Phase transitions for the long-time behavior of interacting diffusions

• 成果类型: Article
• 署名作者: Greven, A.; Den Hollander, F.
• 署名单位: University of Erlangen Nuremberg; Leiden University; Leiden University - Excl LUMC
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000001060
• 发表日期: 2007
• 页码: 1250-1306
• 关键词: ergodic-theorems; nonlinear transformation; attracting orbit; infinite systems; RENORMALIZATION; decay

摘要

Let ({X-i(t)}(i is an element of Z)(d))(t >= 0) be the system of interacting diffusions on [0, infinity) defined by the following collection of coupled stochastic differential equations: dX(i) (t) = Sigma(d)(j is an element of Z) a(i, j)[X-j(t) - X-i(t)]dt + root bX(i)(t)(2)dW(i)(t), i is an element of Z(d), t >= 0. Here, a(., .) is an irreducible random walk transition kernel on Z(d) x Z(d), b is an element of (0, infinity) is a diffusion parameter, and ({W-i(t)}(i is an element of Z)(d))(t >= 0) is a collection of independent standard Brownian motions on R. The initial condition is chosen such that {X-i(0)}(i is an element of Z)(d) is a shift-invariant and shift-ergodic random field on [0, infinity) with mean Theta is an element of (0, infinity) (the evolution preserves the mean). We show that the long-time behavior of this system is the result of a delicate interplay between a(., .) and b, in contrast to systems where the diffusion function is subquadratic. In particular, let (a) over cap (i, j) = 1/2[a(i, j) + a(j, i)], i, j is an element of Z(d), denote the symmetrized transition kernel. We show that: (A) If (a) over cap(., .) is recurrent, then for any b > 0 the system locally dies out. (B) If (a) over cap(., .) is transient, then there exist b(*) >= b(2) > 0 such that: (B1) The system converges to an equilibrium v(Theta) (with mean Theta) if 0 < b < b(*). (B2) The system locally dies out if b > b(*). (B3) v(Theta) has a finite 2nd moment if and only if 0 < b < b(2). (B4) The 2nd moment diverges exponentially fast if and only if b > b(2). The equilibrium v(Theta) is shown to be associated and mixing for all 0 < b < b(*). We argue in favor of the conjecture that b(*) > b(2). We further conjecture that the system locally dies out at b = b(*). For the case where a(., .) is symmetric and transient we further show that: (C) There exists a sequence b(2) >= b(3) >= b(4) >=... > 0 such that: (Cl) v(Theta) has a finite mth moment if and only if 0 < b < b(m). (C2) The mth moment diverges exponentially fast if and only if b > b(m). (C3) b(2) < (m - 1)b(m) < 2. (C4) lim(m ->infinity)(m - 1)b(m) = c = sup(m >= 2)(m - 1)b(m). The proof of these results is based on self-duality and on a representation formula through which the moments of the components are related to exponential moments of the collision local time of random walks. Via large deviation theory, the latter lead to variational expressions for b(*) and the b(m)'s, from which sharp bounds are deduced. The critical value b(*) arises from a stochastic representation of the Palm distribution of the system. The special case where a(., .) is simple random walk is commonly referred to as the parabolic Anderson model with Brownian noise.