Invariant measures for passive tracer dynamics in Ornstein-Uhlenbeck flows. (Q2574567)

Let \(V:\mathbb R\times \mathbb R^ {d}\times \Omega \to \mathbb R^ {d}\) be a zero-mean Gaussian time-space stationary random field over a probability space \((\Omega ,\mathfrak F,\mathbf P)\), \(W\) a standard \(d\)-dimensional Wiener process defined on another probability space \((\varSigma ,\mathfrak S,\mathbf W)\), and let \(\kappa \geq 0\). Let \(x\) be the solution to a stochastic differential equation with a random drift \(dx(t) = V(t,x(t))\,dt + (2\kappa )^ {1/2}\,dW(t)\), \(x(0) = 0\), which is often used in modelling turbulent transport of a passive tracer. In the paper under review, stationarity of the Lagrangian velocity process \(\eta (t) = V(t,x(t))\) defined over the product probability space \((\Omega \times \varSigma ,\mathfrak F\otimes \mathfrak S, \mathbf P\otimes \mathbf W)\) is studied. Suppose that the covariance matrix of the field \(V\) is given by \[ \int _ \Omega V(t,x)\otimes V(s,y)\,d\mathbf P = \int _ {\mathbb R^ {d}} \cos ((x-y)\cdot k)\exp (-r(k)| t-s| )\widehat \Gamma (k)\,dk, \] where \(\widehat \Gamma \) is an even Borel function with values in the set of all real symmetric positive \(d\times d\) matrices, \(\int _ {\mathbb R^ {d}} (1+| k| ^ 2)^ {m}\widehat \Gamma (k)\,dk <\infty \) for all \(m\geq 0\), and \(r\) is an even continuous function such that \(r\geq a\) on \(\mathbb R^ {d}\) for some constant \(a>0\). It is proven that if \(\kappa >0\) and \(a\) is sufficiently large, then there exists a measure \(\mathbf P_ *\) on \((\Omega \times \varSigma , \mathfrak F\otimes \mathfrak S)\) equivalent to \(\mathbf P\otimes \mathbf W\) and such that the process \((\eta (t))_ {t\geq 0}\) is stationary and ergodic under \(\mathbf P_ *\). If \(\kappa =0\) and \(a\) is sufficiently large, then a measure \(\mathbf Q_ *\) on \((\Omega ,\mathfrak F)\) may be found such that \(\mathbf Q_ * \ll \mathbf P\) and \(\eta \) is stationary under \(\mathbf Q_ *\); moreover, it is shown that \(\mathbf Q_ *\) is ergodic under an additional hypothesis that \(\widehat \Gamma \) has a compact support.