Pathwise Uniqueness of the Stochastic Heat Equation with Hölder continuous o diffusion coefficient and colored noise

Abstract

We consider the stochastic heat equation in $\mathbb{R}_+ \times \mathbb{R}^q$ with multiplicative noise: \[ \partial_t u(t,x) = \frac{1}{2} \Delta u(t,x) + b(t,x,u(t,x)) + \sigma(t,x,u(t,x)) \, \dot{W}(t,x) .\] Here, $\dot{W}$ is a centered Gaussian noise which is white in time and colored in space with correlation kernel $k(x,y) \leq const ( |x-y|^{- \alpha } +1)$ for $x,y \in \mathbb{R}^q$ and $\alpha \in (0, 2 \wedge q)$: $E[\dot{W}(t,x)\dot{W}(s,y)] =\delta(s-t) k(x,y).$ Our main result states that if the noise coefficient $\sigma$ is H\"older-continuous of order $\gamma$ in the solution $u$ and satisfies $\alpha < 2(2\gamma -1),$ then the equation has a pathwise unique solution. This was conjectured by Mytnik and Perkins in 2011. Additionally, if $q =1$, we show that the compact support property holds for nonnegative solutions of the stochastic heat equation with $\sigma(t,x,u) = u^\gamma$ for all $\alpha, \gamma \in (0,1)$.