Solutions of SPDE's associated with a stochastic flow

2017/06/20 by Bhar, Suprio, Bhaskaran, Rajeev, Sarkar, Barun
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1706.06262

Abstract

We consider the following stochastic partial differential equation, amp;dY_t=L^∗ Y_tdt+A^∗ Y_t⋅ dB_t
amp;Y₀=ψ, associated with a stochastic flow \X(t,x)\, for t ≥ 0, x ∈ ℝ^d, as in [Rajeev & Thangavelu, Probabilistic representations of solutions of the forward equations, Potential Anal. 28 (2008), no.~2, 139--162]. 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 S^′. 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.

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