The Random Walk Representation for Interacting Diffusion Processes

Abstract

Summary. We investigate a system of lcal interacting diffusion processes with attractive interaction. We show how the random walk representation can be used to express the gradient of the semigroup and to estimates for the time-space correlations. In particular we can answer questions dealing with localization, convergence rates to equilibrium and aging properies of the system.

Persistence Probability for a Class of Gaussian Processes Related to Random Interface Models

“…The term φ t (x) corresponds to the height of the interface at position x ∈ Z d and time t ≥ 0. In particular, we have a random walk representation of space-time correlations of this model (see [8], [12,Proposition 1.3]). Since the energy of the interface φ is determine by its height differences, this model is called the ∇φ interface model and its studies have been active in both of static and dynamic aspects (see [14] and references therein).…”

Persistence Probability for a Class of Gaussian Processes Related to Random Interface Models

“…By standard approximation arguments (1.2) has a unique strong solution. In particular, we have a random walk representation of space-time correlations of this model (see [8], [12,Proposition 1.3]). By these facts, the following holds.…”

Aging for interacting diffusion processes

“…Note that (1.7) follows from a covariance inequality of the type 8) in case c ν (i, j ) = c ν (j, i) 0 are such that sup i k c ν (i, k) ≡ C ν < ∞. Also whenever a measure ν satisfies the FKG property, then (1.8) holds with c ν (i, k) = √ 3 cov ν (x(i), x(k)), cf.…”

“…We provide here a short overview of the random walk representation, cf. [9] or [8], see also [16,18], on which most of our proofs are based. To this end, we denote by (P t , t 0) the semi-group for the SDS (1.1) and by L its generator.…”

“…It is not hard to show that C 1 2 (E r ) is invariant under the semi-group (P t , t 0), and by (1.7) it is also a subset of L 2 (ν). Therefore, while the random walk representation and in particular the decomposition formula (2.3) are derived in [8] only for local functions f, g ∈ C 1 1 (E r ), one can adapt the proof so it applies for all functions in C 1 2 (E r ).…”