θz + ϕ )zdθ. β (cid:90) β β 0 Therefore 1 (cid:48) − (cid:48) G(z, β, β) = [f (θz + ϕ ) f (ϕ )]zdθ, β β (cid:90) (cid:90) 0 so that (cid:107) (cid:107) ≤ 1 (cid:107) (cid:48) − (cid:48) (cid:107) (cid:107) (cid:107) ≤ 1 (cid:107) (cid:107) G(z, β, β) f (θz + ϕ ) f (ϕ ) dθ z c θdθ z 2, β β 0 0 (cid:48) where is the Lipschitz constant of . c f We can now prove Theorem 6.2. ≤ ≤ Proof of Theorem 6.2. Suppose that s t T < τ∞. We have that the mild solution to (6.2) (cid:90)is given by (cid:90) (cid:90) − t − t − − t − z = U (t s)z + U (t r)G(z , β , α)dr+ε U (t r)B(r)dW Q U (t r)dϕ . t α s α r r α r α βr s s s (6.11) 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Proofs

text: In

text: order

text: to

text: prove

text: the

text: results

text: of

text: Section

text: we

text: will

text: need

text: the

text: following

text: lemmas.

text: K

text: R

text: Lemma

text: There

text: exists

text: a

text: constant

text: such

text: that

text: for

text: all

text: α

text: α

text: and

text: h

text: K

text: U

text: (h)

text: U

text: (h)

text: α

text: h,

text: α2

text: L

text: (recall

text: that

text: (U

text: (t))

text: is

text: the

text: semigroup

text: generated

text: by

text: given

text: by

text: α

text: t

text: α

text: Proof.

text: Note

text: that

text: since

text: for

text: U

text: (h)

text: U

text: (h)

text: U

text: (h)

text: U

text: (h)

text: U

text: (t)

text: R

text: L

text: α1−

text: α

text: all

text: The

text: operator

text: is

text: bounded

text: over

text: its

text: domain

text: α

text: f

text: (ϕ

text: f

text: (ϕ

text: α1

text: α1

text: by

text: assumption.

text: We

text: may

text: therefore

text: use

text: the

text: variation

text: of

text: parameters

text: formula

text: E

text: Page

text: to

text: write

text: for

text: any

text: v

text: H

text: h

text: (U

text: (h)

text: U

text: (h))v

text: U

text: (h

text: r)

text: (f

text: (ϕ

text: f

text: (ϕ

text: U

text: (r)vdr.

text: α1

text: α1

text: α1

text: The

text: result

text: now

text: follows

text: from

text: the

text: Lipschitz

text: property

text: of

text: and

text: as

text: well

text: as

text: the

text: f

text: α

text: ϕ

text: R

text: α

text: fact

text: that

text: for

text: all

text: U

text: (t)

text: α

text: α

text: Lemma

text: For

text: G

text: deﬁned

text: by

text: it

text: holds

text: that

text: c

text: R

text: G(z,

text: β,

text: β)

text: z

text: z

text: H,

text: β

text: where

text: is

text: the

text: Lipschitz

text: constant

text: of

text: (which

text: is

text: independent

text: of

text: and

text: c

text: f

text: z

text: β

text: Proof.

text: We

text: ﬁrst

text: note

text: (by

text: Assumption

text: (ii)

text: on

text: that

text: we

text: may

text: write

text: f

text: f(z

text: ϕ

text: f(ϕ

text: f

text: (θz

text: ϕ

text: )zdθ.

text: β

text: β

text: β

text: Therefore

text: G(z,

text: β,

text: β)

text: [f

text: (θz

text: ϕ

text: f

text: (ϕ

text: )]zdθ,

text: β

text: β

text: so

text: that

text: G(z,

text: β,

text: β)

text: f

text: (θz

text: ϕ

text: f

text: (ϕ

text: dθ

text: z

text: c

text: θdθ

text: z

text: β

text: β

text: where

text: is

text: the

text: Lipschitz

text: constant

text: of

text: c

text: f

text: We

text: can

text: now

text: prove

text: Theorem

text: Proof

text: of

text: Theorem

text: Suppose

text: that

text: s

text: t

text: T

text: τ∞.

text: We

text: have

text: that

text: the

text: mild

text: solution

text: to

text: given

text: by

text: t

text: t

text: t

text: z

text: U

text: (t

text: s)z

text: U

text: (t

text: r)G(z

text: β

text: α)dr+ε

text: U

text: (t

text: r)B(r)dW

text: Q

text: U

text: (t

text: r)dϕ

text: t

text: α

text: s

text: α

text: r

text: r

text: α

text: r

text: α

text: βr

text: s

text: s

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:90) → The(cid:10)dominated convergence theorem also im(cid:11)plies that as , h 0 t − → t (cid:104) − (cid:105) P A − z , P A − (f(z + ϕ ) f(ϕ )) dr z , f(z + ϕ ) f(ϕ ) dr. k(r) r r k(r) r r βr βr r r βr βr s s With this in hand, together with the limits calculated for the second, third and fourth terms of (6.23) in the proof Theorem 6.2, we see that taking the as lim sup → in (6.23) yields h 0 (cid:90) (cid:90) (cid:107) (cid:107) ≤ (cid:107) (cid:107)(cid:90) t (cid:107) (cid:107) t(cid:104) (cid:105) z 2 z 2 + Ξ( z )dr + 2 z , κ (r, β , v0)dW Q t s r r 1 r r r s s (cid:90)t (cid:104)(cid:104) − (cid:105) (cid:105) (cid:10) (cid:11) (cid:0) (cid:1) + 2 z , f(z + ϕ ) f(ϕ ) dr r r βr βr s t ∗ + 2 z , κ (z , v0, β ) + Tr κ (r, β , v0)Qκ (r, β , v0) dr. (6.24) r 3 r r r 1 r r 1 r r s Moreover, we can then use (6.17), (6.19) and the deﬁnition of to conclude. κ 3 7 Long-time behavior Again suppose that (ϕ ) ∈R, f and A satisfy Assumptions 2.1, 2.2 and 2.3 (i) re- α α spectively, and that (u ) ≥ = (v0+ϕ ) ≥ is the solution to (4.1) with (deterministic) t t 0 t − 0 t∈0 initial condition such that according to Proposition 4.2. u u ϕ H (cid:55)→ ∗ 0 0 0 ∞ ≥ ∗ Let be any function on such that for all , is a global t β [0, ) t 0 β t R (cid:51) (cid:55)→ (cid:107) − (cid:107) ∗ t minimum of the map α m(t, α) = u ϕ 2. Note that β exists for all ≥ t α t by Lemma 5.1 but it may not be unique. Deﬁne t 0 ∗ − ≥ z := u ϕ ∗, t 0. t t βt The main result of this section is Theorem 7.3, which generalizes the inequality of Theorem 6.6 to arbitrary time. This theorem is a ﬁrst step in the long-time analysis of the system. The global stability results of [5] lends one hope that, for some traveling wave systems, we might be able to get some sort of long-time bound (cid:107) ∗(cid:107) (cid:13) (cid:13) on z 2. In par(cid:13)ticular, o(cid:13)ne may infer from [5, Theorem 3.1] that, under some t ∈ technical assumptions, if (i.e. there is no stochastic term), and is ε = 0 u E u − → → ∞0 0 continuous, then ut ϕβ∗ ∞ 0 (in supremum norm) as t . Coming back to our stochastic setting wt ith , this motivates us to wonder if the stabilizing ε > 0 eﬀect of the internal dynamics of the deterministic system could balance the disorder (cid:107) ∗(cid:107) coming from the noise. In such a case then a long-time bound on z 2 might be t possible. Unfortunately [5] uses the method of comparison of ODE’s, and the bounds are not easy to adapt to our semigroup formalism. Nevertheless, the development, 30 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: convergence

text: theorem

text: also

text: that

text: as

text: h

text: t

text: t

text: P

text: A

text: z

text: P

text: A

text: (f(z

text: ϕ

text: f(ϕ

text: dr

text: z

text: f(z

text: ϕ

text: f(ϕ

text: dr.

text: k(r)

text: r

text: r

text: k(r)

text: r

text: r

text: βr

text: βr

text: r

text: r

text: βr

text: βr

text: s

text: s

text: With

text: this

text: in

text: hand,

text: together

text: with

text: the

text: limits

text: calculated

text: for

text: the

text: second,

text: third

text: and

text: fourth

text: terms

text: of

text: in

text: the

text: proof

text: Theorem

text: we

text: see

text: that

text: taking

text: the

text: as

text: lim

text: sup

text: in

text: yields

text: h

text: t

text: z

text: z

text: Ξ(

text: z

text: )dr

text: z

text: κ

text: (r,

text: β

text: Q

text: t

text: s

text: r

text: r

text: r

text: r

text: r

text: s

text: s

text: z

text: f(z

text: ϕ

text: f(ϕ

text: dr

text: r

text: r

text: βr

text: βr

text: s

text: t

text: z

text: κ

text: (z

text: β

text: Tr

text: κ

text: (r,

text: β

text: (r,

text: β

text: dr.

text: r

text: r

text: r

text: r

text: r

text: r

text: r

text: r

text: s

text: Moreover,

text: we

text: can

text: then

text: use

text: and

text: the

text: deﬁnition

text: of

text: to

text: conclude.

text: κ

text: Long-time

text: behavior

text: Again

text: suppose

text: that

text: (ϕ

text: ∈R,

text: f

text: and

text: A

text: satisfy

text: Assumptions

text: and

text: (i)

text: re-

text: α

text: α

text: spectively,

text: and

text: that

text: (u

text: is

text: the

text: solution

text: to

text: with

text: (deterministic)

text: t

text: t

text: t

text: initial

text: condition

text: such

text: that

text: according

text: to

text: Proposition

text: u

text: u

text: ϕ

text: H

text: Let

text: be

text: any

text: function

text: on

text: such

text: that

text: for

text: all

text: is

text: a

text: global

text: t

text: β

text: t

text: β

text: t

text: R

text: t

text: minimum

text: of

text: the

text: map

text: α

text: m(t,

text: α)

text: u

text: ϕ

text: Note

text: that

text: β

text: exists

text: for

text: all

text: t

text: α

text: t

text: by

text: Lemma

text: but

text: it

text: may

text: not

text: be

text: unique.

text: Deﬁne

text: t

text: z

text: u

text: ϕ

text: t

text: t

text: t

text: βt

text: The

text: main

text: result

text: of

text: this

text: section

text: is

text: Theorem

text: which

text: generalizes

text: the

text: inequality

text: of

text: Theorem

text: to

text: arbitrary

text: time.

text: This

text: theorem

text: is

text: a

text: ﬁrst

text: step

text: in

text: the

text: long-time

text: analysis

text: of

text: the

text: system.

text: The

text: global

text: stability

text: results

text: of

text: lends

text: one

text: hope

text: that,

text: for

text: some

text: traveling

text: wave

text: systems,

text: we

text: might

text: be

text: able

text: to

text: get

text: some

text: sort

text: of

text: long-time

text: bound

text: on

text: z

text: In

text: may

text: infer

text: from

text: Theorem

text: that,

text: under

text: some

text: t

text: technical

text: assumptions,

text: if

text: (i.e.

text: there

text: is

text: no

text: stochastic

text: term),

text: and

text: is

text: ε

text: u

text: E

text: u

text: ∞0

text: continuous,

text: then

text: ut

text: ϕβ∗

text: (in

text: supremum

text: norm)

text: as

text: t

text: Coming

text: back

text: to

text: our

text: stochastic

text: setting

text: wt

text: ith

text: this

text: motivates

text: us

text: to

text: wonder

text: if

text: the

text: stabilizing

text: ε

text: eﬀect

text: of

text: the

text: internal

text: dynamics

text: of

text: the

text: deterministic

text: system

text: could

text: balance

text: the

text: disorder

text: coming

text: from

text: the

text: noise.

text: In

text: such

text: a

text: case

text: then

text: a

text: long-time

text: bound

text: on

text: z

text: might

text: be

text: t

text: possible.

text: Unfortunately

text: uses

text: the

text: method

text: of

text: comparison

text: of

text: ODE’s,

text: and

text: the

text: bounds

text: are

text: not

text: easy

text: to

text: adapt

text: to

text: our

text: semigroup

text: formalism.

text: Nevertheless,

text: the

text: development,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: points in space, then through the Girsanov theorem (u ) ≥ will also be uncorrelated. |D t| →t 0 → We think that this is by no means necessary for as . In fact, it is 0 δ 0 δ,T possible that even if the noise is quite degenerate, the dynamics of and might A f ensure that (u ) ≥ is not. t t 0 Remark 7.6. Suppose that there were to exist constants b, C > 0 such that, for all ∈ , u E − (cid:104) − − − (cid:105) ≤ − (cid:107) − (cid:107) Ξ(u ϕ ) + 2 f(u) f(u ϕ ), u ϕ b u ϕ 2 + C, α α α α (cid:55)→ (cid:107) − (cid:107) where α is a global minimizer of α u ϕ 2. Then a consequence of Theorem α 7.3 would be that E (cid:107) ∗(cid:107) ≤ − − − E (cid:107) ∗(cid:107) − − ∀ ≥ z 2 (1 e bt)C + e bt z 2 + ε2Tr(Q)(1 e bt), t 0. t 0 E (cid:107) ∗(cid:107) That is, we would obtain a bound on z 2 which holds uniformly for all t > t . Unfortunately, at the moment we do not have any examples where the above 0 inequality holds. However, we believe that it might be possible for some traveling waves, particularly if we work in a Hilbert space with weighted inner product, and plan to investigate this in the future. 7.1 Proof of Theorem 7.3 and Lemma 7.12 In order to prove Theorem 7.3, we introduce the following deﬁnitions. ∈ ⊂ ∈ ⇔ ∈ ∃ The set : For deﬁne by i) , ii) a unique E δ (0, 1) E E u E u E δ (cid:55)→ (cid:107) − δ (cid:107) δ ∈ − global minimum Γ(u) of α u ϕ 2, and iii) for all α [Γ(u) δ, Γ(u) + δ], · α (cid:55)→ (cid:107) − (cid:107) γ(α, u) > δ, where we recall that γ( , u) is the ‘curvature’ of the map α u ϕ 2 α given by (5.5). ∈ ⊂ ∈ ⇔ The set EM: For M > 0 and δ(cid:13) (0, 1),(cid:13)let EM E be such that u EM i) ∈ δ(cid:107) − (cid:107) (cid:13) (cid:13) ∈ Rδ\ δ− δ , ii) , and iii) for all , u E u ϕ < M α [Γ(u) δ, Γ(u) + δ] δ 0 (cid:107) − (cid:107) − (cid:107) (cid:48) (cid:107) − u ϕ > u φ + δ3(δ ϕ +(cid:110) 2M) 1. (cid:111) α Γ(u) 0 ≤ (cid:107) (cid:107) ≥ − The stopping time ρδ : For T > 0, we deﬁne ρδ := inf t T : v0 δ 1 , with T T t ρδ = T if the set is empty. T The process (ηδ) ∈ : For T > 0, we now introduce the process (ηδ) ∈ , for any ∈ ∗ t −t [0,T] t {t [0≥,T] ∈ δ (0, γ(β , u ϕ )) in the fol(cid:110)lowing recursive wa(cid:111)y. Let τ0 = inf t 0 : u − } ∧ 0 0 0 ≥ t Eδ 1 ρδ , and for any k 0, let δ T (cid:110) (cid:111) ≥ ∈ − ∧ τ2k+1 = inf t τ2k : u / Eδ 1 ρδ , t δ/2 T ≥ ∈ − ∧ τ2k+2 = inf t τ2k+1 : u Eδ 1 ρδ . t δ T 32

text: points

text: in

text: space,

text: then

text: through

text: the

text: Girsanov

text: theorem

text: (u

text: will

text: also

text: be

text: uncorrelated.

text: |D

text: t|

text: →t

text: We

text: think

text: that

text: this

text: is

text: by

text: no

text: means

text: necessary

text: for

text: as

text: In

text: fact,

text: it

text: is

text: δ

text: δ,T

text: possible

text: that

text: even

text: if

text: the

text: noise

text: is

text: quite

text: degenerate,

text: the

text: dynamics

text: of

text: and

text: might

text: A

text: f

text: ensure

text: that

text: (u

text: is

text: not.

text: t

text: t

text: Remark

text: Suppose

text: that

text: there

text: were

text: to

text: exist

text: constants

text: b,

text: C

text: such

text: that,

text: for

text: all

text: u

text: E

text: Ξ(u

text: ϕ

text: f(u)

text: f(u

text: ϕ

text: u

text: ϕ

text: b

text: u

text: ϕ

text: C,

text: α

text: α

text: α

text: α

text: where

text: α

text: is

text: a

text: global

text: minimizer

text: of

text: α

text: u

text: ϕ

text: Then

text: a

text: consequence

text: of

text: Theorem

text: α

text: would

text: be

text: that

text: E

text: E

text: z

text: e

text: bt)C

text: e

text: bt

text: z

text: e

text: bt),

text: t

text: t

text: E

text: That

text: is,

text: we

text: would

text: obtain

text: a

text: bound

text: on

text: z

text: which

text: holds

text: uniformly

text: for

text: all

text: t

text: t

text: Unfortunately,

text: at

text: the

text: moment

text: we

text: do

text: not

text: have

text: any

text: examples

text: where

text: the

text: above

text: inequality

text: holds.

text: However,

text: we

text: believe

text: that

text: it

text: might

text: be

text: possible

text: for

text: some

text: traveling

text: waves,

text: particularly

text: if

text: we

text: work

text: in

text: a

text: Hilbert

text: space

text: with

text: weighted

text: inner

text: product,

text: and

text: plan

text: to

text: investigate

text: this

text: in

text: the

text: future.

text: Proof

text: of

text: Theorem

text: and

text: Lemma

text: In

text: order

text: to

text: prove

text: Theorem

text: we

text: introduce

text: the

text: following

text: deﬁnitions.

text: The

text: set

text: For

text: deﬁne

text: by

text: i)

text: ii)

text: a

text: unique

text: E

text: δ

text: E

text: E

text: u

text: E

text: u

text: E

text: δ

text: δ

text: δ

text: global

text: minimum

text: Γ(u)

text: of

text: α

text: u

text: ϕ

text: and

text: iii)

text: for

text: all

text: α

text: [Γ(u)

text: δ,

text: Γ(u)

text: δ],

text: α

text: γ(α,

text: u)

text: δ,

text: where

text: we

text: recall

text: that

text: γ(

text: u)

text: is

text: the

text: ‘curvature’

text: of

text: the

text: map

text: α

text: u

text: ϕ

text: α

text: given

text: by

text: The

text: set

text: EM:

text: For

text: M

text: and

text: EM

text: E

text: be

text: such

text: that

text: u

text: EM

text: i)

text: Rδ\

text: δ−

text: δ

text: ii)

text: and

text: iii)

text: for

text: all

text: u

text: E

text: u

text: ϕ

text: M

text: α

text: [Γ(u)

text: δ,

text: Γ(u)

text: δ]

text: δ

text: u

text: ϕ

text: u

text: φ

text: δ3(δ

text: ϕ

text: α

text: Γ(u)

text: The

text: stopping

text: time

text: ρδ

text: For

text: T

text: we

text: deﬁne

text: ρδ

text: inf

text: t

text: T

text: δ

text: with

text: T

text: T

text: t

text: ρδ

text: T

text: if

text: the

text: set

text: is

text: empty.

text: T

text: The

text: process

text: (ηδ)

text: For

text: T

text: we

text: now

text: introduce

text: the

text: process

text: (ηδ)

text: for

text: any

text: t

text: −t

text: t

text: {t

text: δ

text: γ(β

text: u

text: ϕ

text: in

text: the

text: recursive

text: Let

text: τ0

text: inf

text: t

text: u

text: t

text: Eδ

text: ρδ

text: and

text: for

text: any

text: k

text: let

text: δ

text: T

text: inf

text: t

text: u

text: Eδ

text: ρδ

text: t

text: T

text: inf

text: t

text: u

text: Eδ

text: ρδ

text: t

text: δ

text: T


text: ∈ We assumed at the start of the proof that , and for some s = 0 t = T u E 0 δ suﬃciently small . We now treat the more general case when these assumptions do δ (cid:54) ∈ not hold. First, if , then almost surely there exists a such that s = 0 δ > 0 u E s δ (this is noted in the proof of Lemma 7.12). The proof of this case now proceeds ∈ exactly as above, with τ0 redeﬁned to be s. Second, suppose that s = 0 but u / E |D | → → → 0 δ for any δ > 0. Since 0, it follows that τ0 0 as δ 0, and the result still δ,T holds. 7.2 Auxiliary Lemmas ∈ ∈ − (cid:54) Lemma 7.8. Suppose that u EM(cid:13)(cid:13) . Then fo(cid:13)(cid:13)r all α [Γ(u) δ, Γ(u)+δ], α = Γ(u), δ − (cid:107) − (cid:107) − − δ(α Γ(u))2 u ϕ u ϕ > (cid:107) (cid:48) (cid:107) . α Γ(u) δ ϕ + 2M 0 ∈ − (cid:54) Proof. By Taylor’s theorem, for all , , for some α [Γ(u) δ, Γ(u) + δ] α = Γ(u) α¯ (cid:13) (cid:13) (cid:13) (cid:13) between α and (cid:13)Γ(u) (cid:13) (cid:13) (cid:13) (cid:107) − (cid:107) − − − − u ϕ 2 = u ϕ 2 + γ(α¯, u)(α Γ(u))2 > u ϕ 2 + δ(α Γ(u))2, α Γ(u) Γ(u) (cid:13) (cid:13) (cid:13) (cid:13) the last inequality fol(cid:13)lowing fro(cid:13)m the deﬁnition(cid:13) of EM. H(cid:13)ence δ (cid:107) − (cid:107) − − (cid:107) − (cid:107) − − ( u ϕ u ϕ )( u ϕ + u ϕ ) > δ(α Γ(u))2, α Γ(u) α Γ(u) (cid:13) (cid:13) so that (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) − − (cid:107) − (cid:107) − − δ(α Γ(u))2 ≥ δ(α Γ(u))2 u ϕ u ϕ >(cid:107) − (cid:107) − − − . α Γ(u) u ϕ + u ϕ ϕ ϕ + 2 u ϕ α Γ(u) Γ(u) α Γ(u) ∈ − Now again by Taylor’s theorem, for some , it holds that λ [0, 1] ϕ ϕ = − (cid:48) (cid:48) −(cid:13) (cid:13) α Γ(u) ((1 λ)ϕ + λϕ )(α (cid:13)Γ(u)). Thus(cid:13) α Γ(u) − ≤ | − | (cid:107) (cid:48) (cid:107) ϕ ϕ α Γ(u) ϕ . α Γ(u) 0 Therefore, making us(cid:13)(cid:13)e of the de(cid:13)(cid:13)ﬁnition of EδM, (cid:13) (cid:13) (cid:13) (cid:13) − (cid:107) − (cid:107) − − δ(α Γ(u))2 u ϕα u ϕΓ(u) >| − | (cid:107) (cid:48) (cid:107) − α Γ(u) ϕ + 2 u ϕ − 0 Γ(u)− ≥ δ(α Γ(u))2 ≥ δ(α Γ(u))2 | − | (cid:107) (cid:48) (cid:107) (cid:107) (cid:48) (cid:107) . α Γ(u) ϕ + 2M δ ϕ + 2M 0 0 ⊂ Lemma 7.9. If δ < δ , then EM EM. 1 2 δ2 δ1 37 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: We

text: assumed

text: at

text: the

text: start

text: of

text: the

text: proof

text: that

text: and

text: for

text: some

text: s

text: t

text: T

text: u

text: E

text: δ

text: suﬃciently

text: small

text: We

text: now

text: treat

text: the

text: more

text: general

text: case

text: when

text: these

text: assumptions

text: do

text: δ

font: default

text: not

text: hold.

text: First,

text: if

text: then

text: almost

text: surely

text: there

text: exists

text: a

text: such

text: that

text: s

text: δ

text: u

text: E

text: s

text: δ

text: (this

text: is

text: noted

text: in

text: the

text: proof

text: of

text: Lemma

text: The

text: proof

text: of

text: this

text: case

text: now

text: proceeds

text: exactly

text: as

text: above,

text: with

text: τ0

text: redeﬁned

text: to

text: be

text: s.

text: Second,

text: suppose

text: that

text: s

text: but

text: u

text: E

text: |D

text: δ

text: for

text: any

text: δ

text: Since

text: it

text: follows

text: that

text: τ0

text: as

text: δ

text: and

text: the

text: result

text: still

text: δ,T

text: holds.

text: Auxiliary

text: Lemmas

text: Lemma

text: Suppose

text: that

text: u

text: Then

text: all

text: α

text: [Γ(u)

text: δ,

text: Γ(u)+δ],

text: α

text: Γ(u),

text: δ

text: δ(α

text: u

text: ϕ

text: u

text: ϕ

text: α

text: Γ(u)

text: δ

text: ϕ

text: Proof.

text: By

text: Taylor’s

text: theorem,

text: for

text: all

text: for

text: some

text: α

text: [Γ(u)

text: δ,

text: Γ(u)

text: δ]

text: α

text: Γ(u)

text: α¯

text: between

text: α

text: and

text: u

text: ϕ

text: u

text: ϕ

text: γ(α¯,

text: u)(α

text: u

text: ϕ

text: δ(α

text: α

text: Γ(u)

text: Γ(u)

text: the

text: last

text: inequality

text: the

text: of

text: EM.

text: δ

text: u

text: ϕ

text: u

text: ϕ

text: u

text: ϕ

text: u

text: ϕ

text: δ(α

text: α

text: Γ(u)

text: α

text: Γ(u)

text: so

text: that

text: δ(α

text: δ(α

text: u

text: ϕ

text: u

text: ϕ

text: α

text: Γ(u)

text: u

text: ϕ

text: u

text: ϕ

text: ϕ

text: ϕ

text: u

text: ϕ

text: α

text: Γ(u)

text: Γ(u)

text: α

text: Γ(u)

text: Now

text: again

text: by

text: Taylor’s

text: theorem,

text: for

text: some

text: it

text: holds

text: that

text: λ

text: ϕ

text: ϕ

text: α

text: Γ(u)

text: λ)ϕ

text: λϕ

text: )(α

text: α

text: Γ(u)

text: ϕ

text: ϕ

text: α

text: Γ(u)

text: ϕ

text: α

text: Γ(u)

text: Therefore,

text: making

text: of

text: the

text: of

text: EδM,

text: δ(α

text: u

text: ϕα

text: u

text: ϕΓ(u)

text: α

text: Γ(u)

text: ϕ

text: u

text: ϕ

text: Γ(u)−

text: δ(α

text: δ(α

text: α

text: Γ(u)

text: ϕ

text: δ

text: ϕ

text: Lemma

text: If

text: δ

text: δ

text: then

text: EM

text: EM.

text: δ2

text: δ1

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: P ∈ ¯ (7.10) (X + ϕ / E) = 0. t 0 (cid:55)→ (cid:107) − (cid:107) To show (7.10), by Lemma 7.11 it suﬃces to show that i) α X + ϕ ϕ 2 t 0 α has a unique global minimum almost surely, and ii) . α¯ γ(α¯, X + ϕ ) > 0 t 0 To show i), let (cid:104) − (cid:105) − (cid:107) − (cid:107) Y := 2 X , ϕ ϕ , and Z :(cid:0)= Y ϕ (cid:1) ϕ 2 . (7.11) α,t t α 0 α,t α,t α 0 (cid:107) − (cid:107) (cid:107) (cid:107) − (cid:107) (cid:107) − Observe that inf ∈R X + ϕ ϕ 2 = inf ∈R X 2 Z = X 2 sup ∈R Z . α t 0 α α t α,t (cid:55)→ (cid:107) t − α (cid:107) α,t It may thus be seen that α¯ is the unique global minimum of α X + ϕ ϕ 2 (cid:54) t 0 α if and only for all . Z > Z α = α¯ α¯,t α,t R Sin(cid:2)ce X is Gaussian, (Z ) ∈R is(cid:3)a continuous -indexed Gaussian process, for ∈ t α,t α ﬁxed . We have that t [0, T ] − − − (cid:104) − − (cid:105) ∀ ∈ R E (Z Z E[Z Z ])2 = 4 ϕ ϕ , Q (ϕ ϕ ) , α, β , α,t β,t α,t β,t α β t α β where is deﬁned as in Assumption 7.1. By this assumption, the above variance Q t (cid:54) (cid:55)→ is nonzero for all and . Then by [22, Lemma 2.6], has a unique α = β t > 0 α Z α,t supremum almost surely. (cid:48)(cid:48) (cid:54) It remains for us to show ii). It can be seen that this will hold if (the Z = 0 α¯,t derivative with respect to ), almost surely. Since is the unique maximum and by α α¯ (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:48)(cid:48)(cid:48)(cid:48) (cid:48)(cid:48) assumption all exist, if then it must also be the case (Z , Z , Z , Z ) Z = 0 (cid:48) (cid:48)α(cid:48)(cid:48),t α,t α,t α,t α¯,t that (this may be seen by Taylor expanding about ). The Z = Z = 0 Z α¯ α¯,t α¯,t α,t result thus follows from Lemma 7.13 below. ≥ Lemma 7.13. Under Assumption 7.2, for any t 0, the probability that there exists ∈ R an such that α (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (7.12) Z = Z = Z = 0 α,t α,t α,t is zero, where is deﬁned in (7.11). Z α,t Proof. Fix . We will show that the probability that (7.12) holds for any M > 0 ∈ − is zero. The lemma then follows directly from a covering argument. α [ M, M] ∈ { } − − For n > 0 and j 1, . . . , n , let αn = M + 2M(j 1/2)/n. Let Bn be the − j j in(cid:0)terval [αn M/n, αn + M/n]. Fix m > 0. Usin(cid:1)g the result Lemma 7.14 below, we j j ﬁnd that P (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48) (cid:88) (cid:0) ∈ − (cid:1) for some Z = Z = Z = 0, α [ M, M] α,t α,t α,t ≤ P (cid:107) (cid:107) n P (cid:48) (cid:48)(cid:48) (cid:48)(cid:48)(cid:48) ∈ (cid:107) (cid:107) ≤ ( X > m) + Z = Z = Z = 0 for some α Bn, and X m t α,t α,t α,t j t j=1 ≤ P (cid:107) (cid:107) − ( X > m) + C (m + 1)2n 1. t M 40 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P

text: (X

text: ϕ

text: E)

text: t

text: To

text: show

text: by

text: Lemma

text: it

text: suﬃces

text: to

text: show

text: that

text: i)

text: α

text: X

text: ϕ

text: ϕ

text: t

text: α

text: has

text: a

text: unique

text: global

text: minimum

text: almost

text: surely,

text: and

text: ii)

text: α¯

text: γ(α¯,

text: X

text: ϕ

text: t

text: To

text: show

text: i),

text: let

text: Y

text: X

text: ϕ

text: ϕ

text: and

text: Z

text: Y

text: ϕ

text: ϕ

text: α,t

text: t

text: α

text: α,t

text: α,t

text: α

text: Observe

text: that

text: inf

text: ∈R

text: X

text: ϕ

text: ϕ

text: inf

text: ∈R

text: X

text: Z

text: X

text: sup

text: ∈R

text: Z

text: α

text: t

text: α

text: α

text: t

text: α,t

text: t

text: α

text: α,t

text: It

text: may

text: thus

text: be

text: seen

text: that

text: α¯

text: is

text: the

text: unique

text: global

text: minimum

text: of

text: α

text: X

text: ϕ

text: ϕ

text: t

text: α

text: if

text: and

text: only

text: for

text: all

text: Z

text: Z

text: α

text: α¯

text: α¯,t

text: α,t

text: R

text: X

text: is

text: Gaussian,

text: (Z

text: ∈R

text: continuous

text: -indexed

text: Gaussian

text: process,

text: for

text: t

text: α,t

text: α

text: ﬁxed

text: We

text: have

text: that

text: t

text: T

text: R

text: E

text: (Z

text: Z

text: E[Z

text: Z

text: ϕ

text: ϕ

text: Q

text: (ϕ

text: ϕ

text: α,

text: β

text: α,t

text: β,t

text: α,t

text: β,t

text: α

text: β

text: t

text: α

text: β

text: where

text: is

text: deﬁned

text: as

text: in

text: Assumption

text: By

text: this

text: assumption,

text: the

text: above

text: variance

text: Q

text: t

text: is

text: nonzero

text: for

text: all

text: and

text: Then

text: by

text: Lemma

text: has

text: a

text: unique

text: α

text: β

text: t

text: α

text: Z

text: α,t

text: supremum

text: almost

text: surely.

text: It

text: remains

text: for

text: us

text: to

text: show

text: ii).

text: It

text: can

text: be

text: seen

text: that

text: this

text: will

text: hold

text: if

text: (the

text: Z

text: α¯,t

text: derivative

text: with

text: respect

text: to

text: almost

text: surely.

text: Since

text: is

text: the

text: unique

text: maximum

text: and

text: by

text: α

text: α¯

text: assumption

text: all

text: exist,

text: if

text: then

text: it

text: must

text: also

text: be

text: the

text: case

text: (Z

text: Z

text: Z

text: Z

text: Z

text: α,t

text: α,t

text: α,t

text: α¯,t

text: that

text: (this

text: may

text: be

text: seen

text: by

text: Taylor

text: expanding

text: about

text: The

text: Z

text: Z

text: Z

text: α¯

text: α¯,t

text: α¯,t

text: α,t

text: result

text: thus

text: follows

text: from

text: Lemma

text: below.

text: Lemma

text: Under

text: Assumption

text: for

text: any

text: t

text: the

text: probability

text: that

text: there

text: exists

text: R

text: an

text: such

text: that

text: α

text: Z

text: Z

text: Z

text: α,t

text: α,t

text: α,t

text: is

text: zero,

text: where

text: is

text: deﬁned

text: in

text: Z

text: α,t

text: Proof.

text: Fix

text: We

text: will

text: show

text: that

text: the

text: probability

text: that

text: holds

text: for

text: any

text: M

text: is

text: zero.

text: The

text: lemma

text: then

text: follows

text: directly

text: from

text: a

text: covering

text: argument.

text: α

text: M,

text: M]

text: For

text: n

text: and

text: j

text: n

text: let

text: αn

text: M

text: Let

text: Bn

text: be

text: the

text: j

text: j

text: [αn

text: M/n,

text: αn

text: M/n].

text: Fix

text: m

text: the

text: result

text: Lemma

text: below,

text: we

text: j

text: j

text: ﬁnd

text: that

text: P

text: for

text: some

text: Z

text: Z

text: Z

text: α

text: M,

text: M]

text: α,t

text: α,t

text: α,t

text: P

text: n

text: P

text: X

text: m)

text: Z

text: Z

text: Z

text: for

text: some

text: α

text: Bn,

text: and

text: X

text: m

text: t

text: α,t

text: α,t

text: α,t

text: j

text: t

text: P

text: X

text: m)

text: C

text: (m

text: t

text: M

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: It can be shown (see [27, Section 3.1] or [16]) that there exists (again under ∈ R some conditions on the parameters) a smooth function U := (uˆ, vˆ) [L2( )]2 and ∈ R speed such that is a solution to (3.4). Moreover and are both smooth c U uˆ vˆ R functions whose derivatives are all bounded and in L2( ). Thus, again by translation · ∈ R invariance we have that U := U( + α) [L2( )]2 is a solution to α AU + f(U ) = 0 α α (cid:18) (cid:19) ∈ R for all , where α − − (cid:48) − 1 1 ∀ ∈ R AU := cU − U, U [L2( )]2. θ β Once again we are thus in a speciﬁc situation of the general setup described in Section R , this time with H = [L2( )]2 and ϕ := U . Indeed, it is again straightforward to α α ∈ check that Assumptions 2.1, 2.2 and 2.3 (i) are satisﬁed (this time , so that ϕ H 0 → and we can show that Assumption 2.1 (iv) (b) holds). Since as E = H uˆ(x) 0 → ±∞ , we say that the solution is a traveling pulse. x Assumption 2.3 (ii) is again more diﬃcult to check but it is still satisﬁed in the case where the function is replaced by the Heaviside function (see again F [7, 28, 29, 32]). 4 Generalized stochastic traveling wave equation Suppose that (ϕ ) ∈R, f and A satisfy Assumptions 2.1, 2.2 and 2.3 (i) respec- α α tively. Consider the following stochastic evolution equation du = [Au + f(u )]dt + εB(t)dW Q, (4.1) t t t t where ε > 0 and (W Q) ≥ is an H-valued Q-Wiener process on the ﬁltered F {tF t} 0 P probability space (Ω, , ≥ , ) with Q a bounded, symmetric, non-negative t t 0 ∞ deﬁnite linear operator on such that . We work with the following H Tr(Q) < assumptions on the noise: Assumption 4.1. Assume that: ∞ → (i) is continuous, and there exists a constant with B : [0, ) L(H, H) C (cid:107) (cid:107) ≤ ≥ for all . B(t) C t 0 L(H,H) ≥ ∗ ≥ (ii) is a unitary operator on for all i.e. for all . B(t) H t 0 B(t) B(t) = Id t 0 We will work in the general setting, but we will keep the three examples of Section in mind. Proposition 4.2. Suppose that the (deterministic) initial condition u is such that − ∈ ∈ R 0 vα := u ϕ H for some α . Then stochastic evolution equation (4.1) has 0 0 α 7

text: It

text: can

text: be

text: shown

text: (see

text: Section

text: or

text: that

text: there

text: exists

text: (again

text: under

text: R

text: some

text: conditions

text: on

text: the

text: parameters)

text: a

text: smooth

text: function

text: U

text: (uˆ,

text: vˆ)

text: and

text: R

text: speed

text: such

text: that

text: is

text: a

text: solution

text: to

text: Moreover

text: and

text: are

text: both

text: smooth

text: c

text: U

text: uˆ

text: vˆ

text: R

text: functions

text: whose

text: derivatives

text: are

text: all

text: bounded

text: and

text: in

text: Thus,

text: again

text: by

text: translation

text: R

text: invariance

text: we

text: have

text: that

text: U

text: U(

text: α)

text: is

text: a

text: solution

text: to

text: α

text: AU

text: f(U

text: α

text: α

text: R

text: for

text: all

text: where

text: α

text: R

text: AU

text: cU

text: U,

text: U

text: θ

text: β

text: Once

text: again

text: we

text: are

text: thus

text: in

text: a

text: speciﬁc

text: situation

text: of

text: the

text: general

text: setup

text: described

text: in

text: Section

text: R

text: this

text: time

text: with

text: H

text: and

text: ϕ

text: U

text: Indeed,

text: it

text: is

text: again

text: straightforward

text: to

text: α

text: α

text: check

text: that

text: Assumptions

text: and

text: (i)

text: are

text: satisﬁed

text: (this

text: time

text: so

text: that

text: ϕ

text: H

text: and

text: we

text: can

text: show

text: that

text: Assumption

text: (iv)

text: (b)

text: holds).

text: Since

text: as

text: E

text: H

text: uˆ(x)

text: we

text: say

text: that

text: the

text: solution

text: is

text: a

text: traveling

text: pulse.

text: x

text: Assumption

text: (ii)

text: is

text: again

text: more

text: diﬃcult

text: to

text: check

text: but

text: it

text: is

text: still

text: satisﬁed

text: in

text: the

text: case

text: where

text: the

text: function

text: is

text: replaced

text: by

text: the

text: Heaviside

text: function

text: (see

text: again

text: F

text: Generalized

text: stochastic

text: traveling

text: wave

text: equation

text: Suppose

text: that

text: (ϕ

text: ∈R,

text: f

text: and

text: A

text: satisfy

text: Assumptions

text: and

text: (i)

text: respec-

text: α

text: α

text: tively.

text: Consider

text: the

text: following

text: stochastic

text: evolution

text: equation

text: du

text: [Au

text: f(u

text: )]dt

text: εB(t)dW

text: Q,

text: t

text: t

text: t

text: t

text: where

text: ε

text: and

text: (W

text: Q)

text: is

text: an

text: H-valued

text: Q-Wiener

text: process

text: on

text: the

text: ﬁltered

text: F

text: {tF

text: t}

text: P

text: probability

text: space

text: (Ω,

text: with

text: Q

text: a

text: bounded,

text: symmetric,

text: non-negative

text: t

text: t

text: deﬁnite

text: linear

text: operator

text: on

text: such

text: that

text: We

text: work

text: with

text: the

text: following

text: H

text: Tr(Q)

text: assumptions

text: on

text: the

text: noise:

text: Assumption

text: Assume

text: that:

text: (i)

text: is

text: continuous,

text: and

text: there

text: exists

text: a

text: constant

text: with

text: B

text: L(H,

text: H)

text: C

text: for

text: all

text: B(t)

text: C

text: t

text: L(H,H)

text: (ii)

text: is

text: a

text: unitary

text: operator

text: on

text: for

text: all

text: i.e.

text: for

text: all

text: B(t)

text: H

text: t

text: B(t)

text: B(t)

text: Id

text: t

text: We

text: will

text: work

text: in

text: the

text: general

text: setting,

text: but

text: we

text: will

text: keep

text: the

text: three

text: examples

text: of

text: Section

text: in

text: mind.

text: Proposition

text: Suppose

text: that

text: the

text: (deterministic)

text: initial

text: condition

text: u

text: is

text: such

text: that

text: R

text: vα

text: u

text: ϕ

text: H

text: for

text: some

text: α

text: Then

text: stochastic

text: evolution

text: equation

text: has

text: α


text: Illustration: The neural ﬁeld equation Consider again the neural ﬁeld equation (3.1) discussed in Section , but with an (cid:90) (cid:55)→ ∈ C R added continuous deterministic forcing term i.e. t g ( ) t − · − (5.3) ∂ u = u + w( y)F (u (y))dy + g , t t t t t R ≥ for . We can simulate solutions to this equation, both in the case when t 0 − g (x) = 0 and g (x) = 0.5 cos(t)e 10x2, starting from the same initial condition. The t t results are shown in Figure 1. − | | Figure 1: Simulations of the solution to (5.3) with w(x) = 10e 20 x and F (x) = 0.5[1 + − ≡ − tanh(10(x 0.25))]. On the left g 0, while on the right g (x) = 0.5 cos(t)e 10x2. t t (cid:55)→ (cid:55)→ (cid:107) − (cid:107) We can now plot the function α m(t, α) given by (5.2) i.e. α ϕ u 2 − α t where (u ) ≥ is a solution to (5.3) and g (x) = 0.5 cos(t)e 10x2 (see Figure 2). t t 0 t Figure 2 illustrates nicely the fact that the global minimum around at α = 0.5 becomes a local minimum in between and , and therefore that t = 0 t = 0.95 t = 1.0 the position of the global minimum has jumped in between these times. Moreover, at we see that the initial minimum has become a saddle point. t = 1.1 (cid:55)→ 5.1 The dynamics of local minima of α m(t, α) R The aim of this section is to derive an -valued SDE that describes the behavior (cid:55)→ of any local minimum of the function given by (5.2), up until the point α m(t, α) where it is no longer necessarily a local minimum. In order to obtain this equation, ﬁrst suppose that is a local minimum of β · ∈ R0 . The basic idea is then to look for a solution to m(0, ) β t d (cid:107) − (cid:107) − (cid:104) − (cid:48) (cid:105) u ϕ 2 = 2 u ϕ , ϕ = 0, (5.4) dβ t βt t βt βt t 10 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Illustration:

text: The

text: neural

text: ﬁeld

text: equation

text: Consider

text: again

text: the

text: neural

text: ﬁeld

text: equation

text: discussed

text: in

text: Section

text: but

text: with

text: an

text: C

text: R

text: added

text: continuous

text: deterministic

text: forcing

text: term

text: i.e.

text: t

text: g

text: t

text: u

text: u

text: w(

text: y)F

text: (u

text: (y))dy

text: g

text: t

text: t

text: t

text: t

text: t

text: R

text: for

text: We

text: can

text: simulate

text: solutions

text: to

text: this

text: equation,

text: both

text: in

text: the

text: case

text: when

text: t

text: g

text: (x)

text: and

text: g

text: (x)

text: cos(t)e

text: starting

text: from

text: the

text: same

text: initial

text: condition.

text: The

text: t

text: t

text: results

text: are

text: shown

text: in

text: Figure

text: Figure

text: Simulations

text: of

text: the

text: solution

text: to

text: with

text: w(x)

text: x

text: and

text: F

text: (x)

text: On

text: the

text: left

text: g

text: while

text: on

text: the

text: right

text: g

text: (x)

text: cos(t)e

text: t

text: t

text: We

text: can

text: now

text: plot

text: the

text: function

text: α

text: m(t,

text: α)

text: given

text: by

text: i.e.

text: α

text: ϕ

text: u

text: α

text: t

text: where

text: (u

text: is

text: a

text: solution

text: to

text: and

text: g

text: (x)

text: cos(t)e

text: (see

text: Figure

text: t

text: t

text: t

text: Figure

text: illustrates

text: nicely

text: the

text: fact

text: that

text: the

text: global

text: minimum

text: around

text: at

text: α

text: becomes

text: a

text: local

text: minimum

text: in

text: between

text: and

text: and

text: therefore

text: that

text: t

text: t

text: t

text: the

text: position

text: of

text: the

text: global

text: minimum

text: has

text: jumped

text: in

text: between

text: these

text: times.

text: Moreover,

text: at

text: we

text: see

text: that

text: the

text: initial

text: minimum

text: has

text: become

text: a

text: saddle

text: point.

text: t

text: The

text: dynamics

text: of

text: local

text: minima

text: of

text: α

text: m(t,

text: α)

text: R

text: The

text: aim

text: of

text: this

text: section

text: is

text: to

text: derive

text: an

text: -valued

text: SDE

text: that

text: describes

text: the

text: behavior

text: of

text: any

text: local

text: minimum

text: of

text: the

text: function

text: given

text: by

text: up

text: until

text: the

text: point

text: α

text: m(t,

text: α)

text: where

text: it

text: is

text: no

text: longer

text: necessarily

text: a

text: local

text: minimum.

text: In

text: order

text: to

text: obtain

text: this

text: equation,

text: ﬁrst

text: suppose

text: that

text: is

text: a

text: local

text: minimum

text: of

text: β

text: The

text: basic

text: idea

text: is

text: then

text: to

text: look

text: for

text: a

text: solution

text: to

text: β

text: t

text: d

text: u

text: ϕ

text: u

text: ϕ

text: ϕ

text: dβ

text: t

text: βt

text: t

text: βt

text: βt

text: t

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

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text: Figure 3: Nuclear suppression factor R due to collisional and radiation energy loss separately as AA functions of P . Used parameters are: τ = 0.33fm/c, T = 0.375GeV, α = 0.3, T = 0.175GeV . T 0 0 s c has been taken T = 0.16 GeV and an oﬀset 0.05 has been added too. As mentioned before c we have not considered some details of the real situation in our simulations. IV. Conclusions and remarks The ”Iterative Laplace Transform Method” has been introduced as an eﬀective method to solve the Fokker-Planck equation for initial conditions with non-zero values at the boundaries. A numerical algorithm for solving FPE is presented in this paper. Our calculations show that this method is able to solve time evolution of distribution function of HQs successfully. To verify the ability of ILTM and demonstrating its application, we calculated the nuclear suppression factor R . It is shown that there is a good agreement AA between simulation results and reported experimental data. Therefore many problems can be investigated using the ILTM. Eﬀects of a running coupling constant, HQ mass, dead cone eﬀect and LPM eﬀects, QGP viscosity, diﬀerent equations of state as models of QGP are subjects which can be studied in further works. Acknowledgements The authors are very grateful to professor Laszlo Pal Csernai, for many helpful sugges- tions and comments. S. M. P. also gratefully acknowledges warm hospitality of Institute for Physics and Technology, University of Bergen during her visit. This work is supported by the Ferdowsi university of Mashhad with grant NO. 3/27969-1392/10/15. References [1] W. Scheid, M. Muller and W. Greiner, PRL 32, 741 (1974). G. F. Chapline, M. H. Johnsou, E. Teller, M. S. Weiss PRD 8, 135 (1973). L. P. Csernai, et al., PRC 25, 2482 (1982). [2] C .Nonaka and S. A. Bass, Phys. Rev. C 75, 014902 (2007). 11 ##LTFigure## ##LTFigure##

text: Figure

text: Nuclear

label: caption

text: suppression

text: factor

text: R

text: due

text: to

text: collisional

text: and

text: radiation

text: energy

text: loss

text: separately

text: as

text: AA

text: functions

text: of

text: P

text: Used

text: parameters

text: are:

text: τ

text: T

text: α

text: T

text: T

text: s

text: c

text: has

text: been

text: taken

text: T

text: GeV

text: and

text: an

text: oﬀset

text: has

text: been

text: added

text: too.

text: As

text: mentioned

text: before

text: c

text: we

text: have

text: not

text: considered

text: some

text: details

text: of

text: the

text: real

text: situation

text: in

text: our

text: simulations.

text: IV.

text: Conclusions

text: and

text: remarks

text: The

text: ”Iterative

text: Laplace

text: Transform

text: Method”

text: has

text: been

text: introduced

text: as

text: an

text: eﬀective

text: method

text: to

text: solve

text: the

text: Fokker-Planck

text: equation

text: for

text: initial

text: conditions

text: with

text: non-zero

text: values

text: at

text: the

text: boundaries.

text: A

text: numerical

text: algorithm

text: for

text: solving

text: FPE

text: is

text: presented

text: in

text: this

text: paper.

text: Our

text: calculations

text: show

text: that

text: this

text: method

text: is

text: able

text: to

text: solve

text: time

text: evolution

text: of

text: distribution

text: function

text: of

text: HQs

text: successfully.

text: To

text: verify

text: the

text: ability

text: of

text: ILTM

text: and

text: demonstrating

text: its

text: application,

text: we

text: calculated

text: the

text: nuclear

text: suppression

text: factor

text: R

text: It

text: is

text: shown

text: that

text: there

text: is

text: a

text: good

text: agreement

text: AA

text: between

text: simulation

text: results

text: and

text: reported

text: experimental

text: data.

text: Therefore

text: many

text: problems

text: can

text: be

text: investigated

text: using

text: the

text: ILTM.

text: Eﬀects

text: of

text: a

text: running

text: coupling

text: constant,

text: HQ

text: mass,

text: dead

text: cone

text: eﬀect

text: and

text: LPM

text: eﬀects,

text: QGP

text: viscosity,

text: diﬀerent

text: equations

text: of

text: state

text: as

text: models

text: of

text: QGP

text: are

text: subjects

text: which

text: can

text: be

text: studied

text: in

text: further

text: works.

text: Acknowledgements

text: The

text: authors

text: are

text: very

text: grateful

text: to

text: professor

text: Laszlo

text: Pal

text: Csernai,

text: for

text: many

text: helpful

text: sugges-

text: tions

text: and

text: comments.

text: S.

text: M.

text: P.

text: also

text: gratefully

text: acknowledges

text: warm

text: hospitality

text: of

text: Institute

text: for

text: Physics

text: and

text: Technology,

text: University

text: of

text: Bergen

text: during

text: her

text: visit.

text: This

text: work

text: is

text: supported

text: by

text: the

text: Ferdowsi

text: university

text: of

text: Mashhad

text: with

text: grant

text: NO.

text: References

text: W.

text: Scheid,

text: M.

text: Muller

text: and

text: W.

text: Greiner,

text: PRL

text: G.

text: F.

text: Chapline,

text: M.

text: H.

text: Johnsou,

text: E.

text: Teller,

text: M.

text: S.

text: Weiss

text: PRD

text: L.

text: P.

text: Csernai,

text: et

text: al.,

text: PRC

text: C

text: .Nonaka

text: and

text: S.

text: A.

text: Bass,

text: Phys.

text: Rev.

text: C

text: ##LTFigure##

text: ##LTFigure##