Pathwise uniqueness for an SPDE with Hölder continuous coefficient driven by $\alpha $-stable noise

Abstract

In this paper we study the pathwise uniqueness of nonnegative solution to the following stochastic partial differential equation with Hölder continuous noise coefficient: \[ \frac{\partial X_t(x)} {\partial t}=\frac{1} {2} \Delta X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L} _t(x),\quad t>0, ~x\in \mathbb{R} , \] where for $1<\alpha <2$ and $0<\beta <1$, $\dot{L} $ denotes an $\alpha $-stable white noise on $\mathbb{R} _+\times \mathbb{R} $ without negative jumps, $G$ satisfies a condition weaker than Lipschitz and $H$ is nondecreasing and $\beta $-Hölder continuous.

For $G\equiv 0$ and $H(x)=x^\beta $, a weak solution to the above stochastic heat equation was constructed in Mytnik (2002) and the pathwise uniqueness of the nonnegative 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 $\alpha \beta =1$ the solution to the above equation is the density of a super-Brownian motion with $\alpha $-stable branching (see Mytnik (2002)) and our result leads to its pathwise uniqueness for $1<\alpha <\sqrt{5} -1$.

The local Hölder continuity of the solution is also obtained in this paper for fixed time $t>0$.