A NOTE ON THE SMOLUCHOWSKI-KRAMERS APPROXIMATION FOR THE LANGEVIN EQUATION WITH REFLECTION

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1 A NOTE ON THE SMOLUCHOWSKI-KRAMERS APPROXIMATION FOR THE LANGEVIN EQUATION WITH REFLECTION KONSTANTINOS SPILIOPOULOS Deparmen of Mahemaics, Universiy of Maryland, College Park, 2742, Maryland, USA Absrac According o he Smoluchowski-Kramers approximaion, he soluion of he equaion µ q µ = b(q µ ) q µ + Σ(q µ )Ẇ, q µ = q, qµ = p converges o he soluion of he equaion q = b(q ) + Σ(q )Ẇ, q = q as µ. We consider here a similar resul for he Langevin process wih elasic reflecion on he boundary. Keywords: Smoluchowski-Kramers approximaion, reflecion, Langevin equaion, Skorohod reflecion problem. 1 Inroducion The well-known Smoluchowski-Kramers approximaion ([9],[8]) implies ha he soluion of he sochasic differenial equaion (S.D.E.) µ q µ = b(q µ ) q µ + Σ(q µ )Ẇ (1) q µ = q R r q µ = p R r converges in probabiliy as µ o he soluion of he following S.D.E.: q = b(q ) + Σ(q )Ẇ (2) q = q R r, where b = (b 1,..., b r ) (he ranspose of (b 1,..., b r )) wih b j : R r R, j = 1,.., r, Σ = [σ ij ] r i,j wih σ ij : R r R, i, j = 1,.., r have bounded firs derivaives and W = (W 1,..., W r ) is he sandard r-dimensional Wiener process. In oher Elecronic version of an aricle published as [Sochasics and Dynamics, Vol. 7, No. 2, June 27, ] [doi:1.1142/s ] [copyrigh World Scienific Publishing Company] [hp:// 1

2 words, one can prove ha for any δ, T > and q, p R r (see, for example, Lemma 1 in [6]), lim P ( max µ T qµ q > δ) =. (3) Equaion (1) describes he moion of a paricle of mass µ in a force field b(q)+ Σ(q)Ẇ wih a fricion proporional o velociy. The Smoluchowski-Kramers approximaion jusifies he use of equaion (2) o describe he moion of a small paricle. I is easy o see now ha (1) can be equivalenly wrien as: q µ = p µ µṗ µ = b(q µ ) p µ + Σ(q µ )Ẇ (4) q µ = q R r, q µ = p Rr. Le us define R + = {q 1 R : q 1 } and le he configuraion space be D = R + R r 1. In his paper we examine he behavior of he process wih elasic reflecion on he boundary D R r = ( R + R r 1 ) R r of he phase space D R r ha is governed by (4), i.e. of he Langevin process wih reflecion, as µ when Σ is he uni marix. We will show ha he firs componen (he q componen) of he Langevin process wih reflecion a q 1 =, ha is governed by equaion (4), converges in disribuion o he diffusion process wih reflecion on D ha is governed by (2). The mehod is based on properies of he Skorohod reflecion problem and in echniques developed in [2] and in [3]. In secion 2 we define he Langevin process wih reflecion for general diffusion marx Σ wih inpus ha have bounded firs derivaives, in secion 3 we describe he Skorohod reflecion problem and in secion 4 we consider he limi µ when he diffusion marix is he uni marix. We noe here ha he limi when µ for a general diffusion marix as above can be examined similarly. 2 Langevin process wih reflecion and preliminary resuls We begin wih he consrucion of he Langevin process (q µ ; p µ ) in D R r wih elasic reflecion on he boundary. Le b = (b 1,..., b r ) wih b j : D R, j = 1,.., r and Σ = [σ ij ] wih σ ij : D R, i, j = 1,.., r have bounded firs derivaives and Σ be non-degenerae. Le (q, p) D R r be he iniial poin (we 2

3 assume ha (q 1 ) 2 + (p 1 ) 2 ). Then (q µ ; p µ ) is he righ-coninuous Markov process in D R r defined as follows. Consider he following sysem of S.D.E. s: q i,µ = p i,µ µṗ i,µ = p i,µ + b i (q µ ) + r σ ij (q µ )Ẇ j (5) j=1 q i,µ = q i, p i,µ = p i, i = 1,..., r. We define (q µ ; p µ ) o be he soluion o (5) for [, τ µ 1 ), where τ µ 1 = inf{ > : q 1,µ = }. Then define (q µ ; p µ ) for [τ µ 1, τ µ 2 ), where τ µ 2 = inf{ > τ µ 1 : = }, o be he soluion of (5) wih iniial condiions q µ (q µ ; p µ τ µ 1 τ µ 1 ) = (, lim τ µ 1 q 2,µ,..., lim τ µ 1 q r,µ ; lim τ µ 1 p 1,µ, lim τ µ 1 p 2,µ,..., lim p r,µ τ µ ). 1 If < τ µ 1 < τ µ 2 <... < τ µ k and (qµ ; p µ ) for [, τ µ k ) are already defined, hen define (q µ ; p µ ) for [τ µ k, τ µ k+1 ) as soluion of (5) wih iniial condiions (q µ ; p µ ) = (, lim τ µ k τ µ k τ µ k q 2,µ,..., lim τ µ k q r,µ ; lim τ µ k p 1,µ, lim τ µ k p 2,µ,..., lim p r,µ τ µ ) k (see Figure 1 for an illusraion). This consrucion defines he process (q µ ; p µ ) in D R r for all. This follows from Theorem 2.4, which saes ha he process ha we consruced above does no have infiniely many jumps in any finie ime inerval [, T ]. Therefore we have he following definiion: Definiion 2.1. We call he above recursively consruced process, he Langevin process wih elasic reflecion on he boundary D R r. This process has jumps on D R r and is coninuous inside D R r. We will refer o he Langevin process wih reflecion as l.p.r.(q µ ; p µ ). Moreover we will denoe by (q µ,q ; p µ,p ) he rajecories of (q µ ; p µ ) wih iniial posiion (q, p). For easy of noaion we also define x = ( x 1, x 2,..., x r ) and x = ( x 1, x 2,..., x r ) for x R r. Below we see an illusraion of he consrucion above in he (q 1 p 1 ) phase space. Le us give now anoher consrucion of he Langevin process wih reflecion. Consider he following S.D.E. in R 2r : q 1,µ = p 1,µ µ q 1,µ = p 1,µ + sgn(q 1,µ )b 1 ( q µ ) + q 1,µ = q 1, p 1,µ = p 1, r j=1 sgn(q 1,µ )σ 1j ( q µ )Ẇ j q i,µ = p i,µ (6) 3

4 Figure 1: Illusraion of he Langevin process wih reflecion in he (q 1 p 1 ) phase space µṗ i,µ = p i,µ + b i ( q µ ) + r σ ij ( q µ )Ẇ j j=1 q i,µ = q i, p i,µ = p i, i = 2,..., r, where sgn(x) akes wo values, 1 if x and -1 if x <. Lemma 2.2. Equaion (6) has a weak soluion which is unique in he sense of probabiliy law. Proof. The exisence follows from he Girsanov s Theorem on he absolue coninuous change of measures in he space of rajecories (b and Σ are assumed bounded) and he fac ha (6) wih b = has a weak soluion. The uniqueness follows from Proposiion of [7]. Using he processes (q µ,q ; p µ,p ) and (q µ, q ; p µ, p ) we can give anoher consrucion of he Langevin process wih reflecion, as follows. Assume ha q 1 > and p 1 >, The graphs of p 1,µ,p1 and p 1,µ, p1 will be exacly symmeric wih re- and of q 1,µ, q1. = } and ( q µ ; p µ ) be a sochasic process, spec o zero. The same will be rue also for he graphs of q 1,µ,q1 Le τ µ =, τ µ k = inf{ > τ µ k 1 : q1,µ,q1 which is defined as follows: ( q µ ; p µ ) = (q µ,q ( q µ ; p µ ) = (q µ, q ; p µ,p ) for τ µ 2k τ µ, 2k+1 ; p µ, p ) for τ µ 2k+1 τ µ, 2k+2, k =, 1, 2,... (7) 4

5 Process ( q µ ; p µ ) is a process wih reflecion and i can be seen ha ( q µ ; p µ ), which is he same as ( q 1,µ, q 2,µ,, q r,µ ; d d q1,µ, q 2,µ,, q r,µ ), and l.p.r.(q µ ; p µ ) coincide. In he figures below we give an illusraion of he consrucion of ( q 1,µ ; p 1,µ ). The firs figure illusraes wih hick coninuous and doed lines q 1,µ versus. The coninuous line is q 1,µ,q1 versus and he doed is q 1,µ, q1 versus. The second figure illusraes wih hick coninuous and doed lines p 1,µ versus. The coninuous line is p 1,µ,p1 versus and he doed is p 1,µ, p1 versus. 5

6 Figure 2: Illusraion of he process wih reflecion Lemma 2.3. Le T >. The Markov process (q µ ; p µ ) saring a a poin (q, p) differen from he origin O = (,..., ;,..., ), ha saisfies sysem (6), does no reach he origin O in finie ime T, i.e. P ( T s.. (q µ ; p µ ) = O) =. Proof. We easily see ha i is acually enough o consider only (q 1,µ ; p 1,µ ). Le δ 1 be a small number. Define he recangle = {(q, p) R R : q δ 2 2, p δ 2 } and suppose ha he rajecory sars from some poin ouside he recangle, say from (q, ) R 2 \. Le also χ (x) denoe he indicaor funcion of he se. Then E (q,) T χ (q 1 s, p 1 s)ds is he expeced value of he ime,during ime [, T ], ha he process (q 1, p 1 ) wih iniial poin (q, ) spends inside he recangle. If b = and Σ is a marix wih consan enries, (q 1, p 1 ) is a Gaussian process. One can wrie down is densiy explicily (see equaion (6)), which we denoe by ρ( ), and obain he bound T E (q,) χ (qs, 1 p 1 s)ds = T ρ(s, (q, ), y)dsdy A(T, q)δ 3 (8) where A(T, q) is a consan ha depends on T and q. The general case can be reduced o he case wih b = and Σ consan by an absoluely coninuous change of measures in he space of rajecories and by a random ime change. We will esablish now a lower bound for he quaniy E (q,) T χ (qs, 1 p 1 s)ds under he assumpion ha he process (q 1,µ, p 1,µ ) will reach (, ) before ime T wih posiive probabiliy. This will lead o a conradicion. Again by Girsanov s heorem on he absolue coninuiy of measures in he 6

7 space of rajecories i is enough o consider he soluion of he following S.D.E: q 1 = p 1 ṗ 1 = 1 µ r j=1 q 1 = q 1, p 1 = p 1, σ 1j (q µ ) W j (9) where W j = sgn(q1,µ u )dwu. j By he self similariy properies of he Wiener process one can find a Wiener process W 1, such ha 1 r µ j=1 σ 1j(q µ ) W j = W 1, θ(), where θ() = and α 11 = r j,k=1 σ 1jσ 1k. So 1 r µ j=1 σ 1j(q µ ) W j can be obained from W 1, 1 µ 2 α 11 (q µ s )ds via a random ime change. By he law of ieraed logarihm we ge ha for all k [, 1] here exiss a o (k) small enough, such ha P ( 1 2 +k W 1, 1 2 k for [, o (k)]) 1 k. Observe ha if [, o (k)] hen θ() [, c o (k)], where c = 1 µ 2 sup x R α 11 (x). Define also o(k) = min{ o (k), o(k) c }. Then wih probabiliy very close o 1, as k, and for all [, o(k)] i mus hold ha p 1,µ c k and q 1,µ = ds c 1s 1 2 k ds < 2c k, for a consan c 1. p1,µ s Le τ be he firs ime, afer he ime ha he Markov process reached he origin, ha i exis from he recangle, i.e. τ = inf{ > : (q 1, p 1 ) R 2 \ }. Then i follows ha E (q,) T χ (q 1 s, p 1 s)ds > E{τ} P ( T s.. (q 1,µ ; p 1,µ ) = (, )) (1) Define τ q = inf{ > : q 1,µ > δ2 2 } and τ p = inf{ > : p 1,µ > δ 2 }. By he above bounds for q 1,µ and p 1,µ we ge ha τ q > c q δ 4 3 and τ p > c p δ 2, where c q, c p are some consans independen of δ. So he rajecory exis he recangle faser in he direcion of p han in he direcion of q and he exi ime is of order δ 2. Therefore, by his and by (8), we have ha T Bδ 2 < E (q,) χ (qs, 1 p 1 s)ds Aδ 3, (11) which canno hold for consans A and B and small enough δ. So we have a conradicion and hence i is rue ha P ( T s.. (q 1,µ ; p 1µ ) = (, )) =. Theorem 2.4. We have he following wo saemens: 7

8 1. Le T >. The Markov process l.p.r.(q µ ; p µ ) (wih arbirary b) does no reach he origin O = (,..., ;,..., ) in finie ime T, namely P ( T s.. l.p.r.(q µ ; p µ ) = O) =. 2. The sequence of Markov imes {τ µ k } converges o + as k +, i.e. P ( lim k + τ µ k = + ) = 1 Proof. The Langevin process wih reflecion, l.p.r.(q µ ; p µ ), coincides a any ime eiher wih (q µ,q ; p µ,p ) or wih (q µ, q ; p µ, p ). Therefore we have ha: P ( T s.. l.p.r.(q µ ; p µ ) = O) P ( T s.. (q µ,q ; p µ,p ) = O) Hence Lemma 2.3 implies ha + P ( T s.. (q µ, q P ( T s.. l.p.r.(q µ ; p µ ) = O) =. ; p µ, p ) = O). Par (ii) is an easy consequence of par (i). I is easy o see ha {τ µ k } is an unbounded, sricly increasing sequence of Markov imes. Indeed, if on he conrary we assume ha here exiss a N such ha τ µ k N for all k wih posiive probabiliy, hen he rajecories of l.p.r.(q µ ; p µ ) will have limi poins. The only possible limi poin however is he origin (,..., ;,..., ). Bu by par (i) he probabiliy ha wihin any ime T he rajecory will reach he origin is. So {τ µ k } is an unbounded sricly increasing sequence of Markov imes. Therefore we have ha P (lim k + τ µ k = + ) = 1. Therefore he Langevin process wih reflecion has only finiely many jumps in any ime inerval [, T ] wih probabiliy 1. Hence our definiion for he Langevin process wih reflecion is correc. 3 The Skorohod reflecion problem The convergence of he Langevin process wih reflecion ha will be presened in secion 4 relies on resuls abou soluions of he Skorohod reflecion problem, proven in [3] and [1]. Le us firs recall ha D = R + R r 1, D = R + R r 1 and le N(q) be he se of inward normals a q D. Denoe also by D(R +, D) he space of cadlág (righ coninuous wih lef limis) funcions wih values in D, endowed wih he Skorohod opology and by B.V.(R +, D) he se of cadlág funcions wih bounded variaion and values in D. 8

9 Definiion 3.1. Le w be a funcion in D(R +, R r ) such ha w() D. We say ha he pair (q, φ) wih q D(R +, D), φ B.V.(R +, R r ) is a soluion o he Skorohod problem for (D, N, w) if φ = q = w + φ ν(s)d φ s, ν(s) N(q s ), d φ a.e. d φ ( : q D) =, where φ denoes he oal variaion of φ and is called he local ime of he soluion. The following heorem characerizes he coninuiy properies of soluions of he Skorohod reflecion problem. Theorem 3.2. Le W be a compac subse of D(R +, R r ) in he Skorohod opology such ha w() D for every w W. Moreover le Q be he se of (q, φ, φ, w) D(R +, D) B.V.(R +, R r ) B.V.(R +, R + ) D(R +, R r ) such ha (q, φ) is he soluion o he Skorohod problem for (D, N, w) for some w W and q is coninuous. The se D is convex and so Q is a relaively compac subse of D(R +, R 3r+1 ) in he Skorohod opology and for every accumulaion poin of (q, φ, φ, w) in Q we have ha (q, φ) is a soluion o he Skorohod problem for (D, N, w). Proof. This is a special case of heorem 3.2 in [2]. 4 Convergence of he Langevin process wih reflecion In his secion we consider he limi of l.p.r.(q µ ) as µ when he diffusion marix is he uni marix. Below we will assume ha T, where T ia s posiive real number. Consider he sochasic process (q µ ; p µ ) in D R r, which saisfies he following sysem of S.D.E. s: q µ = p µ µṗ µ = p µ + b(q µ ) + Ẇ + ν(q µ ) Ψ µ (12) q µ = q, p µ = p, where q µ = (q 1,µ,, q r,µ ), p µ = (p 1,µ,, p r,µ ), W = (W 1,, W r ), ν(q) denoes he uni inward normal o D a q D, b(q) = (b 1 (q),..., b r (q)) and Ψ µ = µ s ( 2pµ s ν(q s µ )) χ D (q s µ ). I is easy o see ha (12) is pahwise 9

10 equivalen o he Langevin process wih reflecion in D R r of Definiion 2.1. and so i admis a unique weak soluion. We will follow he mehod inroduced in [2]. The main idea is o represen q µ as he firs componen of a soluion o he Skorohod problem for (D, N, H µ + X µ ), where H µ + X µ is a semimaringale. The family {H µ + X µ } urns ou o be igh and his enables us o use Theorem 3.2 o conclude ha he family {q µ } is igh as well. We can suppose ha here is a unique underlying complee probabiliy space (Ω, F, P ). Le F denoe he he σ algebra of F of ses wih P measure or 1 and define he filraion F µ = F σ((q µ s ; p µ s ), s ). Lemma 4.1. For every µ he pair of sochasic processes (q µ, Φ µ ), where Φ µ = ν(q µ s )dψ µ (13) is an almos surely soluion o he Skorohod reflecion problem for (D, N, H µ + X µ ), where H µ = q + µp µp µ X µ = b(q µ s )ds + W (14) Proof. Consider he inegral form of (12). Taking ino accoun ha pµ s ds = q µ q and solving for q µ we see ha: q µ = H µ + X µ + Φ µ Then (q µ, Φ µ ) verifies Definiion 3.1 wih probabiliy 1. Lemma 4.2. For every T > we have ha lim µ E[sup T µp µ 2 ] =. Proof. Assume firs ha b =. Consider equaions (12) and apply he Iô formula for semimaringales o he funcion f(q, p) = p 2 for every pair of imes s, such ha s T. Doing ha we ge p µ 2 = p µ s 2 2 µ s p µ u 2 du + 2 µ s p µ u dw u + 1 r( s) (15) µ 2 I is ineresing o observe ha he local ime Ψ µ does no appear above. This comes from he fac ha under elasic reflecion p µ 2 = p µ 2 for every >. 1

11 Consider now a consan c > and funcions x, g D([, T ], R) wih g() = such ha: x x s c Then one can easily see ha x e c (x + g ) + c s x u du + g g s, s T (16) e c( u) (g g u )du, T (17) By aking expeced value o (15) and applying (17) wih c = 2 µ, g = 1 µ 2 r and x = p µ 2, we ge E p µ 2 e 2 µ ( p µ 2 r) + 2 µ 3 e 2 µ ( u) r( u)du = e 2 µ p 2 + r µ 2 (µ 2 µ 2 e µ ) (18) 2 This implies he saemen of he Lemma for b =. The general case can be reduced o he case wih b = by an absoluely coninuous change of measures in he space of rajecories. The following wo heorems are resaemens of heorems and respecively of [4]. Theorem 4.3. Le {Y n } be a family of processes wih sample pahs in D(R +, D). Assuming ha for every ɛ > and raional here exis a compac se Γ(ɛ, ) D such ha lim inf n P (Y n () Γ(ɛ, )) 1 ɛ, hen he following are equivalen 1. {Y n } is relaively compac. 2. For each T >, here exiss β > and a family of nonnegaive random variables {γ n (δ), < δ < 1} saisfying E( Y n ( + u) Y n () β F n ) E(γ n (δ) F n ), for [, T ] and u [, δ] and in addiion lim δ lim sup n E(γ n (δ)) =. Theorem 4.4. Le {Y n } and Y be processes wih sample pahs in D(R +, D) such ha Y n converges in disribuion o Y. Then Y is almos surely coninuous if and only if e u [sup u Y n () Y n ( ) 1]du. The following lemma shows ha he family {H µ + X µ } is igh in he Skorohod opology. 11

12 Lemma 4.5. The family {H µ + X µ } defined in (14) is relaively compac and all of is accumulaion poins are coninuous. Proof. I is easily seen ha {X µ } is relaively compac and ha all of is accumulaion poins are coninuous. Now Lemma 4.2 suggess ha: lim E[sup H µ 2 ] c (19) µ T lim E[ sup H µ H s µ ] c 1 δ, (2) µ s δ where c, c 1 are posiive consans independen of µ. Chebychev s inequaliy and (19) imply ha lim inf n P ( H1/n () λ) 1 c λ 2. Therefore by his and (2), Theorem 4.3. gives us ha {H µ } is relaively compac. Lasly (2) and Theorem 4.4 implies ha all is accumulaion poins are coninuous. Theorem 4.6. The family {(q µ, Φ µ, Ψ µ, H µ, X µ )} is relaively compac in D(R +, R 4r+1 ). Proof. I follows from Lemma 4.5 and Theorem 3.2. Now ha ighness has been esablished we will proceed wih he idenificaion of he sochasic differenial equaion wih reflecion ha describes he behavior of q µ as µ. Consider he following S.D.E. wih reflecion: q = q + b(q s )ds + W + Φ (21) where Φ = ν(q s)d Φ s, ν(s) N(q s ) and d Φ ({ : q D}) =. I is known ha (21) has a unique weak soluion (q, Φ) ([1]). Theorem 4.7. The family {(q µ, Φ µ )} converges in disribuion o he unique soluion (q, Φ) of (21). Proof. By Theorem 4.6. we have ha he five-uple {(q µ, Φ µ, H µ, X µ, W )} is relaively compac in D(R +, R 5r ). Hence i (or a subsequence) converges in disribuion o a sochasic process {(q, Φ, H, X, W )}. By he Skorohod represenaion heorem, one can find a probabiliy space ( Ω, F, P ) and realizaions {( q µ, Φ µ, H µ, W µ )} and {( q, Φ, H, X, W )} of {(q µ, Φ µ, H µ, X µ, W )} and 12

13 {(q, Φ, H, X, W )} respecively such ha {( q µ, Φ µ, H µ, X µ, W µ )} converges P - almos surely o {( q, Φ, H, X, W )}. Therefore by Theorem 3.2. ( q, Φ) is a soluion o he Skorohod problem for (D, N, H + X) P almos surely. Now by he convergence of q µ o q we ge ha X mus be given by: X = b( q s )ds + W Finally Lemma 4.2 and is proof imply ha H = q. We would like o noe here ha one could prove he convergence in disribuion of he Langevin procces wih reflecion o he corresponding diffusion process wih reflecion using he Smoluchowski-Kramers approximaion. However he beauy and generaliy of he resuls of [3] resuled in using he mehod ha was presened here. Acknowledgmens I would like o hank my advisor Mark Freidlin for posing he problem and for his valuable help and Dwijavani Ahreya, Hyejin Kim and James (J.T.) Halber for heir helpful suggesions. References [1] R.F. Anderson, S.Orey, 1976, Small Random Perurbaions of Dynamical Sysems wih Reflecing Boundary, Nagoya Mah. J.,Vol 6, pp [2] C. Consanini, 1991, Diffusion approximaion for a class of ranspor processes wih physical reflecion boundary condiions, Annals of Probabiliy, 19, pp [3] C. Consanini, 1992, The Skorohod oblique reflecion problem wih applicaion o sochasic differenial equaions, Probabiliy Theory Relaed Fields, 91, pp [4] S.N. Ehier, T.G. Kurz, 1986, Markov processes: Characerizaion and Convergence, Wiley, New York. [5] M. Freidlin, 1985, Funcional Inegraion and Parial Diffferenial Equaions, Princeon Universiy Press. [6] M. Freidlin, 24, Some Remarks on he Smoluchowski-Kramers Approximaion, Journal of Saisical Physics, Vol. 117, No. 314, pp

14 [7] I.Karazas, S.E.Shreve, 1994, Brownian Moion and Sochasic Calculus, Second ediion, Springer. [8] H. Kramers, 194, Brownian Moion in a field of force and he diffusion model of chemical reacions, Physica, Vol. 7, pp [9] M. Smoluchowski, 1916, Drei Vorrage über Diffusion Brownsche and Koagulaion von Kolloideilchen, Phys. Z., Vol 17, pp [1] H. Tanaka, 1979, Sochasic Differenial Equaions wih Reflecing Boundary Condiions in Convex Regions, Hiroshima Mah. J.9,