Stochastic PDEs in S′ for SDEs driven by L ́evy noise and SPDEs associated with a stochastic flow

Abstract: In this talk we will show that, a finite dimensional stochastic differential equation driven by a L ́evy process can be formulated as a stochastic partial differential equation (SPDE) driven by the same L ́evy process. We show existence of solutions to these SPDEs, as in the diffusion case, via the Ito formula for translations of the initial condition by a finite dimensional semi-martingale. We prove the uniqueness of these SPDEs by using the `Monotonicity inequality', proved earlier in the diffusion case. As a consequence, the solutions that we construct have the `translation invariance' property. 

In the second part, we will discuss about the solutions of SPDEs associated with a stochastic flow. We define strong as well as mild solutions of the corresponding SPDE and study their equivalence in the multi Hilbertian space \(\mathcal{S}^\prime\).

This is a joint work with Rajeev Bhaskaran (Indian Statistical Institute, Bangalore Centre) and Suprio Bhar (Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur).