Lagrangian dynamics for a passive tracer in a class of Gaussian Markovian flows

We formulate a stochastic differential equation describing the Lagrangian environment process of a passive tracer in Ornstein–Uhlenbeck velocity fields. We subsequently prove a local existence and uniqueness result when the velocity field is regular. When the Ornstein–Uhlenbeck velocity field is only spatially Hölder continuous we construct and identify the probability law for the Lagranging process under a condition on the time correlation function and the Hölder exponent.