text: parency. In Proc. of the IEEE Conference on Computer Vi- sion and Pattern Recognition, 2000. 2 [27] R. Wan, B. Shi, L.-Y. Duan, A.-H. Tan, and A. C. Kot. Benchmarking single-image reﬂection removal algorithms. In Proc. of the IEEE International Conference on Computer Vision, 2017. 1, 5, 7 [28] R. Wan, B. Shi, T. A. Hwee, and A. C. Kot. Depth of ﬁeld guided reﬂection removal. In IEEE International Conference on Image Processing, 2016. 2 [29] T. Xue, M. Rubinstein, C. Liu, and W. T. Freeman. A com- putational approach for obstruction-free photography. ACM Transactions on Graphics, 34(4):79, 2015. 2 [30] Q. Yan, Y. Xu, X. Yang, and T. Nguyen. Separation of weak reﬂection from a single superimposed image. IEEE Signal Processing Letters, 21(10):1173–1176, 2014. 2 [31] C.-Y. Yang, S. Liu, and M.-H. Yang. Structured face halluci- nation. In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition, 2013. 2 [32] B. Zhou, A. Lapedriza, A. Khosla, A. Oliva, and A. Torralba. Places: A 10 million image database for scene recognition. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 2017. 2 [33] J.-Y. Zhu, T. Park, P. Isola, and A. A. Efros. Unpaired image- to-image translation using cycle-consistent adversarial net- works. arXiv preprint arXiv:1703.10593, 2017. 2

text: parency.

text: In

text: Proc.

text: of

text: the

text: IEEE

text: Conference

text: on

text: Computer

text: Vi-

text: sion

text: and

text: Pattern

text: Recognition,

text: R.

text: Wan,

text: B.

text: Shi,

text: L.-Y.

text: Duan,

text: A.-H.

text: Tan,

text: and

text: A.

text: C.

text: Kot.

text: Benchmarking

text: single-image

text: reﬂection

text: removal

text: algorithms.

text: In

text: Proc.

text: of

text: the

text: IEEE

text: International

text: Conference

text: on

text: Computer

text: Vision,

text: R.

text: Wan,

text: B.

text: Shi,

text: T.

text: A.

text: Hwee,

text: and

text: A.

text: C.

text: Kot.

text: Depth

text: of

text: ﬁeld

text: guided

text: reﬂection

text: removal.

text: In

text: IEEE

text: International

text: Conference

text: on

text: Image

text: Processing,

text: T.

text: Xue,

text: M.

text: Rubinstein,

text: C.

text: Liu,

text: and

text: W.

text: T.

text: Freeman.

text: A

text: com-

text: putational

text: approach

text: for

text: obstruction-free

text: photography.

text: ACM

text: Transactions

text: on

text: Graphics,

text: Q.

text: Yan,

text: Y.

text: Xu,

text: X.

text: Yang,

text: and

text: T.

text: Nguyen.

text: Separation

text: of

text: weak

text: reﬂection

text: from

text: a

text: single

text: superimposed

text: image.

text: IEEE

text: Signal

text: Processing

text: Letters,

text: C.-Y.

text: Yang,

text: S.

text: Liu,

text: and

text: M.-H.

text: Yang.

text: Structured

text: face

text: halluci-

text: nation.

text: In

text: Proc.

text: of

text: the

text: IEEE

text: Conference

text: on

text: Computer

text: Vision

text: and

text: Pattern

text: Recognition,

text: B.

text: Zhou,

text: A.

text: Lapedriza,

text: A.

text: Khosla,

text: A.

text: Oliva,

text: and

text: A.

text: Torralba.

text: Places:

text: A

text: million

text: image

text: database

text: for

text: scene

text: recognition.

text: IEEE

text: Transactions

text: on

text: Pattern

text: Analysis

text: and

text: Machine

text: Intelli-

text: gence,

text: J.-Y.

text: Zhu,

text: T.

text: Park,

text: P.

text: Isola,

text: and

text: A.

text: A.

text: Efros.

text: Unpaired

text: image-

text: to-image

text: translation

text: using

text: cycle-consistent

text: adversarial

text: net-

text: works.

text: arXiv

text: preprint


text: COUPLINGS IN DISTANCE OF TWO BROWNIAN LP MOTIONS AND THEIR LE´VY AREA MICHEL BONNEFONT AND NICOLAS JUILLET Abstract. We study co-adapted couplings of (canonical hypoelliptic) diﬀu- sions on the (subRiemannian) Heisenberg group, that we call (Heisenberg) Brow- nian motions and are the joint laws of a planar Brownian motion with its L´evy area. We show that contrary to the situation observed on Riemannian manifolds of non-negative Ricci curvature, for any co-adapted coupling, two Heisenberg Brownian motions starting at two given points can not stay at bounded dis- ≥ tance for all time 0. Actually, we prove the stronger result that they can t ≥ not stay bounded in Lp for p 2. We also study the coupling by reﬂection, and ≤ show that it stays bounded in Lp for 0 p < 1. Finally, we explain how the results generalise to the Heisenberg groups of higher dimension 1. Introduction The ﬁrst goal of this paper is to answer a question, concerning heat diﬀusion on the Heisenberg group, that is implicitly raised by Kuwada in [15, Remark 4.4]. Let us start with some background that motivates the problem. All deﬁnitions will be given later. In the literature, ∞-Wasserstein control for a diﬀusion has been L used to deduce 1-gradient estimates of its associated semigroup (see for example L [23] and the references therein). Kuwada extends this result to -Wasserstein Lp ≥ control and -gradient estimates for all 1 with 1 + 1 = 1 and, using Lq p, q Kantorovich duality, proves that, conversely, -gradient pestimqates allow one to Lq obtain -Wasserstein control for the diﬀusion. Lp On the Heisenberg group H, the following 1-gradient bound was established L by H.Q Li [17] (see also [1]) generalising [8] ∀ ∈ C∞(H ) ∀ ≥ 0 ∀ ∈ H |∇ ( )| ≤ (|∇ |)( ) f 1 , t , a , hP f a CP hf a , c t t where 1 is constant, denotes the heat semigroup associated to half the C > P ∇ t sub-Laplacian and the horizontal gradient (see Section for the deﬁnitions). h Consequently, Kuwada’s result implies that the heat diﬀusion of the Heisenberg group possesses a ∞-Wasserstein control at any time. Therefore, for each 0 ∈ H L a, a Date: Version of May 30, 2020. 2010 Mathematics Subject Classiﬁcation. 60H10, 60J60, 60J65, 53C17, 22E25. Key words and phrases. Heisenberg group; Co-adapted coupling; Wasserstein distance; Hy- poelliptic diﬀusion. 1 ##LTLine## ##LTLine## ##LTLine##

text: COUPLINGS

label: paragraph

text: IN

text: DISTANCE

text: OF

text: TWO

text: BROWNIAN

text: LP

text: MOTIONS

text: AND

text: THEIR

text: LE´VY

text: AREA

text: MICHEL

text: BONNEFONT

text: AND

text: NICOLAS

text: JUILLET

text: Abstract.

text: We

label: abstract

text: study

text: co-adapted

text: couplings

text: of

text: (canonical

text: hypoelliptic)

text: diﬀu-

text: sions

text: on

text: the

text: (subRiemannian)

text: Heisenberg

text: group,

text: that

text: we

text: call

text: (Heisenberg)

text: Brow-

text: nian

text: motions

text: and

text: are

text: the

text: joint

text: laws

text: of

text: a

text: planar

text: Brownian

text: motion

text: with

text: its

text: L´evy

text: area.

text: We

text: show

text: that

text: contrary

text: to

text: the

text: situation

text: observed

text: on

text: Riemannian

text: manifolds

text: of

text: non-negative

text: Ricci

text: curvature,

text: for

text: any

text: co-adapted

text: coupling,

text: two

text: Heisenberg

text: Brownian

text: motions

text: starting

text: at

text: two

text: given

text: points

text: can

text: not

text: stay

text: at

text: bounded

text: dis-

text: tance

text: for

text: all

text: time

text: Actually,

text: we

text: prove

text: the

text: stronger

text: result

text: that

text: they

text: can

text: t

text: not

text: stay

text: bounded

text: in

text: Lp

text: for

text: p

text: We

text: also

text: study

text: the

text: coupling

text: by

text: reﬂection,

text: and

text: show

text: that

text: it

text: stays

text: bounded

text: in

text: Lp

text: for

text: p

text: Finally,

text: we

text: explain

text: how

text: the

text: results

text: generalise

text: to

text: the

text: Heisenberg

text: groups

text: of

text: higher

text: dimension

text: Introduction

text: The

text: ﬁrst

text: goal

text: of

text: this

text: paper

text: is

text: to

text: answer

text: a

text: question,

text: concerning

text: heat

text: diﬀusion

text: on

text: the

text: Heisenberg

text: group,

text: that

text: is

text: implicitly

text: raised

text: by

text: Kuwada

text: in

text: Remark

text: Let

text: us

text: start

text: with

text: some

text: background

text: that

text: motivates

text: the

text: problem.

text: All

text: deﬁnitions

text: will

text: be

text: given

text: later.

text: In

text: the

text: literature,

text: ∞-Wasserstein

text: control

text: for

text: a

text: diﬀusion

text: has

text: been

text: L

text: used

text: to

text: deduce

text: estimates

text: of

text: its

text: associated

text: semigroup

text: (see

text: for

text: example

text: L

text: and

text: the

text: references

text: therein).

text: Kuwada

text: extends

text: this

text: result

text: to

text: -Wasserstein

text: Lp

text: control

text: and

text: -gradient

text: estimates

text: for

text: all

text: with

text: and,

text: using

text: Lq

text: p,

text: q

text: Kantorovich

text: duality,

text: proves

text: that,

text: conversely,

text: -gradient

text: pestimqates

text: allow

text: one

text: to

text: Lq

text: obtain

text: -Wasserstein

text: control

text: for

text: the

text: diﬀusion.

text: Lp

text: On

text: the

text: Heisenberg

text: group

text: H,

text: the

text: following

text: bound

text: was

text: established

text: L

text: by

text: H.Q

text: Li

text: (see

text: also

text: generalising

text: C∞(H

text: H

text: f

text: t

text: a

text: hP

text: f

text: a

text: CP

text: hf

text: a

text: c

text: t

text: t

text: where

text: is

text: constant,

text: denotes

text: the

text: heat

text: semigroup

text: associated

text: to

text: half

text: the

text: C

text: P

text: t

text: sub-Laplacian

text: and

text: the

text: horizontal

text: gradient

text: (see

text: Section

text: for

text: the

text: deﬁnitions).

text: h

text: Consequently,

text: Kuwada’s

text: result

text: implies

text: that

text: the

text: heat

text: diﬀusion

text: of

text: the

text: Heisenberg

text: group

text: possesses

text: a

text: ∞-Wasserstein

text: control

text: at

text: any

text: time.

text: Therefore,

text: for

text: each

text: H

text: L

text: a,

text: a

text: Date:

text: Version

text: of

text: May

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Key

text: words

text: and

text: phrases.

text: Heisenberg

text: group;

text: Co-adapted

text: coupling;

text: Wasserstein

text: distance;

text: Hy-

text: poelliptic

text: diﬀusion.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 MICHEL BONNEFONT AND NICOLAS JUILLET Z t E[ 2] ≥ 2 (14) R ds R0t. 0 s Since 0, (13) and (14) provide a contradiction in (12) as goes to inﬁnity. R0 > t (cid:3) Remark 3.1. The proof of Theorem 1.1 can be improved to show that any co- adapted Heisenberg Brownian motions can not stay bounded in 4. In this proof, L the only place where we fully use the fact that is uniformly bounded almost surely R is (10). At the other places we merely need thatt E[ 2] and E[ 2] are bounded. But R Z E[ 4] ≤ 4 for every ≥ 0 is a suﬃcient assumptiotn for (10)tand, hence, for the R C t prootf. (cid:20)Z (cid:21) Indeed, by Lemma 2.8 one has E[ 4] = 4 + E t 12 2(1 − 1 1( )) + 4 2(1 − 2 2( )) R R0 R K , s R K , s ds . t 0 s s ≥ This quantity is uniformly bounded by 4 for every 0 so that (10) holds. (the C t bound in (10) can even be divided by two: 4 8 in place of 4 4). C / C / To go beyond Theorem 1.1, we conduct a precise study of the expected total variation (or length in 1) of the martingale part and of the drift part of ( ) , L Z ≥0 t t the relative L´evy area. We need (13) from the proof of Theorem 1.1, and Lemma 3.2, the statement and the proof of which are postponed at the end of the section. q Proof of Theorem 1.2. Let be sup max(E( 2) E(| |)) and as before, as- ∞ (cid:16) (cid:17) C (cid:18)Z t≥0 Rt , Zt (cid:19) sume + by contradiction. First recall C < E 2 = 2 + E t 2[(1 − 1 1) + (1 − 2 2)] ≥ 0 R R0 J , J , ds , t 0 which gives (cid:18)Z (cid:19) E +∞[(1 − 1 1) + (1 − 2 2)] ≤ C2 +∞ (15) J , J , ds < . 2 0 { ≥ } Let := inf 0 = be the hitting time of . We show that we can T t , R R0 R0 assume P( = +∞) 0t. Sup2pose for the rest of this pa2ragraph P( +∞) = 1 T > T < Z and let the ﬁnite random variable deﬁned by S = T 2(1 − 1 1) S K , ds. 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MICHEL

text: BONNEFONT

text: AND

text: NICOLAS

text: JUILLET

text: Z

label: equation

text: t

text: E[

text: R

text: ds

text: s

text: Since

text: and

text: provide

text: a

text: contradiction

text: in

text: as

text: goes

text: to

text: inﬁnity.

text: t

text: Remark

text: The

text: proof

text: of

text: Theorem

text: can

text: be

text: improved

text: to

text: show

text: that

text: any

text: co-

text: adapted

text: Heisenberg

text: Brownian

text: motions

text: can

text: not

text: stay

text: bounded

text: in

text: In

text: this

text: proof,

text: L

text: the

text: only

text: place

text: where

text: we

text: fully

text: use

text: the

text: fact

text: that

text: is

text: uniformly

text: bounded

text: almost

text: surely

text: R

text: is

text: At

text: the

text: other

text: places

text: we

text: merely

text: need

text: thatt

text: E[

text: and

text: E[

text: are

text: bounded.

text: But

text: R

text: Z

text: E[

text: for

text: every

text: is

text: a

text: suﬃcient

text: assumptiotn

text: for

text: hence,

text: for

text: the

text: R

text: C

text: t

text: prootf.

text: Indeed,

text: by

text: Lemma

text: one

text: has

text: E[

text: E

text: t

text: R

text: R

text: K

text: s

text: R

text: K

text: s

text: ds

text: t

text: s

text: s

text: This

text: quantity

text: is

text: uniformly

text: bounded

text: by

text: for

text: every

text: so

text: that

text: holds.

text: (the

text: C

text: t

text: bound

text: in

text: can

text: even

text: be

text: divided

text: by

text: two:

text: in

text: place

text: of

text: C

text: C

text: To

text: go

text: beyond

text: Theorem

text: we

text: conduct

text: a

text: precise

text: study

text: of

text: the

text: expected

text: total

text: variation

text: (or

text: length

text: in

text: of

text: the

text: martingale

text: part

text: and

text: of

text: the

text: drift

text: part

text: of

text: L

text: Z

text: ≥0

text: t

text: t

text: the

text: relative

text: L´evy

text: area.

text: We

text: need

text: from

text: the

text: proof

text: of

text: Theorem

text: and

text: Lemma

text: the

text: statement

text: and

text: the

text: proof

text: of

text: which

text: are

text: postponed

text: at

text: the

text: end

text: of

text: the

text: section.

text: q

text: Proof

text: of

text: Theorem

text: Let

text: be

text: sup

text: max(E(

text: E(|

text: and

text: as

text: before,

text: as-

text: C

text: Rt

text: Zt

text: sume

text: by

text: contradiction.

text: First

text: recall

text: C

text: E

text: E

text: t

text: R

text: J

text: J

text: ds

text: t

text: which

text: gives

text: E

text: J

text: J

text: ds

text: Let

text: inf

text: be

text: the

text: hitting

text: time

text: of

text: We

text: show

text: that

text: we

text: can

text: T

text: t

text: R

text: assume

text: P(

text: for

text: the

text: rest

text: of

text: this

text: P(

text: T

text: T

text: Z

text: and

text: let

text: the

text: ﬁnite

text: random

text: variable

text: deﬁned

text: by

text: S

text: T

text: S

text: K

text: ds.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 MICHEL BONNEFONT AND NICOLAS JUILLET We hence obtain (18) taking great enough in (19) and applying Lemma 3.2 to h − ( ) . Note that the proof of Lemma 3.2 is independent from the rest M + M ≥0 t h t h of the present proof. { ∞} Proof of (19): considering only the event = + for the martingale part of T described in Lemma 2.6, one has Z t (cid:18) (cid:19) Z h i − h i ≥ 1 R0 2 t+h 1 + K2,2 (20) M t+h M t {T=+∞} 2 2 ds. t " # Now, since by (15) it hZolds − E t+h 1 1 K2,2 d ≤ 2 4 {T=+R∞} 2 s C / , t { ≥ } { ∞} taking the complementary set of t+h 1+K2,2 ds h in T = + and using Markov inequality, one obtains: t 2 2 ! Z P 1 t+h 1 + K2,2 ≥ h ! {T=+∞} Z 2 ds 2 t − = − P 1 t+h 1 K2,2 h ≥ − C2 · 2 (21) q {T=+∞} 2 ds > 2 q 4 . h t Hence, in (20) we consider the probability that the right-hand side is greater · than ( 2)2 ( 2), which, with (21), gives the wanted estimate (19) for every R0/ h/ ≥ 5 2 . (cid:3) h C q Lemma 3.2. Let ( ) be a continuous martingale with = 0 and be in N ≥0 N0 p t t ]0 1[. Then there exists 0 such that for every positive real numbers and , , a > β h p the estimate P (h i ≥ ) ≥ N h β q p implies E[| |] ≥ N a β. h p Proof. Let be the quadratic variation of φ N h i ( ) = φ t N , t { ≥ ≥ } ∧ and consider the hitting time = inf 0 : ( ) . Set ( ) = ( ) . The τ t φ t β ψ t φ t β Dambins theorem shows that there exists a standard Brownian motion ( ) W ≥0 ≥ t t such that for every 0, t ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MICHEL

text: BONNEFONT

text: AND

text: NICOLAS

text: JUILLET

text: We

text: hence

text: obtain

text: taking

text: great

text: enough

text: in

text: and

text: applying

text: Lemma

text: to

text: h

text: Note

text: that

text: the

text: proof

text: of

text: Lemma

text: is

text: independent

text: from

text: the

text: rest

text: M

text: M

text: ≥0

text: t

text: h

text: t

text: h

text: of

text: the

text: present

text: proof.

text: Proof

text: of

text: considering

text: only

text: the

text: event

text: for

text: the

text: martingale

text: part

text: of

text: T

text: described

text: in

text: Lemma

text: one

text: has

text: Z

text: t

text: Z

text: h

text: i

text: h

text: i

text: t+h

text: M

text: t+h

text: M

text: t

text: {T=+∞}

text: ds.

text: t

text: Now,

text: since

text: by

text: it

text: hZolds

text: E

text: t+h

text: d

text: {T=+R∞}

text: s

text: C

text: t

text: taking

text: the

text: complementary

text: set

text: of

text: t+h

text: ds

text: h

text: in

text: T

text: and

text: using

text: Markov

text: inequality,

text: one

text: obtains:

text: t

text: Z

text: P

text: t+h

text: h

text: {T=+∞}

text: Z

text: ds

text: t

text: P

text: t+h

text: h

text: q

text: {T=+∞}

text: ds

text: q

text: h

text: t

text: Hence,

text: in

text: we

text: consider

text: the

text: probability

text: that

text: the

text: right-hand

text: side

text: is

text: greater

text: than

text: which,

text: with

text: gives

text: the

text: wanted

text: estimate

text: for

text: every

text: h/

text: h

text: C

text: q

text: Lemma

text: Let

text: be

text: a

text: continuous

text: martingale

text: with

text: and

text: be

text: in

text: N

text: ≥0

text: p

text: t

text: t

text: Then

text: there

text: exists

text: such

text: that

text: for

text: every

text: positive

text: real

text: numbers

text: and

text: a

text: β

text: h

text: p

text: the

text: estimate

text: P

text: (h

text: i

text: N

text: h

text: β

text: q

text: p

text: implies

text: E[|

text: N

text: a

text: β.

text: h

text: p

text: Proof.

text: Let

text: be

text: the

text: quadratic

text: variation

text: of

text: φ

text: N

text: h

text: i

text: φ

text: t

text: N

text: t

text: and

text: consider

text: the

text: hitting

text: time

text: inf

text: Set

text: The

text: τ

text: t

text: φ

text: t

text: β

text: ψ

text: t

text: φ

text: t

text: β

text: Dambins

text: theorem

text: shows

text: that

text: there

text: exists

text: a

text: standard

text: Brownian

text: motion

text: W

text: ≥0

text: t

text: t

text: such

text: that

text: for

text: every

text: t

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: Lp COUPLING OF TWO BROWNIAN MOTIONS AND THEIR LE´VY AREA 17 h i E [| |] ≥ E [| |] = E | | (22) N N ∧ W ( ) . t t τ ψ t Let now be the event { ∈ Ω : ( ) ≥ } and recall the assumption P( ) ≥ . A ω φ h β A p One has h i h i q E | | ≥ E | | · 1 = E [| | · 1 ] ≥ (23) W ( ) W ( ) W a β ψ h ψ h A β A p where the constant h is given by i Z a p = E | | 1 = Φ−1(1+2p) | |e√−x2/2 d ap G {|G|≤Φ−1(1+2p)} Φ−1(1−p) x 2π x. 2 with a standard normal random variable and Φ its cumulative distribution G function. The lower bound in (23) is obtained for P( ) = and the normal random A p variable of variance concentrated as much as possible close to zero on event W β β (cid:3) . Equation (22) for = and (23) ﬁnally provide the wanted estimate. A t h Remark 3.3. The major constraint for generalising Theorem 1.2 and its proof to a -Wasserstein control for 2 is that E(| |) is replaced by E(| | 2). Here Lp p < Z Z p/ 7→ | | t t 2 is not convex when 2 and Jensen’s inequality does not apply. x x p/ p < 4. Coupling by reflection In this section, we study precisely the coupling by reﬂection. We recall that (B ) and (B0) are two Heisenberg Brownian motions coupled by reﬂection ≥0 ≥0 if atntd only if (t t) and ( 0) are two Brownian motions on R2 coupled by B ≥0 B ≥0 reﬂection. This mtetans that thte tcoupling matrices are given by 1 1 = −1, 2 2 = K , K , ≥ 1, 1 2 = 2 1 = 0 for and by the matrix = Id for where K , { ≥K , } t < τ Z J 2 t τ = inf 0 : = 0 is the hitting time of 0 for ( ) . We recall τ s R R ≥0 s t t ∧ = + 2 and = + t τ( + 2 ) ˜ R R0 C ∧ Z Z0 R0 C ∧ dC t t τ t s τ s 0 ˜ where ( ) and ( ) are two independent Brownian motions (starting in 0) and C C s s { ≥ s s − } with = inf 0 : = 2 . τ s C R0/ s For simplicity, in the following we only consider the case 0 and = 0. R0 > Z0 Proposition 4.1. With the above notation, assume 0 and = 0, Let 0, R0 > Z0 p > then there exists some constants 0 00 0 such that C , C , C > p p p E[ ] = R R0 t ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lp

text: COUPLING

text: OF

text: TWO

text: BROWNIAN

text: MOTIONS

text: AND

text: THEIR

text: LE´VY

text: AREA

text: h

text: i

text: E

text: E

text: E

text: N

text: N

text: W

text: t

text: t

text: τ

text: ψ

text: t

text: Let

text: now

text: be

text: the

text: event

text: Ω

text: and

text: recall

text: the

text: assumption

text: P(

text: A

text: ω

text: φ

text: h

text: β

text: A

text: p

text: One

text: has

text: h

text: i

text: h

text: i

text: q

text: E

text: E

text: E

text: W

text: W

text: W

text: a

text: β

text: ψ

text: h

text: ψ

text: h

text: A

text: β

text: A

text: p

text: where

text: the

text: constant

text: h

text: is

text: given

text: by

text: i

text: Z

text: a

text: p

text: E

text: d

text: ap

text: G

text: x

text: 2π

text: x.

text: with

text: a

text: standard

text: normal

text: random

text: variable

text: and

text: Φ

text: its

text: cumulative

text: distribution

text: G

text: function.

text: The

text: lower

text: bound

text: in

text: is

text: obtained

text: for

text: P(

text: and

text: the

text: normal

text: random

text: A

text: p

text: variable

text: of

text: variance

text: concentrated

text: as

text: much

text: as

text: possible

text: close

text: to

text: zero

text: on

text: event

text: W

text: β

text: β

text: Equation

text: for

text: and

text: ﬁnally

text: provide

text: the

text: wanted

text: estimate.

text: A

text: t

text: h

text: Remark

text: The

text: major

text: constraint

text: for

text: generalising

text: Theorem

text: and

text: its

text: proof

text: to

text: a

text: -Wasserstein

text: control

text: for

text: is

text: that

text: E(|

text: is

text: replaced

text: by

text: E(|

text: Here

text: Lp

text: p

text: Z

text: Z

text: p/

text: 7→

text: t

text: t

text: is

text: not

text: convex

text: when

text: and

text: Jensen’s

text: inequality

text: does

text: not

text: apply.

text: x

text: x

text: p/

text: p

text: Coupling

text: by

text: reflection

text: In

text: this

text: section,

text: we

text: study

text: precisely

text: the

text: coupling

text: by

text: reﬂection.

text: We

text: recall

text: that

text: (B

text: and

text: are

text: two

text: Heisenberg

text: Brownian

text: motions

text: coupled

text: by

text: reﬂection

text: ≥0

text: ≥0

text: if

text: atntd

text: only

text: if

text: (t

text: t)

text: and

text: are

text: two

text: Brownian

text: motions

text: on

text: coupled

text: by

text: B

text: ≥0

text: B

text: ≥0

text: reﬂection.

text: This

text: mtetans

text: that

text: thte

text: tcoupling

text: matrices

text: are

text: given

text: by

text: K

text: K

text: for

text: and

text: by

text: the

text: matrix

text: Id

text: for

text: where

text: K

text: ≥K

text: t

text: τ

text: Z

text: J

text: t

text: τ

text: inf

text: is

text: the

text: hitting

text: time

text: of

text: for

text: We

text: recall

text: τ

text: s

text: R

text: R

text: ≥0

text: s

text: t

text: t

text: and

text: t

text: τ(

text: R

text: C

text: Z

text: C

text: dC

text: t

text: t

text: τ

text: t

text: s

text: τ

text: s

text: where

text: and

text: are

text: two

text: independent

text: Brownian

text: motions

text: (starting

text: in

text: and

text: C

text: C

text: s

text: s

text: s

text: s

text: with

text: inf

text: τ

text: s

text: C

text: s

text: For

text: simplicity,

text: in

text: the

text: following

text: we

text: only

text: consider

text: the

text: case

text: and

text: Proposition

text: With

text: the

text: above

text: notation,

text: assume

text: and

text: Let

text: p

text: then

text: there

text: exists

text: some

text: constants

text: such

text: that

text: C

text: C

text: C

text: p

text: p

text: p

text: E[

text: R

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Lp COUPLING OF TWO BROWNIAN MOTIONS AND THEIR LE´VY AREA 19 E(| | | = ) = E( | (cid:18)(Z )| 2| =(cid:19) ) ! (26) W ( ) p τ u c T t p/ τ u T t p 2 = c E "u(cid:18)RZ2ds p/ (cid:19)|τ = u # p 0 s = 2 E 1 ˜2 p/2 | = c pup R dλ τ u p 0 λ ˜ where we have introduced the normalised process ( ) (deﬁned almost R ∈[0 1] λ λ , surely, since the hitting time is almost surely ﬁnite) in such a way it hits 0 at τ time 1: 1 ˜ √ ∈ = [0 1] R R , λ , . λ 2 τλ τ ˜ Note that R0 = 2R√0 . τ ˜ It is then well-known that, conditioned on = , the entire process ( ) τ u R ∈[0 1] → ∞ λ λ , converges in law when to a normal positive Brownian excursion ( ) . u X ∈[0 1] → ∞ s s , Moreover, as proven in Lemma 4.3, when , u "(cid:18)Z (cid:19) # "(cid:18)Z (cid:19) # E 1 ˜2 p/2 | = → E 1 2 p/2 (27) R dλ τ u X ds . 0 λ 0 s Finally with (26) denoting the limit in (27) by , the ﬁrst integral in (25) E p satisﬁes the following equivalence: Z Z ( ) = t E(| | | = ) ( ) ∼ 2 t ( ) h1 t W ( ) p τ u f u du →∞ p c E upf u du T t τ t p p τ 0 0 From the density estimate of f in (24) we have upf (u) ∼+∞ √R0 up−3/2. There- τ τ 2 2 fore: π •• If p > 1/2, the function h1(t) is equivalent to (2pp−−13//22)R√0πcupE3/p2 tp−1/2 at +∞, if p = 1/2, it is equivalent to R√0cpEp ln t, • 2 π u3/2 if 0 1 2, it converges to a positive constant. < p < / ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lp

text: COUPLING

text: OF

text: TWO

text: BROWNIAN

text: MOTIONS

text: AND

text: THEIR

text: LE´VY

text: AREA

text: E(|

text: E(

text: W

text: p

text: τ

text: u

text: c

text: T

text: t

text: p/

text: τ

text: u

text: T

text: t

text: p

text: c

text: E

text: p/

text: u

text: p

text: s

text: E

text: ˜2

text: c

text: pup

text: R

text: dλ

text: τ

text: u

text: p

text: λ

text: where

text: we

text: have

text: introduced

text: the

text: normalised

text: process

text: (deﬁned

text: almost

text: R

text: λ

text: λ

text: surely,

text: since

text: the

text: hitting

text: time

text: is

text: almost

text: surely

text: ﬁnite)

text: in

text: such

text: a

text: way

text: it

text: hits

text: at

text: τ

text: time

text: R

text: R

text: λ

text: λ

text: τλ

text: τ

text: Note

text: that

text: τ

text: It

text: is

text: then

text: well-known

text: that,

text: conditioned

text: on

text: the

text: entire

text: process

text: τ

text: u

text: R

text: λ

text: λ

text: converges

text: in

text: law

text: when

text: to

text: a

text: normal

text: positive

text: Brownian

text: excursion

text: u

text: X

text: s

text: s

text: Moreover,

text: as

text: proven

text: in

text: Lemma

text: when

text: u

text: E

text: ˜2

text: E

text: R

text: dλ

text: τ

text: u

text: X

text: ds

text: λ

text: s

text: Finally

text: with

text: denoting

text: the

text: limit

text: in

text: by

text: the

text: ﬁrst

text: integral

text: in

text: E

text: p

text: satisﬁes

text: the

text: following

text: equivalence:

text: Z

text: Z

text: t

text: E(|

text: t

text: t

text: W

text: p

text: τ

text: u

text: f

text: u

text: du

text: p

text: c

text: E

text: upf

text: u

text: du

text: T

text: t

text: τ

text: t

text: p

text: p

text: τ

text: From

text: the

text: density

text: estimate

text: of

text: f

text: in

text: we

text: have

text: upf

text: (u)

text: There-

text: τ

text: τ

text: fore:

text: π

text: If

text: p

text: the

text: function

text: is

text: equivalent

text: to

text: at

text: if

text: p

text: it

text: is

text: equivalent

text: to

text: ln

text: t,

text: π

text: if

text: it

text: converges

text: to

text: a

text: positive

text: constant.

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Lp COUPLING OF TWO BROWNIAN MOTIONS AND THEIR LE´VY AREA 9 Proof. The ﬁrst two relations follow from the fact that is an orthogonal matrix. Q − For the last relation, one can note that 1 2 2 1 = tr( ) = tr( ) with J , J , JM KQT MQ the matrix ( 0 −1 ). Now a computation gives = (det ) and the last M QT MQ Q M 1 0 (cid:3) relation follows from the fact is actually a rotation matrix. Q The stochastic processes 2 and are semimartingales deﬁned for all time R Z ≥ 0. In the next statement, wt e providt e stochastic diﬀerential equations for their t evolution. 6 Lemma 2.6. With the above notation, when = 0, the processes 2 and solve the stochastic diﬀerentiqal equation: (cid:16) Rt Rt (cid:17) Zt d(Rt2) = 2Rtq2(1 − K1,1) dCt + 2(1 − K1,1) + 2(1 − K2,2) dt 1 dZ = Rt 2(1 + K2,2) dC˜ + (K1,2 − K2,1)dt t 2 t 2 where ( ) and ( ˜ ) are some 1-dimensional Brownian motions whose co- C ≥0 C ≥0 t t t t variation satisqﬁes: q h 2(1 − K1,1)dC , 2(1 + K2,2)dC˜ i = (K1,2 − K2,1)dt. (7) t t Remark 2.7. Actually the stochastic process ( 2 ) is perfectly deﬁned for all R , Z ≥0 ≥ 0 (even when = 0). The technical problemt intLtemma 2.6 is that the matrix t R t 6 and thus the matrix are only deﬁned for = 0. However, the matrix is Q K R J t t t t deﬁned for every value of and we have:  R (cid:16) (cid:17) t  ( 2) = ( 0 ) + 2(1 − 1 1) + 2(1 − 2 2) d R σ B , B , J dC J , J , dt t R t t t t 1 = ( 0 ) ˜ + ( 1 2 − 2 1) dZ σ B , B , J dC J , J , dt t Z t t t t 2 ( where and are deﬁned by: σ σ q R Z 0 if = 0 σR(Bt, Bt0, Jt) = ( 2R 2(1 − (P T J P )1,1) if BBt 6= BBt0 t t t t t t and q 0 if = 0 σZ(Bt, Bt0, Jt) = R2t 2(1 + (PtT JtPt)2,2) if BBtt 6= BBt0t. Note ﬁnally that the fact that and vanish for = 0 is rather clear from σ σ R R Z t their expressions in Lemma 2.6. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lp

text: COUPLING

text: OF

text: TWO

text: BROWNIAN

text: MOTIONS

text: AND

text: THEIR

text: LE´VY

text: AREA

text: Proof.

text: The

text: ﬁrst

text: two

text: relations

text: follow

text: from

text: the

text: fact

text: that

text: is

text: an

text: orthogonal

text: matrix.

text: Q

text: For

text: the

text: last

text: relation,

text: one

text: can

text: note

text: that

text: tr(

text: tr(

text: with

text: J

text: J

text: JM

text: KQT

text: MQ

text: the

text: matrix

text: −1

text: Now

text: a

text: computation

text: gives

text: (det

text: and

text: the

text: last

text: M

text: QT

text: MQ

text: Q

text: M

text: relation

text: follows

text: from

text: the

text: fact

text: is

text: actually

text: a

text: rotation

text: matrix.

text: Q

text: The

text: stochastic

text: processes

text: and

text: are

text: semimartingales

text: deﬁned

text: for

text: all

text: time

text: R

text: Z

text: In

text: the

text: next

text: statement,

text: wt

text: e

text: providt

text: e

text: stochastic

text: diﬀerential

text: equations

text: for

text: their

text: t

text: evolution.

text: Lemma

text: With

text: the

text: above

text: notation,

text: when

text: the

text: processes

text: and

text: solve

text: the

text: stochastic

text: diﬀerentiqal

text: equation:

text: Rt

text: Rt

text: Zt

text: dCt

text: dt

text: dZ

text: Rt

text: dC˜

text: t

text: t

text: where

text: and

text: are

text: some

text: Brownian

text: motions

text: whose

text: co-

text: C

text: ≥0

text: C

text: ≥0

text: t

text: t

text: t

text: t

text: variation

text: satisqﬁes:

text: q

text: h

text: i

text: t

text: t

text: Remark

text: Actually

text: the

text: stochastic

text: process

text: is

text: perfectly

text: deﬁned

text: for

text: all

text: R

text: Z

text: ≥0

text: (even

text: when

text: The

text: technical

text: problemt

text: intLtemma

text: is

text: that

text: the

text: matrix

text: t

text: R

text: t

text: and

text: thus

text: the

text: matrix

text: are

text: only

text: deﬁned

text: for

text: However,

text: the

text: matrix

text: is

text: Q

text: K

text: R

text: J

text: t

text: t

text: t

text: t

text: deﬁned

text: for

text: every

text: value

text: of

text: and

text: we

text: have:

text: R

text: t

text: d

text: R

text: σ

text: B

text: B

text: J

text: dC

text: J

text: J

text: dt

text: t

text: R

text: t

text: t

text: t

text: t

text: dZ

text: σ

text: B

text: B

text: J

text: dC

text: J

text: J

text: dt

text: t

text: Z

text: t

text: t

text: t

text: t

text: where

text: and

text: are

text: deﬁned

text: by:

text: σ

text: σ

text: q

text: R

text: Z

text: if

text: σR(Bt,

text: Jt)

text: (P

text: T

text: J

text: P

text: if

text: BBt

text: t

text: t

text: t

text: t

text: t

text: t

text: and

text: q

text: if

text: σZ(Bt,

text: Jt)

text: (PtT

text: if

text: BBtt

text: Note

text: ﬁnally

text: that

text: the

text: fact

text: that

text: and

text: vanish

text: for

text: is

text: rather

text: clear

text: from

text: σ

text: σ

text: R

text: R

text: Z

text: t

text: their

text: expressions

text: in

text: Lemma

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: { } where yˆ 1, S(x ) is the prediction for Y based on S(x ) but excluding the explanatory i i i i variables d(x ) from the model. We interpret (5) as the fraction of explained variation in the i response Y by the explanatory variables d but unexplained by the spatial random eﬀects S. In the next example, we compute (3) and (5) by plugging-in the maximum likelihood estimates of the regression coeﬃcients. These are obtained using the Monte Carlo likelihood method (Christensen, 2004) implemented in the R package PrevMap (Giorgi & Diggle, 2017). We | simulate from S (y, d) using a Laplace sampling technique described in detail in Section 2.1 of Giorgi & Diggle (2017). 3 Example: River-blindness mapping in Liberia River-blindness is an infectious disease caused by the parasite Onchocerca volvulus and is transmitted by a black ﬂy of the genus Simulium. We analyse data from 90 communities in Liberia, where people were tested by palpation for the presence of skin nodules caused by the disease; for an Africa-wide analysis of these data, see Zour´e et al. (2014). Let x be the location of the i-th sampled community, where y out of n randomly selected i i i individuals tested positive. Our model for the data is a GLGM, where the Y conditionally on i S(x ) are mutually independent Binomial variables with number of trials n and probability i i (cid:26) (cid:27) of having skin nodules p(x ), such that i p(x ) log − i = β + β x + β x + S(x ), (6) 0 1 i,1 2 i,2 i 1 p(x ) i where x and x are the abscissa and ordinate components of the geographical location x . i,1 i,2 i The reason for using a linear trend in x is shown in Figure 1, where the map of the empirical i nodule prevalence shows an increase in the values as we move further from the coast in the north-east direction. Finally, we model S(x) as a zero-mean Gaussian process with isotropic exponential covariance function having variance σ2 and scale parameter φ. The maximum likelihood estimates of the model parameters and their 95% conﬁdence intervals are reported in Table 1. We observe that the use of the explanatory variables leads to a remarkable reduction in the values of the estimated σ2 and φ. The ﬁtted GLGM explains about 59% of the variation in nodule prevalence compared to 27% from a classical GLM where S(x) = 0 for all x. However, the small value of 1% for the coeﬃcient of partial determination, R˜2, indicates that the point estimates from the GLGM with covariates, given by (6), are strongly similar to a model without covariates, where β = β = 0. Nonetheless, Figure 2 1 2 shows that the standard errors for the estimated nodule prevalence (computed using Monte | Carlo samples from S (y, d) while pugging-in the Monte Carlo maximum likelihood estimates) at the observed locations from the model with covariates are smaller almost everywhere than those from the model with only the intercept. More precisely, the largest relative reduction 4 ##LTLine##

text: where

text: yˆ

text: S(x

text: is

text: the

text: prediction

text: for

text: Y

text: based

text: on

text: S(x

text: but

text: excluding

text: the

text: explanatory

text: i

text: i

text: i

text: i

text: variables

text: d(x

text: from

text: the

text: model.

text: We

text: interpret

text: as

text: the

text: fraction

text: of

text: explained

text: variation

text: in

text: the

text: i

text: response

text: Y

text: by

text: the

text: explanatory

text: variables

text: d

text: but

text: unexplained

text: by

text: the

text: spatial

text: random

text: eﬀects

text: S.

text: In

text: the

text: next

text: example,

text: we

text: compute

text: and

text: by

text: plugging-in

text: the

text: maximum

text: likelihood

text: estimates

text: of

text: the

text: regression

text: coeﬃcients.

text: These

text: are

text: obtained

text: using

text: the

text: Monte

text: Carlo

text: likelihood

text: method

text: (Christensen,

text: implemented

text: in

text: the

text: R

text: package

text: PrevMap

text: (Giorgi

text: Diggle,

text: We

text: simulate

text: from

text: S

text: (y,

text: d)

text: using

text: a

text: Laplace

text: sampling

text: technique

text: described

text: in

text: detail

text: in

text: Section

text: of

text: Giorgi

text: Diggle

text: Example:

text: River-blindness

text: mapping

text: in

text: Liberia

text: River-blindness

text: is

text: an

text: infectious

text: disease

text: caused

text: by

text: the

text: parasite

text: Onchocerca

text: volvulus

text: and

text: is

text: transmitted

text: by

text: a

text: black

text: ﬂy

text: of

text: the

text: genus

text: Simulium.

text: We

text: analyse

text: data

text: from

text: communities

text: in

text: Liberia,

text: where

text: people

text: were

text: tested

text: by

text: palpation

text: for

text: the

text: presence

text: of

text: skin

text: nodules

text: caused

text: by

text: the

text: disease;

text: for

text: an

text: Africa-wide

text: analysis

text: of

text: these

text: data,

text: see

text: Zour´e

text: et

text: al.

text: Let

text: x

text: be

text: the

text: location

text: of

text: the

text: i-th

text: sampled

text: community,

text: where

text: y

text: out

text: of

text: n

text: randomly

text: selected

text: i

text: i

text: i

text: individuals

text: tested

text: positive.

text: Our

text: model

text: for

text: the

text: data

text: is

text: a

text: GLGM,

text: where

text: the

text: Y

text: conditionally

text: on

text: i

text: S(x

text: are

text: mutually

text: independent

text: Binomial

text: variables

text: with

text: number

text: of

text: trials

text: n

text: and

text: probability

text: i

text: i

text: of

text: having

text: skin

text: nodules

text: p(x

text: such

text: that

text: i

text: p(x

text: log

text: i

text: β

text: β

text: x

text: β

text: x

text: S(x

text: i

text: p(x

text: i

text: where

text: x

text: and

text: x

text: are

text: the

text: abscissa

text: and

text: ordinate

text: components

text: of

text: the

text: geographical

text: location

text: x

text: i

text: The

text: reason

text: for

text: using

text: a

text: linear

text: trend

text: in

text: x

text: is

text: shown

text: in

text: Figure

text: where

text: the

text: map

text: of

text: the

text: empirical

text: i

text: nodule

text: prevalence

text: shows

text: an

text: increase

text: in

text: the

text: values

text: as

text: we

text: move

text: further

text: from

text: the

text: coast

text: in

text: the

text: north-east

text: direction.

text: Finally,

text: we

text: model

text: S(x)

text: as

text: a

text: zero-mean

text: Gaussian

text: process

text: with

text: isotropic

text: exponential

text: covariance

text: function

text: having

text: variance

text: σ2

text: and

text: scale

text: parameter

text: φ.

text: The

text: maximum

text: likelihood

text: estimates

text: of

text: the

text: model

text: parameters

text: and

text: their

text: conﬁdence

text: intervals

text: are

text: reported

text: in

text: Table

text: We

text: observe

text: that

text: the

text: use

text: of

text: the

text: explanatory

text: variables

text: leads

text: to

text: a

text: remarkable

text: reduction

text: in

text: the

text: values

text: of

text: the

text: estimated

text: σ2

text: and

text: φ.

text: The

text: ﬁtted

text: GLGM

text: explains

text: about

text: of

text: the

text: variation

text: in

text: nodule

text: prevalence

text: compared

text: to

text: from

text: a

text: classical

text: GLM

text: where

text: S(x)

text: for

text: all

text: x.

text: However,

text: the

text: small

text: value

text: of

text: for

text: the

text: coeﬃcient

text: of

text: partial

text: determination,

text: indicates

text: that

text: the

text: point

text: estimates

text: from

text: the

text: GLGM

text: with

text: covariates,

text: given

text: by

text: are

text: strongly

text: similar

text: to

text: a

text: model

text: without

text: covariates,

text: where

text: β

text: β

text: Nonetheless,

text: Figure

text: shows

text: that

text: the

text: standard

text: errors

text: for

text: the

text: estimated

text: nodule

text: prevalence

text: (computed

text: using

text: Monte

text: Carlo

text: samples

text: from

text: S

text: (y,

text: d)

text: while

text: pugging-in

text: the

text: Monte

text: Carlo

text: maximum

text: likelihood

text: estimates)

text: at

text: the

text: observed

text: locations

text: from

text: the

text: model

text: with

text: covariates

text: are

text: smaller

text: almost

text: everywhere

text: than

text: those

text: from

text: the

text: model

text: with

text: only

text: the

text: intercept.

text: More

text: precisely,

text: the

text: largest

text: relative

text: reduction

text: ##LTLine##


text: l l l l l l ll l l l l l l l l ll lll lll llllllll lllll llll ll l ll lllll l lll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 3e+05 4e+05 5e+05 6e+05 X Coord Figure 1: Map of the empirical nodule prevalence. The radius of each point is proportional the quintile class within which the associated prevalence falls, with larger radiuses corresponding to higher quintile clasees. Prevalence standard errors l l ll l l l ll l ll ll lll llllllllllllllllllllllllllll lllllllllll l lllllllll l l l l l ll l l l l llll lll ll ll l l 0.020 0.025 0.030 0.035 0.040 0.045 GLGM with covariates Figure 2: Standard errors for the estimated nodule prevalence from a Binomial geostatistical model without covariates against one with covariates as deﬁned in Section . The solid line is the identity line. 5 setairavoc tuohtiw MGLG drooC Y 540.0 040.0 530.0 030.0 520.0 020.0 50+e9 50+e8 50+e7 50+e6 50+e5 ##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##

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font: default

text: l

text: l

text: l

text: l

text: l

text: ll

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: ll

text: lll

text: lll

text: llllllll

text: lllll

text: llll

text: ll

text: l

text: ll

text: lllll

text: l

text: lll

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: l

text: X

text: Coord

text: Figure

text: Map

text: of

text: the

text: empirical

text: nodule

text: prevalence.

text: The

text: radius

text: of

text: each

text: point

text: is

text: proportional

text: the

text: quintile

text: class

text: within

text: which

text: the

text: associated

text: prevalence

text: falls,

text: with

text: larger

text: radiuses

text: corresponding

text: to

text: higher

text: quintile

text: clasees.

text: Prevalence

text: standard

text: errors

text: l

text: l

text: ll

text: l

text: l

text: l

text: ll

text: l

text: ll

text: ll

text: lll

text: llllllllllllllllllllllllllll

text: lllllllllll

text: l

text: lllllllll

text: l

text: l

text: l

text: l

text: l

text: ll

text: l

text: l

text: l

text: l

text: llll

text: lll

text: ll

text: ll

text: l

text: l

text: GLGM

text: with

text: covariates

text: Figure

text: Standard

text: errors

text: for

text: the

text: estimated

text: nodule

text: prevalence

text: from

text: a

text: Binomial

text: geostatistical

text: model

text: without

text: covariates

text: against

text: one

text: with

text: covariates

text: as

text: deﬁned

text: in

text: Section

text: The

text: solid

text: line

text: is

text: the

text: identity

text: line.

text: setairavoc

text: tuohtiw

text: MGLG

text: drooC

text: Y

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: X → (cid:0) (cid:1) (cid:0) (cid:1) and the term D (w) satisﬁes the expansion, as τ t, t,x − X (τ t)2 − D (w) (τ) = D v t, x, P (t) + o(τ t)2 . (4.10) t,x P 2 X X Proof. Since t and x are kept ﬁxed throughout this proof, we write (w) for (w). Recall ∈ (cid:90) t,x (cid:0) (cid:1) that, for τ [t, t + ∆t], X τ X − (w)(τ) = x + v t, (w)(s), P (t) + (s t)w ds . t Fix a direction δ ∈ RN \ {0}. F(cid:13)(cid:13)irst we show the boundedne(cid:13)(cid:13)ss of the diﬀerence quotient w X − X (w + εδ )(τ) (w)(τ) w . ε (cid:13) ∈ (cid:13) For(cid:13)τ [t, t + ∆t], we have (cid:13) (cid:90) (cid:13) (cid:13) 1 X (cid:13) (cid:0) − X (cid:1) (cid:0) (cid:1)(cid:13) (w(cid:13) + εδw)(τ) (w)(τ) (cid:13) ε ≤1 (cid:90) τ(cid:13)(cid:13) v(cid:0) t, X (w + εδ )(s), P (t) + (s − t)(w + εδ (cid:1)) −(cid:0)v t, X (w)(s), P (t) + (s − t)w ds (cid:1)(cid:13)(cid:13) (cid:13) w w (cid:13) ε t ≤1 τ(cid:90) v (cid:13)(cid:13)t, X(cid:0) (w + εδ )(s),P (t)+(s − t)(w + εδ (cid:1)) −v(cid:0) t, X (w)(s),P (t)+(s − t)(w(cid:1)(cid:13)(cid:13)+ εδ ) ds (cid:13) w w (cid:13) w ε t 1 τ X (cid:34)(cid:90) (cid:13) − − X(cid:13) (cid:90) − (cid:35) (cid:13) (cid:13) + v t, (w)(s), P (t) + (s t)(w + εδ ) v t, (w)(s), P (t) + (s t)w ds w ε t X − X ≤(cid:107) (cid:107) (cid:34)(cid:90) τ (cid:13) (w + εδw)(s) (w)(s)(cid:13) τ − (cid:107)(cid:35) (cid:107) v C1([0,T]×RN×RN;RN) (cid:13) ε (cid:13) ds + (s t) δw ds t t X − X ≤(cid:107) (cid:107) τ (w + εδ )(s) (w)(s) (cid:107) (cid:107) v C1([0,T]×RN×RN;RN) w ε ds + (∆t)2 δw . t (cid:13) (cid:13) (cid:13) (cid:13) Hence an application of Gr¨onwall Lemma, see, e.g., [2, Chapter 3, Lemma 3.1] ensures that X − X (w + εδ )(τ) (w)(τ) ≤ (cid:107) (cid:107) w K (∆t)3 δ exp (K ∆t) , (4.11) 1 w 1 ε (cid:107) (cid:107) (cid:13) (cid:13) where K1 = v C1([0,T]×RN×RN;RN(cid:13)). Consequently (cid:13) X − X lim sup (w + εδ )(τ) (w)(τ) = 0 . (4.12) ε→0 τ∈[t,t+∆t] w X ∈ R \{ } We now prove the existence of directional derivatives of along the direction δ N 0 . → w Calling τ Y (τ) the solution to the Cauchy problem (4.9), we have X − X (w + εδ )(τ) (w)(τ) − w Y (τ) δ w ε 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: and

text: the

text: term

text: D

text: (w)

text: satisﬁes

text: the

text: expansion,

text: as

text: τ

text: t,

text: t,x

text: X

text: (τ

text: D

text: (w)

text: (τ)

text: D

text: v

text: t,

text: x,

text: P

text: (t)

text: o(τ

text: t,x

text: P

text: X

text: X

text: Proof.

text: Since

text: t

text: and

text: x

text: are

text: kept

text: ﬁxed

text: throughout

text: this

text: proof,

text: we

text: write

text: (w)

text: for

text: (w).

text: Recall

text: t,x

text: that,

text: for

text: τ

text: [t,

text: t

text: ∆t],

text: X

text: τ

text: X

text: (w)(τ)

text: x

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: t

text: Fix

text: a

text: direction

text: δ

text: RN

text: we

text: show

text: the

text: of

text: the

text: diﬀerence

text: quotient

text: w

text: X

text: X

text: (w

text: εδ

text: )(τ)

text: (w)(τ)

text: w

text: ε

text: [t,

text: t

text: ∆t],

text: we

text: have

text: X

text: X

text: εδw)(τ)

text: (w)(τ)

text: ε

text: ≤1

text: t,

text: X

text: (w

text: εδ

text: )(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: t,

text: X

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: w

text: w

text: ε

text: t

text: ≤1

text: v

text: (w

text: εδ

text: )(s),P

text: (t)+(s

text: t)(w

text: εδ

text: t,

text: X

text: (w)(s),P

text: (t)+(s

text: εδ

text: ds

text: w

text: w

text: w

text: ε

text: t

text: τ

text: X

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: w

text: ε

text: t

text: X

text: X

text: τ

text: (w

text: εδw)(s)

text: τ

text: v

text: ε

text: ds

text: (s

text: t)

text: δw

text: ds

text: t

text: t

text: X

text: X

text: τ

text: (w

text: εδ

text: )(s)

text: (w)(s)

text: v

text: w

text: ε

text: ds

text: δw

text: t

text: Hence

text: an

text: application

text: of

text: Gr¨onwall

text: Lemma,

text: see,

text: e.g.,

text: Chapter

text: Lemma

text: ensures

text: that

text: X

text: X

text: (w

text: εδ

text: )(τ)

text: (w)(τ)

text: w

text: K

text: δ

text: exp

text: (K

text: ∆t)

text: w

text: ε

text: where

text: v

text: Consequently

text: X

text: X

text: lim

text: sup

text: (w

text: εδ

text: )(τ)

text: (w)(τ)

text: ε→0

text: τ∈[t,t+∆t]

text: w

text: X

text: R

text: We

text: now

text: prove

text: the

text: existence

text: of

text: directional

text: derivatives

text: of

text: along

text: the

text: direction

text: δ

text: N

text: w

text: Calling

text: τ

text: Y

text: (τ)

text: the

text: solution

text: to

text: the

text: Cauchy

text: problem

text: we

text: have

text: X

text: X

text: (w

text: εδ

text: )(τ)

text: (w)(τ)

text: w

text: Y

text: (τ)

text: δ

text: w

text: ε

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:104) (cid:105) (cid:0) (cid:1) (cid:0) (cid:1) 1 (cid:90)τ X(cid:0) − (cid:1) − X − = v t, (w + εδ )(s), P (t) + (s t)(w + εδ ) v t, (w)(s), P (t) + (s t)w ds w w ε t − (cid:90) τ X (cid:0) − (cid:1) D v t, (w)(s), P (t) + (s t)w Y (s) ds δ x w t −(cid:90) τ(cid:104) (cid:0)− X − (cid:1) (s t)D v t, (w)(s), P (t) + (s t)w ds δ P w t 1 τ X (cid:0) − (cid:1)(cid:105) = v t, (w + εδ )(s), P (t) + (s t)(w + εδ ) w w ε (cid:90)t (cid:0) − X (cid:1)− v t, (w)(s), P (t) + (s t)(w + δ ) ds w − (cid:90)τ (cid:104) (cid:0) X − (cid:1) (cid:0) (cid:1)(cid:105) D v t, (w)(s), P (t) + (s t)w Y (s) ds δ x w t 1(cid:90) τ X (cid:0) − (cid:1) − X − + v t, (w)(s), P (t) + (s t)(w + εδ ) v t, (w)(s), P (t) + (s t)w ds w ε t −(cid:90) (cid:90)τ (cid:18)− X − (cid:0) (cid:1) (s t)D v t, (w)(s), P (t) + (s t)w ds δ P w t τ 1 X − (cid:19)X − = D v t, ϑ (w + εδ )(s) + (1 ϑ) (w)(s), P (t) + (s t)(w + εδ ) dϑ x w w t 0 X − X (cid:90) (cid:0) × (w + εδw)(s) (w)(cid:1)(s) ds ε − (cid:90) τ (cid:18)X(cid:90) − (cid:0) (cid:1) D v t, (w)(s), P (t) + (s t)w Y (s) ds δ x w t τ − 1 X − (cid:19) − (cid:0) (cid:1) + (s t) D v t, (w)(s), P (t) + (s t)(w + (1 ϑ)εδ ) dϑ P w t 0 (cid:90) (cid:90) (cid:18) − X − (cid:0) (cid:1) D v t, (w)(s), P (t) + (s t)w ds δ P w τ 1 X − (cid:19)X − = D v t, ϑ (w + εδ )(s) + (1 ϑ) (w)(s), P (t) + (s t)(w + εδ ) dϑ x w w t 0 X − X (cid:90) × (w + εδ )(s) (w)(s) (cid:0) w (cid:1) ds ε X − X ∓ (cid:90) τ (cid:0) X − (cid:1) (w + εδw)(s) (w)(s) D v t, (w)(s), P (t) + (s t)w ds x ε t − (cid:90) τ (cid:18)X(cid:90) − (cid:0) (cid:1) D v t, (w)(s), P (t) + (s t)w Y (s) ds δ x w t τ − 1 X − (cid:19) − (cid:0) (cid:1) + (s t) D v t, (w)(s), P (t) + (s t)(w + (1 ϑ)εδ ) dϑ P w t 0 (cid:90) (cid:90) (cid:18) − X − (cid:0) (cid:1) D v t, (w)(s), P (t) + (s t)w ds δ P w τ 1 X − X (cid:19) − (cid:0) (cid:1) = D v t, ϑ (w + εδ )(s) + (1 ϑ) (w)(s), P (t) + (s t)(w + εδ ) dϑ x w w t 0 X − X − X − (w + εδ )(s) (w)(s) w D v t, (w)(s), P (t) + (s t)w ds x ε 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: v

text: t,

text: (w

text: εδ

text: )(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: w

text: w

text: ε

text: t

text: τ

text: X

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: Y

text: (s)

text: ds

text: δ

text: x

text: w

text: t

text: X

text: (s

text: t)D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: δ

text: P

text: w

text: t

text: τ

text: X

text: v

text: t,

text: (w

text: εδ

text: )(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: w

text: w

text: ε

text: X

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: δ

text: ds

text: w

text: X

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: Y

text: (s)

text: ds

text: δ

text: x

text: w

text: t

text: τ

text: X

text: X

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: w

text: ε

text: t

text: X

text: (s

text: t)D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: δ

text: P

text: w

text: t

text: τ

text: X

text: D

text: v

text: t,

text: ϑ

text: (w

text: εδ

text: )(s)

text: ϑ)

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: dϑ

text: x

text: w

text: w

text: t

text: X

text: X

text: (w

text: εδw)(s)

text: ds

text: ε

text: τ

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: Y

text: (s)

text: ds

text: δ

text: x

text: w

text: t

text: τ

text: X

text: (s

text: t)

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: ϑ)εδ

text: dϑ

text: P

text: w

text: t

text: X

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: δ

text: P

text: w

text: τ

text: X

text: D

text: v

text: t,

text: ϑ

text: (w

text: εδ

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text: ϑ)

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: dϑ

text: x

text: w

text: w

text: t

text: X

text: X

text: (w

text: εδ

text: )(s)

text: (w)(s)

text: w

text: ds

text: ε

text: X

text: X

text: τ

text: X

text: (w

text: εδw)(s)

text: (w)(s)

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: x

text: ε

text: t

text: τ

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: Y

text: (s)

text: ds

text: δ

text: x

text: w

text: t

text: τ

text: X

text: (s

text: t)

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: ϑ)εδ

text: dϑ

text: P

text: w

text: t

text: X

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: δ

text: P

text: w

text: τ

text: X

text: X

text: D

text: v

text: t,

text: ϑ

text: (w

text: εδ

text: )(s)

text: ϑ)

text: (w)(s),

text: P

text: (t)

text: (s

text: t)(w

text: εδ

text: dϑ

text: x

text: w

text: w

text: t

text: X

text: X

text: X

text: (w

text: εδ

text: )(s)

text: (w)(s)

text: w

text: D

text: v

text: t,

text: (w)(s),

text: P

text: (t)

text: (s

text: t)w

text: ds

text: x

text: ε

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:18) (cid:90) (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) − τ − ρ (τ, x) = ρ t, X (t; τ, x) exp div v t, X (s; τ, x), P (t) + (s t) w ds (2.7) w t,w x t,w t where (cid:40) (cid:0) (cid:1) (cid:48) − → ξ = v t, ξ, P (t) + (τ t)w ∈ τ X (τ; t¯, x) solves for t¯, τ [t, t + ∆t] . (2.8) t,w ξ(t¯) = x ∈ R R L ∞ ∈ R Moreover,(cid:13)if ρ¯ C1,1( N; ) and N(cid:13)(spt ρ¯) < + , for all w , w N (cid:13) (cid:13) 1 2 (cid:18) (cid:19) (cid:16) (cid:17) − ρw1(t + ∆t) ρw2(t + ∆t) L1(RN;R) ≤ (cid:107) (cid:107) L (cid:107) (cid:107) grad ρ¯ ∞ R R N spt ρ¯, CeC∆t∆t + (1 + C ∆t) ρ¯ R R (2.9) x L ( N; N) L1( N; ) × (cid:107) − (cid:107) Ce2C∆t (∆t)2 w w 1 2   where       (cid:107) (cid:107) v ∞ ×R × R ,  (cid:107) L (cid:107)([t,t+∆t] N B(P(t),∆tU); N)    C = max  D(cid:107)xv L∞(cid:107) ([t,t+∆t]×RN×B(P(t),∆tU);RN×N),  (2.10) (cid:107)12 DP v L∞(cid:107)([t,t+∆t]×RN×RN;RN×N), DP divx v L∞([t,t+∆t]×RN×RN;RN) (cid:48) (Above and in the sequel, ξ = dξ ). The proof of Lemma 2.3 is deferred to Section . dτ In the case of the functional (1.2),(cid:90)a natural choice for the agent P at time t is then to choose a speed w on the time interval [t, t + ∆t] to minimize the quantity J (w) = ρ (t + ∆t, x) ψ(x) dx . (2.11) t,∆t w R N ∈ ∈ Proposition 2.4. Fix positive T, U, ∆t ]0, T [, and ﬁx a boundedly supported initial datum ρ¯ R R ∈ R ∈ × R × R R ∈ ∞ R R L1( N; ), P¯ N, a speed law v C0,1([0, T ] N N; N) and a weight ψ L ( N; ). ∈ ∈ − Then, with the notation in (2.1) and (2.6), fo(cid:90)r any t [0, T [ and ∆t ]0, T t] the map J R → R : N t,∆t → w ρ (t + ∆t, x) ψ(x) dx w R N is well deﬁned and Lipschitz continuous. The main theorem now follows, providing explicit information on a non anticipative optimal choice of w. 5 ##LTLine## ##LTLine##

text: τ

text: ρ

text: (τ,

text: x)

text: ρ

text: t,

text: X

text: (t;

text: τ,

text: x)

text: exp

text: div

text: v

text: t,

text: X

text: (s;

text: τ,

text: x),

text: P

text: (t)

text: (s

text: t)

text: w

text: ds

text: w

text: t,w

text: x

text: t,w

text: t

text: where

text: ξ

text: v

text: t,

text: ξ,

text: P

text: (t)

text: (τ

text: t)w

text: τ

text: X

text: (τ;

text: t¯,

text: x)

text: solves

text: for

text: t¯,

text: τ

text: [t,

text: t

text: ∆t]

text: t,w

text: ξ(t¯)

text: x

text: R

text: R

text: L

text: R

text: ρ¯

text: N;

text: and

text: ρ¯)

text: for

text: all

text: w

text: w

text: N

text: ∆t)

text: ∆t)

text: L

text: grad

text: ρ¯

text: R

text: R

text: N

text: spt

text: ρ¯,

text: CeC∆t∆t

text: C

text: ∆t)

text: ρ¯

text: R

text: R

text: x

text: L

text: N;

text: N)

text: N;

text: w

text: w

text: where

text: v

text: ×R

text: R

text: L

text: N

text: B(P(t),∆tU);

text: N)

text: C

text: max

text: ([t,t+∆t]×RN×B(P(t),∆tU);RN×N),

text: DP

text: v

text: DP

text: divx

text: v

text: L∞([t,t+∆t]×RN×RN;RN)

text: (Above

text: and

text: in

text: the

text: sequel,

text: ξ

text: dξ

text: The

text: proof

text: of

text: Lemma

text: is

text: deferred

text: to

text: Section

text: dτ

text: In

text: the

text: case

text: of

text: the

text: functional

text: natural

text: choice

text: for

text: the

text: agent

text: P

text: at

text: time

text: t

text: is

text: then

text: to

text: choose

text: a

text: speed

text: w

text: on

text: the

text: time

text: interval

text: [t,

text: t

text: ∆t]

text: to

text: minimize

text: the

text: quantity

text: J

text: (w)

text: ρ

text: (t

text: ∆t,

text: x)

text: ψ(x)

text: dx

text: t,∆t

text: w

text: R

text: N

text: Proposition

text: Fix

text: positive

text: T,

text: U,

text: ∆t

text: T

text: and

text: ﬁx

text: a

text: boundedly

text: supported

text: initial

text: datum

text: ρ¯

text: R

text: R

text: R

text: R

text: R

text: R

text: R

text: R

text: N;

text: P¯

text: N,

text: a

text: speed

text: law

text: v

text: T

text: N

text: N;

text: N)

text: and

text: a

text: weight

text: ψ

text: L

text: N;

text: Then,

text: with

text: the

text: notation

text: in

text: and

text: any

text: t

text: T

text: and

text: ∆t

text: T

text: t]

text: the

text: map

text: J

text: R

text: R

text: N

text: t,∆t

text: w

text: ρ

text: (t

text: ∆t,

text: x)

text: ψ(x)

text: dx

text: w

text: R

text: N

text: is

text: well

text: deﬁned

text: and

text: Lipschitz

text: continuous.

text: The

text: main

text: theorem

text: now

text: follows,

text: providing

text: explicit

text: information

text: on

text: a

text: non

text: anticipative

text: optimal

text: choice

text: of

text: w.

text: ##LTLine##

text: ##LTLine##


text: ∈ ∈ × R × R R ∈ Theorem 2.5. Fix positive T , U, and ∆t ]0, T [. Let v C2([0, T ] N N; N), ψ ∞ R R ∈ R R L ( N; ) and a boundedly supported ρ¯ C1( N; ). Deﬁne ρ as the solution to (2.1) and ∈ R ρw as the solution to (2.6), for a w N. T(cid:90)he map J R → R : N t,∆t → (2.12) w ρ (t + ∆t, x) ψ(x) dx w R N admits the expansion J J J · → (w + δ ) = (w) + grad (w) δ + o(δ ) as w 0 (2.13) t,∆t w t,∆t w t,∆t w w → where, as ∆t 0, (cid:90) (cid:20) (cid:21) (cid:0) (cid:1) (cid:0) (cid:1) J (∆t)2 − grad (w)= grad ρ(t, x)D v t, x, P (t) ρ(t, x) grad div v t, x, P (t) ψ(x) dx w t,∆t 2 R x P P x N (2.14) +o(∆t)2. The proof is deferred to Section . On the basis of Theorem 2.5, the deﬁnition of an eﬀective non anticipative strategy for P can be easily achieved as follows. Split the interval [0, T ] in smaller i portions [t , t [, where t = (cid:96) ∆t. On each of them, deﬁne u (t) = w , where w minimizes on (cid:96) (cid:96)+1 J (cid:96) i (cid:96) (cid:96) B(0, U) the(cid:90)cos(cid:20)t t ,∆t deﬁned in (2.12). The leading term in the right hand side o(cid:21)f (2.14) is (cid:96) (cid:0) (cid:1) (cid:0) (cid:1) independent of w, so that for ∆t small it is reasonable to choose (cid:13) (cid:13) (cid:13)(cid:90) (cid:20) − (cid:21) (cid:13) −(cid:13)(cid:13)U R gradx ρ(t(cid:96), x) DP v(cid:0) t(cid:96), x, Pi(t(cid:96)(cid:1)) ρ(t(cid:96), x) gradP divxv(cid:0) t(cid:96), x, Pi(t(cid:96)(cid:1)) ψ(x) dx(cid:13)(cid:13) w = (cid:13) N (cid:13) (cid:96) − grad ρ(t , x) D v t , x, P (t ) ρ(t , x) grad div v t , x, P (t ) ψ(x) dx x (cid:96) P (cid:96) i (cid:96) (cid:96) P x (cid:96) i (cid:96) R N as long as the denominator above does not vanish, in which case we set w = 0. Remark that, (cid:96) through the term ρ , the right hand side above depends on all the past values w , . . . , w − (cid:96) → 0 (cid:96) 1 attained by u . Formally, in the limit ∆t 0, the above relations thus leads to a delayed i integrodiﬀerential equation. 3 Examples This section presents a few numerical integrations of the game (1.3)–(1.2) in which a strategy is chosen as described in Section . As the function v in (1.3), we choose(cid:88) (cid:0) (cid:1) k (cid:107) − (cid:107) − v(t, x, P ) = a x P (P x) , (3.1) i i i i=1 6 ##LTLine## ##LTLine## ##LTLine##

text: R

text: R

text: R

text: Theorem

text: Fix

text: positive

text: T

text: U,

text: and

text: ∆t

text: T

text: Let

text: v

text: T

text: N

text: N;

text: N),

text: ψ

text: R

text: R

text: R

text: R

text: L

text: N;

text: and

text: a

text: boundedly

text: supported

text: ρ¯

text: N;

text: Deﬁne

text: ρ

text: as

text: the

text: solution

text: to

text: and

text: R

text: ρw

text: as

text: the

text: solution

text: to

text: for

text: a

text: w

text: N.

text: map

text: J

text: R

text: R

text: N

text: t,∆t

text: w

text: ρ

text: (t

text: ∆t,

text: x)

text: ψ(x)

text: dx

text: w

text: R

text: N

text: admits

text: the

text: expansion

text: J

text: J

text: J

text: (w

text: δ

text: (w)

text: grad

text: (w)

text: δ

text: o(δ

text: as

text: w

text: t,∆t

text: w

text: t,∆t

text: w

text: t,∆t

text: w

text: w

text: where,

text: as

text: ∆t

text: J

text: grad

text: (w)=

text: grad

text: ρ(t,

text: x)D

text: v

text: t,

text: x,

text: P

text: (t)

text: ρ(t,

text: x)

text: grad

text: div

text: v

text: t,

text: x,

text: P

text: (t)

text: ψ(x)

text: dx

text: w

text: t,∆t

text: R

text: x

text: P

text: P

text: x

text: N

text: The

text: proof

text: is

text: deferred

text: to

text: Section

text: On

text: the

text: basis

text: of

text: Theorem

text: the

text: deﬁnition

text: of

text: an

text: eﬀective

text: non

text: anticipative

text: strategy

text: for

text: P

text: can

text: be

text: easily

text: achieved

text: as

text: follows.

text: Split

text: the

text: interval

text: T

text: in

text: smaller

text: i

text: portions

text: [t

text: t

text: where

text: t

text: ∆t.

text: On

text: each

text: of

text: them,

text: deﬁne

text: u

text: (t)

text: w

text: where

text: w

text: minimizes

text: on

text: J

text: i

text: U)

text: t

text: ,∆t

text: deﬁned

text: in

text: The

text: leading

text: term

text: in

text: the

text: right

text: hand

text: side

text: is

text: independent

text: of

text: w,

text: so

text: that

text: for

text: ∆t

text: small

text: it

text: is

text: reasonable

text: to

text: choose

text: R

text: gradx

text: x)

text: DP

text: x,

text: x)

text: gradP

text: x,

text: ψ(x)

text: w

text: N

text: grad

text: ρ(t

text: x)

text: D

text: v

text: t

text: x,

text: P

text: (t

text: ρ(t

text: x)

text: grad

text: div

text: v

text: t

text: x,

text: P

text: (t

text: ψ(x)

text: dx

text: x

text: P

text: i

text: P

text: x

text: i

text: R

text: N

text: as

text: long

text: as

text: the

text: denominator

text: above

text: does

text: not

text: vanish,

text: in

text: which

text: case

text: we

text: set

text: w

text: Remark

text: that,

text: through

text: the

text: term

text: ρ

text: the

text: right

text: hand

text: side

text: above

text: depends

text: on

text: all

text: the

text: past

text: values

text: w

text: w

text: attained

text: by

text: u

text: Formally,

text: in

text: the

text: limit

text: ∆t

text: the

text: above

text: relations

text: thus

text: leads

text: to

text: a

text: delayed

text: i

text: integrodiﬀerential

text: equation.

text: Examples

text: This

text: section

text: presents

text: a

text: few

text: numerical

text: integrations

text: of

text: the

text: game

text: in

text: which

text: a

text: strategy

text: is

text: chosen

text: as

text: described

text: in

text: Section

text: As

text: the

text: function

text: v

text: in

text: we

text: k

text: v(t,

text: x,

text: P

text: a

text: x

text: P

text: (P

text: x)

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 17 − where x = mλ 2. The recursive relation for p (x) can also be derived on more formal grounds [8]. κ n − Note that n denotes the number of bonds in a chain, and not the number of beads in the chain (n = L 1).For arbitrary angles between the bonds it is not possible to describe the roots of this polynomial, except for oligomers (see below). But, if all angles are the same we can bring (C5) into the form of the Chebyshev polynomials of the second kind U (x) by substituting x˜ = x/2 cos θ: n p (x˜) = cosn θ U (x˜), n n − x mλ 2 (C6) (cid:16) (cid:17) x˜ = = κ . 2 cos θ 2 cos θ The roots of U (x) are x = cos k π ; k = 1, ..., n, which gives us the eigenvalues of the linear chain with n k n+1 constant angle as: (cid:18) (cid:20) (cid:21)(cid:19) 2κ kπ λ = ω2 = 1 + cos θ cos , k (cid:113)k m n + 1 2 ω (cid:54) D(ω) = (cid:32) (cid:114) (cid:33) (θ = π/2), π 4κ2 cos2 θ − (ω2 − 2κ)2 (C7) m2 m − 2κ ρ(ω) = δ ω (θ = π/2). m This result can also be derived using a diﬀerent approach which exploits the periodicity of the chain with constant angle, as was done by Kirkwood [13]. If we assume that the chain points along the x-axis we can identify the previously introduced along-chain and perpendicular motion as A and B in Eq. 6 of Kirkwood [13]. By using the dispersion relation found in this work, we can solve for those two quantities and ﬁnd the weight functions: | | − A 2 cos θ + 1 ω2 + 2ω(cos θ 1) X (ω) = | | | | = m , l A 2 + B 2 cos θ 2ω2 | | − − (C8) B 2 cos θ 1 ω2 2κ(cos θ + 1) X (ω) = | | | | = m . t A 2 + B 2 cos θ 2ω2 As mentioned before, for the ﬂexible case with distributed angles an analytical solution is not accessible for a chain of arbitrary length. But we can give the eigenvalues in the case of short chains with L = 2, 3, 4, 5: m L = 2 : ω2 = 2, κ (cid:112) m ± L = 3 : ω2 = 2 cos θ , κ 1 (cid:113) m ±(cid:112) (C9) L = 4 : ω2 = 2, 2 cos2 θ + cos2 θ , κ 1 2 m ±√1 ± − L = 5 : ω2 = cos2 θ + cos2 θ + cos2 θ (cos2 θ + cos2 θ + cos2 θ )2 4 cos2 θ cos2 θ . κ 2 1 2 3 1 2 3 1 3 ∗ fxundeﬁned [1]ifx1 fnum [1]1ﬁrstoftwo fx [1]secondoftwo noop [0]secondoftworef[1]@startlink1@hrefhref[1]1@endlinkanitize@url [0]startlink[1]endlink[0]rl [1]href 1bib@innerbibempty ##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: where

text: x

text: mλ

text: The

text: recursive

text: relation

text: for

text: p

text: (x)

text: can

text: also

text: be

text: derived

text: on

text: more

text: formal

text: grounds

text: κ

text: n

text: Note

text: that

text: n

text: denotes

text: the

text: number

text: of

text: bonds

text: in

text: a

text: chain,

text: and

text: not

text: the

text: number

text: of

text: beads

text: in

text: the

text: chain

text: (n

text: L

text: arbitrary

text: angles

text: between

text: the

text: bonds

text: it

text: is

text: not

text: possible

text: to

text: describe

text: the

text: roots

text: of

text: this

text: polynomial,

text: except

text: for

text: oligomers

text: (see

text: below).

text: But,

text: if

text: all

text: angles

text: are

text: the

text: same

text: we

text: can

text: bring

text: into

text: the

text: form

text: of

text: the

text: Chebyshev

text: polynomials

text: of

text: the

text: second

text: kind

text: U

text: (x)

text: by

text: substituting

text: x˜

text: cos

text: θ:

text: n

text: p

text: (x˜)

text: cosn

text: θ

text: U

text: (x˜),

text: n

text: n

text: x

text: mλ

text: x˜

text: κ

text: cos

text: θ

text: cos

text: θ

text: The

text: roots

text: of

text: U

text: (x)

text: are

text: x

text: cos

text: k

text: π

text: k

text: n,

text: which

text: gives

text: us

text: the

text: eigenvalues

text: of

text: the

text: linear

text: chain

text: with

text: n

text: k

text: constant

text: angle

text: as:

text: 2κ

text: kπ

text: λ

text: ω2

text: cos

text: θ

text: cos

text: k

text: m

text: n

text: ω

text: D(ω)

text: (θ

text: π

text: θ

text: m

text: 2κ

text: ρ(ω)

text: δ

text: ω

text: (θ

text: m

text: This

text: result

text: can

text: also

text: be

text: derived

text: using

text: a

text: diﬀerent

text: approach

text: which

text: exploits

text: the

text: periodicity

text: of

text: the

text: chain

text: with

text: constant

text: angle,

text: as

text: was

text: done

text: by

text: Kirkwood

text: If

text: we

text: assume

text: that

text: the

text: chain

text: points

text: along

text: the

text: x-axis

text: we

text: can

text: identify

text: the

text: previously

text: introduced

text: along-chain

text: and

text: perpendicular

text: motion

text: as

text: A

text: and

text: B

text: in

text: Eq.

text: of

text: Kirkwood

text: By

text: using

text: the

text: dispersion

text: relation

text: found

text: in

text: this

text: work,

text: we

text: can

text: solve

text: for

text: those

text: two

text: quantities

text: and

text: ﬁnd

text: the

text: weight

text: functions:

text: A

text: cos

text: θ

text: ω2

text: θ

text: X

text: (ω)

text: m

text: l

text: A

text: B

text: cos

text: θ

text: B

text: cos

text: θ

text: ω2

text: θ

text: X

text: (ω)

text: m

text: t

text: A

text: B

text: cos

text: θ

text: As

text: mentioned

text: before,

text: for

text: the

text: ﬂexible

text: case

text: with

text: distributed

text: angles

text: an

text: analytical

text: solution

text: is

text: not

text: accessible

text: for

text: a

text: chain

text: of

text: arbitrary

text: length.

text: But

text: we

text: can

text: give

text: the

text: eigenvalues

text: in

text: the

text: case

text: of

text: short

text: chains

text: with

text: L

text: m

text: L

text: ω2

text: κ

text: m

text: L

text: ω2

text: cos

text: θ

text: κ

text: m

text: L

text: ω2

text: θ

text: θ

text: κ

text: m

text: ±√1

text: L

text: ω2

text: θ

text: θ

text: θ

text: θ

text: θ

text: θ

text: θ

text: θ

text: κ

text: fxundeﬁned

text: fnum

text: fx

text: noop

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text: ##LTLine##

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text: ##LTLine##

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text: 4 a b increasing chain length increasing chain length Tg* Flexible chains Stiﬀ chains c d increasing increasing stiﬀness stiﬀness Chain length = 3 Chain length = 50 T* T* FIG. 2. Volume-temperature curves for polymer cooling as functions of chain length and chain stiﬀness. The volume is rescaled with bead number N and size σ3. a) Increasing chain length for κ /κ = 0. Highlighted is the glass transition temperature p bend LJ ∗ T , where the polymer transitions from a melt to a glassy state; b) Increasing chain length for κ /κ = 17.5; c) Increasing g bend LJ chain stiﬀness for chain length 3; d) Increasing chain stiﬀness for chain length 50. Colors in (a) and (b) refer, from blue to red, to chain lengths 2, 3, 4, 5, 20 and 50. Colors in (c) and (d) refer, from blue to red, to chain stiﬀnesses κ /κ = 0.0175, bend LJ 0.0525, 0.175, 0.525, 1.75, 5.25 and 17.5. ∗ T = k T/ε . Two additional rescaled temperatures could be deﬁned using ε or ε as the reference energy, B LJ FENE bend but we ﬁnd that the most convenient description and characterisation of the transition to glassy behavior is obtained using ε . LJ Initial loose polymer conﬁgurations are generated within a cubic periodic domain using a non-overlapping random- × walk algorithm. We use a system of N = 5 103 beads, in chains of uniform length L, which we vary from 2 p to 50. The value of N is chosen following the entanglement critical of Ref [24], and moreover we demonstrate the p sensitivity to N in Figure 9. For each value of L we generate 5 realisations of the system for the purposes of ensemble p averaging. We comment on the variation between realisations elsewhere [22]. The system is ﬁrst equilibrated in a ∗ melted state at T = 1.2, maintaining zero external pressure using a Nose-Hoover barostat with damping parameter ∗ ∗ ∼ O of 100τ. The system is subsequently cooled to T = 0.1 by decreasing T at rate 1/τ , with τ (105)τ. Since c c ∗ T = 0.1 is below the glass transition for all of the polymers considered in this work, this cooling procedure allows us to measure T . For determining the vibrational properties, though, it is necessary to go to lower temperatures. To g ∗ reach temperatures closer to T = 0, we subsequently relax the system further by applying the gradient method to (cid:104) (cid:105) ∼ ∗ the simulation conﬁguration at constant volume. We used the ﬂuctuation of net forces acting on beads f2 T as a i ∗ measure for temperature. By comparing the forces with the reference value from T = 0.1 we get the temperature of ∗ (cid:104) (cid:105) (cid:104) ∗ (cid:105) ∗ − the relaxed conﬁguration by T = 0.1 f2 / f2(T = 0.1) . By this protocol, a target temperature of T = 10 4 was i i reached for each realization. Further decreasing the temperature does not lead to changes in the VDOS or structural quantities. III. STRUCTURE OF COARSE-GRAINED POLYMER GLASSES A. Changes in T with chain length and stiﬀness g Ensuring that the external pressure remains zero, the system undergoes a decrease in volume V as it is cooled, ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: a

text: b

text: increasing

text: chain

text: length

text: increasing

text: chain

text: length

text: Tg*

text: Flexible

text: chains

text: Stiﬀ

text: chains

text: c

text: d

text: increasing

text: increasing

text: stiﬀness

text: stiﬀness

text: Chain

text: length

text: Chain

text: length

text: T*

text: T*

text: FIG.

text: Volume-temperature

text: curves

text: for

text: polymer

text: cooling

text: as

text: functions

text: of

text: chain

text: length

text: and

text: chain

text: stiﬀness.

text: The

text: volume

text: is

text: rescaled

text: with

text: bead

text: number

text: N

text: and

text: size

text: a)

text: Increasing

text: chain

text: length

text: for

text: κ

text: /κ

text: Highlighted

text: is

text: the

text: glass

text: transition

text: temperature

text: p

text: bend

text: LJ

text: T

text: where

text: the

text: polymer

text: transitions

text: from

text: a

text: melt

text: to

text: a

text: glassy

text: state;

text: b)

text: Increasing

text: chain

text: length

text: for

text: κ

text: /κ

text: c)

text: Increasing

text: g

text: bend

text: LJ

text: chain

text: stiﬀness

text: for

text: chain

text: length

text: d)

text: Increasing

text: chain

text: stiﬀness

text: for

text: chain

text: length

text: Colors

text: in

text: (a)

text: and

text: (b)

text: refer,

text: from

text: blue

text: to

text: red,

text: to

text: chain

text: lengths

text: and

text: Colors

text: in

text: (c)

text: and

text: (d)

text: refer,

text: from

text: blue

text: to

text: red,

text: to

text: chain

text: stiﬀnesses

text: κ

text: /κ

text: bend

text: LJ

text: and

text: T

text: k

text: T/ε

text: Two

text: additional

text: rescaled

text: temperatures

text: could

text: be

text: deﬁned

text: using

text: ε

text: or

text: ε

text: as

text: the

text: reference

text: energy,

text: B

text: LJ

text: FENE

text: bend

text: but

text: we

text: ﬁnd

text: that

text: the

text: most

text: convenient

text: description

text: and

text: characterisation

text: of

text: the

text: transition

text: to

text: glassy

text: behavior

text: is

text: obtained

text: using

text: ε

text: LJ

text: Initial

text: loose

text: polymer

text: conﬁgurations

text: are

text: generated

text: within

text: a

text: cubic

text: periodic

text: domain

text: using

text: a

text: non-overlapping

text: random-

text: walk

text: algorithm.

text: We

text: use

text: a

text: system

text: of

text: N

text: beads,

text: in

text: chains

text: of

text: uniform

text: length

text: L,

text: which

text: we

text: vary

text: from

text: p

text: to

text: The

text: value

text: of

text: N

text: is

text: chosen

text: following

text: the

text: entanglement

text: critical

text: of

text: Ref

text: and

text: moreover

text: we

text: demonstrate

text: the

text: p

text: sensitivity

text: to

text: N

text: in

text: Figure

text: For

text: each

text: value

text: of

text: L

text: we

text: generate

text: realisations

text: of

text: the

text: system

text: for

text: the

text: purposes

text: of

text: ensemble

text: p

text: averaging.

text: We

text: comment

text: on

text: the

text: variation

text: between

text: realisations

text: elsewhere

text: The

text: system

text: is

text: ﬁrst

text: equilibrated

text: in

text: a

text: melted

text: state

text: at

text: T

text: maintaining

text: zero

text: external

text: pressure

text: using

text: a

text: Nose-Hoover

text: barostat

text: with

text: damping

text: parameter

text: O

text: of

text: The

text: system

text: is

text: subsequently

text: cooled

text: to

text: T

text: by

text: decreasing

text: T

text: at

text: rate

text: with

text: τ

text: Since

text: c

text: c

text: T

text: is

text: below

text: the

text: glass

text: transition

text: for

text: all

text: of

text: the

text: polymers

text: considered

text: in

text: this

text: work,

text: this

text: cooling

text: procedure

text: allows

text: us

text: to

text: measure

text: T

text: For

text: determining

text: the

text: vibrational

text: properties,

text: though,

text: it

text: is

text: necessary

text: to

text: go

text: to

text: lower

text: temperatures.

text: To

text: g

text: reach

text: temperatures

text: closer

text: to

text: T

text: we

text: subsequently

text: relax

text: the

text: system

text: further

text: by

text: applying

text: the

text: gradient

text: method

text: to

text: the

text: simulation

text: conﬁguration

text: at

text: constant

text: volume.

text: We

text: used

text: the

text: ﬂuctuation

text: of

text: net

text: forces

text: acting

text: on

text: beads

text: T

text: as

text: a

text: i

text: measure

text: for

text: temperature.

text: By

text: comparing

text: the

text: forces

text: with

text: the

text: reference

text: value

text: from

text: T

text: we

text: get

text: the

text: temperature

text: of

text: the

text: relaxed

text: conﬁguration

text: by

text: T

text: By

text: this

text: protocol,

text: a

text: target

text: temperature

text: of

text: T

text: was

text: i

text: i

text: reached

text: for

text: each

text: realization.

text: Further

text: decreasing

text: the

text: temperature

text: does

text: not

text: lead

text: to

text: changes

text: in

text: the

text: VDOS

text: or

text: structural

text: quantities.

text: III.

text: STRUCTURE

text: OF

text: COARSE-GRAINED

text: POLYMER

text: GLASSES

text: A.

text: Changes

text: in

text: T

text: with

text: chain

text: length

text: and

text: stiﬀness

text: g

text: Ensuring

text: that

text: the

text: external

text: pressure

text: remains

text: zero,

text: the

text: system

text: undergoes

text: a

text: decrease

text: in

text: volume

text: V

text: as

text: it

text: is

text: cooled,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##