On the sector condition and homogenization of diffusions with a Gaussian drift (original) (raw)
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Forum Mathematicum, 2000
A stochastic dynamics (X(t)) t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ of all locally finite subsets (configurations) in R d and which has a Gibbs measure µ as an invariant measure. We assume that µ corresponds to a symmetric pair potential φ(x − y). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics-the so-called gradient stochastic dynamics, or interacting Brownian particles-has been investigated. By using the theory of Dirichlet forms from [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form E Γ µ on L 2 (Γ; µ), and under general conditions on the potential φ, prove its closability. For a potential φ having a "weak" singularity at zero, we also write down an explicit form of the generator of E Γ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form E Γ µ , there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, ∞), D ′ ), where D ′ is the dual space of D:=C ∞ 0 (R d ).
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We study a diffusion with a random, time dependent drift. We prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition. This result generalizes the results of [13, 7]. ޒ d V(t, x; ω) • ∇ x ϕ(x) dx = 0 P − a.s., with P the underlying probability measure. In order to guarantee the existence of the solution of (1.1) we will also assume that V(t, x; ω) is (P −a.s.) locally Lipschitz in x. We are interested in proving an invariance principle for x(t), i.e. the convergence in distribution of the process εx(ε −2 t) to a Brownian motion with a certain co-variance matrix, sometimes referred to as the effective diffusivity, D ≥ 2I. This problem has been widely studied under various conditions on the random flow. Typically the flow is assumed to be the divergence of a stationary random anti-symmetric matrix valued field H(t, x; ω) = {H p,q (t, x; ω)}-the so-called stream matrix: V(t, x; ω) = ∇ x • H(t, x; ω) .
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A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals. Classification (1991): 60K35, 60J60, 60J15 is one-sided bounded. We show that clustering is universal in the diffusion coefficient. This had been conjectured in Cox, Greven and Shiga [CGS95a] (see also Shiga [Shi92]). On the way, we obtain a new proof, in the case where the state space of a component is compact, based on the interacting Fisher-Wright diffusion where a well-known duality is available.
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