On the existence of invariant measure for Lagrangian velocity in compressible environments
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 \(P_*\) on \((\Omega,\mathcal V)\) such that the stochastic process \(\eta_{t} = u(t,x(t))\) is strictly stationary on the probability space \((\Omega\times\Sigma, P_*\otimes W)\). Moreover, the measure \(P_*\) is equivalent to \(P\) and the process \(\eta\) is ergodic in the following sense: if \(A\subseteq C(\mathbb R_+;\mathbb R^{d})\) is a Borel set such that the symmetric difference \(\{\eta_{\bullet +h}\in A \} \triangle \{\eta_\bullet \in A\}\) is a \(P_*\otimes W\)-null set for any \(h\geq 0\), then \(P_*\otimes W\{\eta_\bullet \in A\}\) is either 0 or 1. As a consequence, the existence of the Stokes drift and the effective diffusivity is established.
scientific article; zbMATH DE number 1754121
scientific article