# Solutions of SPDE's associated with a stochastic flow

## TL;DR

This paper studies solutions to a class of stochastic partial differential equations linked with stochastic flows, establishing properties of solutions, their equivalence, and uniqueness under certain conditions.

## Contribution

It introduces the concept of 'local compact support' for solutions, defines mild solutions, and proves their equivalence and uniqueness in a multi Hilbertian space.

## Key findings

- Strong solutions are locally of compact support.
- Mild and strong solutions are equivalent in the space alS'.

## Abstract

We consider the following stochastic partial differential equation, \begin{align*} &dY_t=L^\ast Y_tdt+A^\ast Y_t\cdot dB_t\\ &Y_0=\psi, \end{align*} associated with a stochastic flow $\{X(t,x)\}$, for $t \geq 0$, $x \in \mathbb{R}^d$, as in [Rajeev \& Thangavelu, \emph{{Probabilistic representations of solutions of the forward equations}}, Potential Anal. \textbf{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 $\mathcal{S}^\prime$. We show uniqueness of solutions in the case when $\psi$ is smooth via the `monotonicity inequality' for $(L^\ast,A^\ast)$, which is a known criterion for uniqueness.