Nonuniqueness for a parabolic spde with 3/4 - ε-hölder diffusion coefficients

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) ∂u/∂t = Δ/2 u(t, x) + |u(t, x)|^γ W(t,x), u(0, x) = 0. Here Ẇ is a space-time white noise on ℝ₊ × ℝ. More precisely, we show the above stochastic PDE has a nonzero solution for 0 < γ < 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 ∂u/∂t = Δ/2 u(t, x) +σ (u(t, x)) Ẇ(t,x) if σ is Hölder continuous of index γ >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 γ = 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.