Weak solutions and invariant measures of stochastic Oldroyd-B type model driven by jump noise

Abstract

In this work, we consider sub-critical and critical models for viscoelastic flows driven by pure jump Lévy noise. Due to the elastic property, the noise in the equation for the stress tensor is considered in the Marcus canonical form. We investigate existence of a weak martingale solution for stochastic Oldroyd-B models, with full dissipation in whole of Rd,d=2,3. The key ingredients of the proof are classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Pathwise uniqueness, existence of a strong solution and uniqueness in law for the two-dimensional model are also shown. We also prove, in a Poincaré domain in two-dimensions, existence of an invariant measure using bw-Feller property of the associated Markov semigroup.