On the existence of invariant measure for Lagrangian velocity in compressible environments
type: Item
instance of: scholarly article
title: On the existence of invariant measure for Lagrangian velocity in compressible environments
zbMATH Open document ID: 0996.60065
DOI: 10.1023/A:1013762406825
published in: Journal of Statistical Physics
publication date: 2002-06-11T00:00:00Z
review text: A stochastic differential equation \[ dx(t) = u(t,x(t)) dt + \sqrt{2\kappa} dw, \quad x(0) = 0,\tag{1} \] in \(\mathbb R^{d}\) is considered as a model of the passive tracer motion in a turbulent flow. In (1), it is supposed that \(w\) is a standard \(d\)-dimensional Wiener process and \(u\) a stationary mixing random field independent of \(w\). Long time, large scale behaviour of solutions to (1) has been studied up to now mainly under the incompressibility hypothesis \(\text{div} u \equiv 0\). To make extensions to the compressible case possible, the authors prove that there exists a regular invariant measure for (1). More precisely: let the Wiener process \(w\) be defined on a probability space \((\Sigma,W)\), let \((\Omega, \mathcal V,P)\) be a probability space such that \(\Omega\) is a Polish space, \(\mathcal V\) the completion of its Borel \(\sigma\)-algebra, and let \((\tau_{t,x}\), \(t\in\mathbb R\), \(x\in \mathbb R^{d})\) be a group of measure preserving transformations acting on \(\Omega\). Define the random field \(u\) by \(u(t,x,\omega) = \mathbf u(\tau_{t,x}\omega)\), \(\mathbf u: \Omega\to\mathbb R^{d}\) being a centered random variable. Suppose that the trajectories of \(u\) are uniformly bounded, Hölder continuous, and have continuous bounded derivatives in the space variables. If \(t_1,\dots,t_{N}\in \mathbb R\), \(x_1,\dots, x_{N}\in\mathbb R^{d}\), \((t_{i},x_{i})\neq (t_{j},x_{j})\) for \(i\neq j\), then it is assumed that the law of the random vector \((u(t_1,x_1),\ldots,u(t_{N},x_{N}))\) is absolutely continuous with respect to Lebesgue measure on \(\mathbb R^{Nd}\). Finally, let the random field \(u\) have a finite decorrelation time: there exists a \(T>0\) such that the \(\sigma\)-algebras generated by \(u(s,x)\), \(-\infty<s\leq t\), \(x\in \mathbb R^{d}\), and by \(u(s,x)\), \(t+T\leq s<+\infty\), \(x\in \mathbb R^{d}\), respectively, are independent for all \(t\in\mathbb R\). It is shown that there exists a probability measure \(P_*\) o…
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 76F25
Mathematics Subject Classification ID: 37A50
zbMATH DE Number: 1754121
zbMATH Keywords: passive tracer
zbMATH Keywords: turbulent transport
zbMATH Keywords: stationary processes
author: Tomasz Komorowski
author: Grzegorz Krupa
MaRDI profile type: Publication
full work available at URL: a:1013762406825
OpenAlex ID: W1592395073
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description: scientific article; zbMATH DE number 1754121