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September 2014 Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon $-Hölder diffusion coefficients
Carl Mueller, Leonid Mytnik, Edwin Perkins
Ann. Probab. 42(5): 2032-2112 (September 2014). DOI: 10.1214/13-AOP870


Motivated by Girsanov’s nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\bigl|u(t,x)\bigr|^{\gamma}\dot{W}(t,x),\qquad u(0,x)=0.\] 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 \[\frac{\partial u}{\partial t}=\frac{\Delta}{2}u(t,x)+\sigma \bigl(u(t,x)\bigr)\dot{W}(t,x)\] 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.


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Carl Mueller. Leonid Mytnik. Edwin Perkins. "Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon $-Hölder diffusion coefficients." Ann. Probab. 42 (5) 2032 - 2112, September 2014.


Published: September 2014
First available in Project Euclid: 29 August 2014

zbMATH: 1301.60080
MathSciNet: MR3262498
Digital Object Identifier: 10.1214/13-AOP870

Primary: 60H15
Secondary: 35K05 , 35R60

Keywords: heat equation , Stochastic partial differential equations , White noise

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 5 • September 2014
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