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Solutions of SPDE’s Associated with a Stochastic Flow

  • Suprio Bhar
  • Rajeev BhaskaranEmail author
  • Barun Sarkar
Article
  • 6 Downloads

Abstract

We consider the following stochastic partial differential equation,
$$\begin{array}{@{}rcl@{}} &&dY_{t} = L^{\ast} Y_{t} dt + A^{\ast} Y_{t} \cdot dB_{t}\\ &&Y_{0} = \psi, \end{array} $$
associated with a stochastic flow {X(t,x)}, for t ≥ 0, \(x \in \mathbb {R}^{d}\), as in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162, 2008). We show that the strong solutions constructed there are ‘locally of compact support’. Using this notion,we define the mild solutions of the above equation and show the equivalence between strong and mild solutions in the multi Hilbertian space \(\mathscr{S}^{\prime }\). We show uniqueness of solutions in the case when ψ is smooth via the ‘monotonicity inequality’ for (L,A), which is a known criterion for uniqueness.

Keywords

\(\mathscr{E}^{\prime }\) valued process Hermite-Sobolev space Mild solution Strong solution Monotonicity inequality \(\mathscr{S}^{\prime }\) valued processes locally of compact support Stochastic flow Martingale representation 

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Notes

Acknowledgements

The first author was a Post Doctoral Fellow at Tata Institute of Fundamental Research, Centre For Applicable Mathematics, Bangalore, India (Post Bag No 6503, GKVK Post Office, Sharada Nagar, Chikkabommsandra, Bangalore 560065, India) at the time of the work. The second author would like to thank P.Fitzsimmons for some discussions relating to mild solutions of SPDE’s. He would also like to thank V.Mandrekar and L.Gawarecki for discussions relating to the notion of distribution valued processes that are ‘locally of compact support’.

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIndian Institute of Technology KanpurKalyanpurIndia
  2. 2.Indian Statistical InstituteBangaloreIndia

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