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arXiv 1706.06262math.PR
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与随机流相关的SPDE的解

Solutions of SPDE's associated with a stochastic flow

Suprio Bhar, Rajeev Bhaskaran, Barun Sarkar

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摘要：

我们考虑与随机流$\{X(t,x)\}$相关联的以下随机偏微分方程\begin{align*} &dY_t=L^\ast Y_tdt+A^\ast Y_t\cdot dB_t\\ &Y_0=ψ, \end{align*}，对于$t \geq 0$，$x \in \mathbb{R}^d$，如[Rajeev \& Thangavelu，\emph{Probabilistic representations of solutions of the forward equations}，Potential Anal. \textbf{28}（2008年），编号~ 2，139- 162]。我们证明了强解的构造有“局部紧支持”。利用这一概念，我们定义了上述方程的温和解，并证明了在多Hilbert空间$\mathcal{S}^\prime$中强解与温和解的等价性。我们显示的情况下，$ψ$是光滑的“单调性不等式”的$(L^\ast,A^\ast)$，这是一个已知的唯一性标准的解决方案的唯一性。

英文摘要：

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=ψ, \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 $ψ$ is smooth via the `monotonicity inequality' for $(L^\ast,A^\ast)$, which is a known criterion for uniqueness.

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