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.