Pathwise uniqueness for a SPDE with H\"older continuous coefficient driven by \alpha-stable noise

TL;DR

This paper proves pathwise uniqueness for a class of SPDEs driven by lpha-stable noise with Hf6lder continuous coefficients, extending previous results and establishing local Hf6lder continuity of solutions.

Contribution

It establishes pathwise uniqueness for certain lpha-stable driven SPDEs with Hf6lder coefficients, solving an open problem for specific parameter ranges.

Findings

Pathwise uniqueness proven for lpha-stable noise with f6lder coefficients.

Results apply to super-Brownian motion with lpha-stable branching.

Local Hf6lder continuity of solutions established.

Abstract

In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H\"older continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} \Delta X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $\alpha$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $\beta$-H\"older continuous for $1<\alpha<2$ and $0<\beta<1$. For $G\equiv0$ and $H(x)=x^\beta$, in Mytnik (2002) a weak solution to the above SPDE was constructedand the pathwise uniqueness of the solution was left as an open problem. In this paper we give an affirmative answer to this problem for certain values of $\alpha$ and $\beta$. In particular, for $\alpha\beta=1$, where the solution to the above equation is the density of a super-Brownian motion with $\alpha$-stable branching (see also Mytnik (2002)), our result leads to its pathwise uniqueness for $1<\alpha<\sqrt{5}-1$. The local H\"older continuity of the solution is also obtained in this paper for fixed time $t>0$.