A slow transient diusion in a drifted stable potential

Transcription

1 A slow transient diusion in a drifted stable potential Arvind Singh Université Paris VI Abstract We consider a diusion process X in a random potential V of the form V x = S x δx, where δ is a positive drift and S is a strictly stable process of index α (1, 2 with positive jumps Then the diusion is transient and X t / log α t converges in law towards an exponential distribution This behaviour contrasts with the case where V is a drifted Brownian motion and provides an example of a transient diusion in a random potential which is as "slow" as in the recurrent setting Keywords diusion with random potential, stable processes MSC 2 6K37, 6J6, 6F5 arvindsingh@ensfr 1 Introduction Let (V(x, x R be a two-sided stochastic process dened on some probability space (Ω, F, P We call a diusion in the random potential V a formal solution X of the SDE: dxt = dβ t 1 2 V (X t dt X =, where β is a standard Brownian motion independent of V Of course, the process V may not be dierentiable (for example when V is a Brownian motion and we should formally consider X as a diusion whose conditional generator given V is 1 d 2 ev(x dx ( e V(x d dx Such a diusion may be explicitly constructed from a Brownian motion through a random change of time and a random change of scale This class of processes has Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 175 rue du Chevaleret, 7513 Paris, France 1

2 been widely studied for the last twenty years and bears a close connection with the model of the random walk in random environment (RWRE, see [17] and [12] for a survey on RWRE and [11], [12] for the connection between the two models This model exhibits many interesting features For instance, when the potential process V is a Brownian motion, the diusion X is recurrent and Brox [2] proved that X t / log 2 t converges to a non-degenerate distribution Thus, the diusion is much "slower" than in the trivial case V = (then X is simply a Brownian motion We point out that Brox's theorem is the analogue of Sinai's famous theorem for RWRE [13] (see also [4] and [8] Just as for the RWRE, this result is a consequence of a so-called "localization phenomenon": the diusion is trapped in some valleys of its potential V Brox's theorem may also be extended to a wider class of potentials For instance, when V is a strictly stable process of index α (, 2], Schumacher [11] proved that X t log α t law t b, where b is a non-degenerate random variable, whose distribution depends on the parameters of the stable process V There is also much interest concerning the behaviour of X in the transient case When the potential is a drifted Brownian motion ie V x = B x κ x where B is a 2 two-sided Brownian motion and κ >, then the associated diusion X is transient toward + and its rate of growth is polynomial and depends on κ Precisely, Kawazu and Tanaka [7] proved that If < κ < 1, then 1 t κ X t converges in law towards a Mittag-Leer distribution of index κ If κ = 1, then log t t X t converges in probability towards 1 4 If κ > 1, then 1 t X t converges almost surely towards κ 1 4 In particular, when κ < 1, the rate of growth of X is sub-linear Rened results on the rates of convergence for this process were later obtained by Tanaka [16] and Hu et al [6] In fact, this behaviour is not specic to diusions in a drifted Brownian potential More generally, it is proved in [15] that if V is a two-sided Lévy process with no positive jumps and if there exists κ > such E[e κv 1 ] = 1, then the rate of growth of X t is linear when κ > 1 and of order t κ when < κ < 1 (see also [3] for a law of large numbers in a general Lévy potential These results are the analogues of those previously obtained by Kesten et al [9] for the discrete model of the RWRE In this paper, we study the asymptotic behaviour of a diusion in a drifted stable potential Precisely, let (S x, x R denote a two-sided càdlàg stable process with index α (1, 2 By two-sided, we mean that (a The process (S x, x is strictly stable with index α (1, 2, in particular S = (b For all x R, the process (S x+x S x, x R has the same law as S 2

3 It is well known that the Lévy measure Π of S has the form Π(dx = ( c + 1 x>} + c 1 x<} dx x α+1 (1 where c + and c are two non-negative constants such that c + +c > In particular, the process (S x, x has no positive jumps (resp no negative jumps if and only if c + = (resp c = Given δ >, we consider a diusion X in the random potential V x = S x δx Since the index α of the stable process S is larger than 1, we have E[V x ] = δx, and therefore lim V x = and lim V x = + x + x almost surely This two facts easily imply that X is transient towards + (see the beginning of Section 21 We have already mentioned that, when S has no positive jumps (ie c + =, the rate of transience of X is given in [15] and X t has polynomial growth Thus, we here assume that S possesses positive jumps Theorem 1 Assume that c + >, then X t log α t law t E ( c + where E(c + /α denotes an exponential law with parameter c + /α This result also holds with sup s t X s or inf s t X s in place of X t The asymptotic behaviour of X is in this case very dierent from the one observed when V is a drifted Brownian motion Here, the rate of growth is very slow: it is the same as in the recurrent setting We also note that neither the rate of growth nor the limiting law depend on the value of the drift parameter δ Theorem 1 has a simple heuristic explanation: the "localisation phenomenon" for the diusion X tells us that the time needed to reach a positive level x is approximatively exponentially proportional to the biggest ascending barrier of V on the interval [, x] In the case of a Brownian potential, or more generally a spectrally negative Lévy potential, the addition of a negative drift somehow "kills" the ascending barriers, thus accelerating the diusion and leading to a polynomial rate of transience However, in our setting, the biggest ascending barrier on [, x] of the stable process S is of the same order as its biggest jump on this interval Since the addition of a drift has no inuence on the jumps of the potential process, the time needed to reach level x still remains of the same order as in the recurrent case (ie when the drift is zero and yields a logarithmic rate of transience α, 3

4 2 Proof of the theorem 21 Representation of X and of its hitting times In the remainder of this paper, we indierently use the notation V x or V(x Let us rst recall the classical representation of the diusion X in the random potential V from a Brownian motion through a random change of scale and a random change of time (see [2] or [12] for details Let (B t, t denote a standard Brownian motion independent of V and let σ stand for its hitting times: σ(x = inf(t, B t = x Dene the scale function of the diusion X, A(x = x e V y dy for x R (2 Since lim x + V x /x = δ and lim x V x /x = δ almost surely, it is clear that A( = lim A(x < and lim A(x = x + x Let A -1 : (, A( R denote the inverse of A and dene T(t = t e 2V(A-1 (B s ds for t < σ(a( almost surely Similarly, let T -1 denote the inverse of T According to Brox [2] (see also [12], the diusion X in the random potential V may be represented in the form X t = A -1 ( B T -1 (t (3 It is now clear that, under our assumptions, the diusion X is transient toward + We will study X via its hitting times H dened by H(r = inf(t, X t = r for r Let (L(t, x, t, x R stand for the bi-continuous version of the local time process of B In view of (3, we can write H(r = T (σ(a(r = σ(a(r e 2V(A-1 (B s ds = A(r Making use of the change of variable x = A(y, we get where H(r = I 1 (r I 2 (r e 2V(A-1 (x L(σ(A(r, xdx e Vy L(σ(A(r, A(ydy = I 1 (r + I 2 (r (4 = = e Vy L(σ(A(r, A(ydy, e Vy L(σ(A(r, A(ydy 4

5 22 Proof of Theorem 1 Given a càdlàg process (Z t, t, we denote by t Z = Z t Z t the size of the jump at time t We also use the notation Z t to denote the largest positive jump of Z before time t, Z t = sup s Z s t Let Z # t stand for the largest ascending barrier on [, t], namely: We also dene the functionals: Z t = sup Z s s [,t] Z t = inf s [,t] Z s Z t Z # t = sup (Z y Z x x y t (running unilateral maximum (running unilateral minimum = sup Z s (running bilateral supremum s [,t] We start with a simple lemma concerning the uctuations of the potential process Lemma 1 There exist two constants c 1, c 2 > such that for all a, x > PV # x a} e c 1 x a α, (5 and whenever a x is suciently large, PV x > a} c 2 x a α (6 Proof Recall that V x = S x δx In view of the form of the density of the Lévy measure of S given in (1, we get ( ( PV # x a} PV c + x a} = exp x dy = exp c+ x a yα+1 α a α This yields (5 From the scaling property of the stable process S, we also have } PV x > a} = P x 1 α sup S t δx 1 1 α t > a P S 1 > a } δx 1 1 t [,1] x 1 α α Notice further that a/x 1/α δx 1 1/α > a/(2x 1/α whenever a/x is large enough Therefore, making use of a classical estimate concerning the tail distribution of the stable process S (cf Proposition 4, p221 of [1], we nd that PV x > a} P S 1 > a } P S 2x 1 1 > a } + P S α 2x 1 1 < a } x c α 2x 1 2 α a α 5

6 Proposition 1 There exists a constant c 3 > such that, for all r suciently large and all x, PV # r x+log 4 r} c 3 e log2 r Plog I 1 (r x} PV # r x log 4 r}+c 3 e log2 r Proof This estimate was rst proved by Hu and Shi (see Lemma 41 of [5] when the potential process is close to a standard Brownian motion A similar result is given in Proposition 32 of [14] when V is a random walk in the domain of attraction of a stable law As explained by Shi [12], the key idea is the combined use of Ray-Knight's Theorem and Laplace's method However, in our setting, additional diculties appear since the potential process is neither at on integer interval nor continuous We shall therefore give a complete proof but one can still look in [5] and [14] for additional details Recall that I 1 (r = e V y L(σ(A(r, A(ydy, where L is the local time of the Brownian motion B (independent of V Let (U(t, t denote a two-dimensional squared Bessel process starting from zero, also independent of V According to the rst Ray-Knight Theorem (cf Theorem 22 p455 of [1], for any x > the process (L(σ(x, x y, y x has the same law as (U(y, y x Therefore, making use of the scaling property of the Brownian motion and the independence of V and B, for each xed r >, the random variable I 1 (r has the same law as ( Ĩ 1 (r = A(r e V y A(r A(y U dy A(r We simply need to prove the proposition for Ĩ1 instead of I 1 In the rest of the proof, we assume that r is very large Proof of the upper bound Dene the event } E 1 = sup t (,1] U(t t log ( r 8 t According to Lemma 61 of [5], PE1} c c 4 e r/2 for some constant c 4 > On E 1, we have ( Ĩ 1 (r r e V y 8A(r (A(r A(y log dy = r r 2 e V# r ( e V z V y dz y log ( 8A(r log A(r A(y A(r A(y ( 8A(r A(r A(y dy dy 6

7 Notice also that A(r = ev z dz re V r and similarly A(r A(y (r yev r Therefore ( 8A(r r ( 8r log dy r(v r V A(r A(y r + log dy r y = r(v r V r log 8 Dene the set E 2 = V r V r e log3 r } In view of Lemma 1, PE2} c P V r > 1 } r 2 elog3 e log2 r Therefore, P(E 1 E 2 c } 2e log2 r and on E 1 E 2, Ĩ 1 (r r 3 (e log3 r log 8e V# r e log4 r+v # r This completes the proof of the upper bound Proof of the lower bound Dene by induction γ γ k+1 =, = inf(t > γ n, V t V γk 1 The sequence (γ k+1 γ k, k is iid and distributed as γ 1 = inft > : V t 1} We denote by x the integer part of x We also use the notation ɛ = e log3 r Consider the following events E 3 E 4 = γ r 2 > r }, = γ k γ k 1 2ɛ for all k = 1, 2, r 2 } With the help of Markov's inequality, we get P E c 3} = P e γ r 2 e r } e r E [ e γ r 2 ] = e r E [ e γ 1 ] r 2 e r, where we used that r is very large and that E [e γ 1 ] < 1 for the last inequality (because γ 1 is non-negative and not identically zero We also have r 2 PE4} c Pγ k γ k 1 < 2ɛ} r 2 Pγ 1 < 2ɛ} k=1 r 2 PV 2ɛ 1} e log2 r, where we used Lemma 1 for the last inequality Dene also E 5 = V x V r < 1 for all x [r 2ɛ, r]} 7

8 x + x + +ɛ V # r 1 1 γ 1 γ 2 γ 6 γ 3 γ 4 γ 5 r x x +ɛ r 2ɛ r Figure 1: Sample path of V on E 6 From time reversal, the processes (V t, t 2ɛ and (V r V (r t, t 2ɛ have the same law Thus, PE c 5} PV 2ɛ 1} e log2 r Setting E 6 = E 3 E 4 E 5, we get PE c 6} 3e log2 r Moreover, it is easy to check (see gure 1 that on E 6, we can always nd x, x + such that: x x + r 2ɛ, for any a [x, x + ɛ], V x V a 2, for any b [x +, x + + ɛ], V x+ V b 2, V x+ V x V # r 4 Let us also dene ( A(r A(y E 7 = E 6 inf U y [x,x +ɛ] A(r E 8 = V # r 3 log 2 r } A(r A(x } e 2 log2 r, A(r 8

9 We nally set E 9 = E 7 E 8 Then on E 9, we have, for all r large enough, x +ɛ ( Ĩ 1 (r A(r e V y A(r A(y U dy A(r x x +ɛ e Vx 2 2 log2 r x = e Vx 2 2 log2 r log 3 r (A(r A(x dy x e Vy dy x+ +ɛ e V x 2 2 log 2 r log 3 r x + e Vx + Vx 4 2 log2 r 2 log 3 r e V# r log 4 r e V y dy This proves the lower bound on E 9 It simply remains to show that PE9} c c 5 e log2 r According to Lemma 61 of [5], for any < a < b and any η >, we have } P inf U(t ηb 2 ( η η + 2 exp a<t<b 2(1 a/b Therefore, making use of the independence of V and U, we nd where PE9} c PE6} c + PE8} c + PE7 c E 6 E 8 } [ ] PE6} c + PE8} c + 2e log2 r + 2E e 1 2 J(re 2 log2 r 1 E6 E 8, J(r = A(r A(x A(x + ɛ A(x We have already proved that PE6} c 3e log2 r Using Lemma 1, we also check that PE8} c e log2 r Thus, it remains to show that [ ] E e 1 2 J(re 2 log2 r 1 E6 E 8 c 6 e log2 r (7 Notice that, on E 6, and also A(r A(x = Therefore, on E 6 E 8, x e Vy dy A(x + ɛ A(x = x +ɛ x+ +ɛ x + e Vy dy e log3 r+v x+ 2, x e Vy dy e log3 r+v x +2 J(r e Vx + Vx 4 e V# r 8 e V r 8 e 3 log2 r 8 which clearly yields (7 and the proof of the proposition is complete 9

10 Lemma 2 We have V # r r 1/α law r S 1 Proof Let f : [, 1] R be a deterministic càdlàg function For λ, dene We rst show that f λ (x = f(x λx lim f # λ (1 = f (1 (8 λ It is clear that f (1 = f λ (1 f # λ (1 for any λ > Thus, we simply need to prove that lim sup f # λ (1 f (1 Let η > and set so that A(η, λ B(η, λ = sup f λ (y f λ (x : x y 1 and y x η}, = sup f λ (y f λ (x : x y 1 and y x > η}, f # λ Notice that A(η, λ A(η where (1 = maxa(η, λ, B(η, λ} (9 A(η = A(η, = sup f(y f(x : x y 1 and y x η} Since f is càdlàg, we have lim η A(η = f (1 Thus, for any ε >, we can nd η > small enough such that Notice also that lim sup A(η, λ f (1 + ε (1 λ B(η, λ sup f(y f(x η λ : x y 1 and y x > η } f # (1 η λ which implies lim B(η, λ = (11 λ The combination of (9, (1 and (11 yields (8 Making use of the scaling property of the stable process S, for any xed r >, (V y, y r law = (r 1/α S y δry, y 1 Therefore, setting R(z = (S z # 1, we get the equality in law: V # r law = R(δr 1 1/α (12 r1/α Making use of (8, we see that R(z converges almost surely towards S 1 as z goes to innity Since α > 1 and δ >, we also have δr 1 1/α as r goes to innity and we conclude from (12 that V # r law r 1/α r S 1 1

11 Proof of Theorem 1 Recall that the random variable S 1 denotes the largest positive jump of S over the interval [, 1] In view of the Lévy measure of S given by (1, this random variable has a continuous density Thus, on the one hand, the combination of Proposition 1 and Lemma 2 readily shows that log(i 1 (r r 1/α law r S 1 (13 On the other hand, the random variables A( = lim x A(x and e V y dy have the same law We have already noticed that these random variables are almost surely nite Since the function L(t, is, for any xed t, continuous with compact support, we get I 2 (r = e Vy L(σ(A(r, A(ydy sup L(σ(A(, z e Vy dy < z (,] Therefore, Combining (4, (13 and (14, we deduce that sup I 2 (r < almost surely (14 r log(h(r r 1/α law r S 1 which, from the denition of the hitting times H, yields ( sup s t X α s law 1 log α t t Moreover, according to the density of the Lévy measure of S, we have ( PS c + 1 x} = exp dy = exp ( c+ yα+1 αx α x Therefore, the random variable (1/S 1 α has an exponential distribution with parameter c + /α so the proof of the theorem for sup s t X s is complete We nally use the classical argument given by Kawazu and Tanaka, p21 [7] to obtain the corresponding results for X t and inf s t X s S 1 Acknowledgments I would like to thank Yueyun Hu for his precious advices References [1] Bertoin, J (1996 Lévy processes, Cambridge Univ Press, Cambridge [2] Brox, Th (1986 A one-dimensional diusion process in a Wiener medium, Ann Probab 14, no 4,

12 [3] Carmona, P (1997 The mean velocity of a Brownian motion in a random Lévy potential, Ann Probab 25, no 4, [4] Golosov, A O (1986 Limit distributions for a random walk in a critical onedimensional random environment, Uspekhi Mat Nauk 41, no 2(248, [5] Hu, Y, Shi, Z (1998 The limits of Sinai's simple random walk in random environment, Ann Probab 26, no 4, [6] Hu, Y, Shi, Z and Yor, M (1999 Rates of convergence of diusions with drifted Brownian potentials, Trans Amer Math Soc 351, no 1, [7] Kawazu, K and Tanaka, H (1997 A diusion process in a Brownian environment with drift, J Math Soc Japan 49, no 2, [8] Kesten, H (1986 The limit distribution of Sinai's random walk in random environment, Phys A 138, no 1-2, [9] Kesten, H, Kozlov, M V and Spitzer, F (1975 A limit law for random walk in a random environment, Compositio Math 3, [1] Revuz, D and Yor, M (1999 Continuous martingales and Brownian motion, Third edition, Springer, Berlin [11] Schumacher, S (1985 Diusions with random coecients, in Particle systems, random media and large deviations (Brunswick, Maine, 1984, , Contemp Math, 41, Amer Math Soc, Providence, RI [12] Shi, Z (21 Sinai's walk via stochastic calculus, in Milieux aléatoires, 5374, Soc Math France, Paris [13] Sinai, Ya G (1982 The limiting behavior of a one-dimensional random walk in a random environment, Teor Veroyatnost i Primenen 27, no 2, [14] Singh, A (27 Limiting behavior of a diusion in an asymptotically stable environment, Ann Inst H Poincaré Probab Statist 43, no 1, [15] Singh, A (27 Rates of convergence of a transient diusion in a spectrally negative Lévy potential, to appear in Ann Probab [16] Tanaka, H (1997 Limit theorems for a Brownian motion with drift in a white noise environment, Chaos Solitons Fractals 8, no 11, [17] Zeitouni, O (24 Random walks in random environment, in Lectures on probability theory and statistics, , Lecture Notes in Math, 1837, Springer, Berlin 12