Nonuniqueness for a parabolic SPDE with $3/4 -ε$-Hölder diffusion coefficients

Details Nonuniqueness for a parabolic SPDE with $3/4 -ε$-Hölder diffusion coefficients Nonuniqueness for a parabolic SPDE with $3/4 -ε$-Hölder diffusion coefficients

2012-01-13

arxiv.org/abs/1201.2767v1

Mathematics

Probability

74 pages

Scientific paper

Motivated by Girsanov's nonuniqueness examples for SDE's, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) $$ \frac{\partial u}{\partial t}= 1/2\Delta u(t,x) + |u(t,x)|^\gamma \dot{W}(t,x), u(0,x) = 0.$$ Here $\dot W$ is a space-time white noise on $\R_+\times\R$. More precisely, we show the above stochastic PDE has a non-zero 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 SDE's was recently shown in [MP11] for SPDE's by establishing pathwise uniqueness of solutions to $$\frac{\partial u}{\partial t}=1/2\Delta u(t,x) + \sigma (u(t,x))\dot{W}(t,x)$$ if $\sigma$ is H\"older 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 $\gamma=1/2$ 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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