Lagrangian dynamics for a passive tracer in a class of Gaussian Markovian flows.
An ordinary differential equation \(dx(t)/dt = V(t,x(t))\), \(x(0) = 0\), in \(\mathbb R^ {d}\), where \(V\) is a time-space stationary random field, is often studied in connection with modelling the transport of a passive tracer. In the paper under review, random fields \(V\) corresponding to stationary infinite-dimensional Ornstein-Uhlenbeck processes, defined by an equation \[ dV = -AV\,dt + B\,dW \tag{1} \] in a suitable Hilbert space, are considered. In (1), \(W\) is a standard cylindrical Wiener process on \(L^ 2(\mathbb R^ {d})\) and the linear operators \(A\), \(B\) are related to the spectral density of the random field \(V\). Let \(\eta \) be the Lagrangian environment process, defined formally as \(\eta (t,\cdot ) = V(t,x(t)+\cdot )\), \(t\geq 0\). The authors prove that for regular velocity fields \(V\) the process \(\eta \) satisfies an equation \[ d\eta (t) = (-A\eta (t) + \eta (t,0) \cdot \nabla _ {x}\eta (t))\,dt + B\,d\widetilde W\tag{2} \] for a certain cylindrical Wiener process \(\widetilde W\). On the other hand, it is shown that, again for \(V\) and \(f\) regular, there exists a unique local strong solution to (2) satisfying an initial condition \(\eta (0) = f\). Moreover, for \(\mu \)-almost every \(f\) the solution does not blow up in finite time, \(\mu \) denoting the invariant measure for (1). Further, irregular velocity fields which are only spatially Hölder continuous are treated. Approximations \(V_ {k}\) to \(V\) are found such that processes \(\eta _ {k}\) solving equations like (2) but corresponding to \(V_ {k}\) converge weakly on \(C(\mathbb R_ {+}; C(\mathbb R^ {d};\mathbb R^ {d}))\) to a probability measure which may be identified with the law of the Lagrangian environment process associated with an irregular velocity field.
scientific article; zbMATH DE number 2138937
scientific article