NONUNIQUENESS FOR A PARABOLIC SPDE WITH 3/4-ε-HOLDER DIFFUSION COEFFICIENTS

• 成果类型: Article
• 署名作者: Mueller, Carl; Mytnik, Leonid; Perkins, Edwin
• 署名单位: University of Rochester; Technion Israel Institute of Technology; University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP870
• 发表日期: 2014
• 页码: 2032-2112
• 关键词: partial-differential-equations; inequalities; uniqueness

摘要

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) partial derivative u/partial derivative t = Delta/2 u(t, x) + vertical bar u(t, x)vertical bar(gamma) W(t, x), u(0, x) = 0 Here W is a space time white noise on R x R. More precisely, we show the above stochastic PDE has a nonzero solution for 0 < y < 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 partial derivative u/partial derivative t = Delta/2 u(t, x) + sigma (u(t, x))W(t, x) if a is Holder 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.