Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\epsilon$-Hölder diffusion coefficients

Abstract

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\fracΔ{2}u(t,x) +\bigl|u(t,x)\bigr|^γ\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb{R}}_{+}\times \mathbb{R}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<\gamma <3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab. Theory Related Fields 149 (2011) 1-96] for SPDE's by establishing pathwise uniqueness of solutions to \[\frac{\partial u}{\partial t}=\fracΔ{2}u(t,x)+σ\bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $\sigma$ is Hölder continuous of index $\gamma >3/4$. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDE's is therefore similar to their finite dimensional counterparts, but with the index $3/4$ in place of $1/2$. The case $\gamma =1/2$ of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks.

Published in at http://dx.doi.org/10.1214/13-AOP870 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case
• Non-uniqueness for non-negative solutions of parabolic stochastic partial differential equations

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