A FORWARD-BACKWARD SDE FROM THE 2D NONLINEAR STOCHASTIC HEAT EQUATION

• 成果类型: Article
• 署名作者: Dunlap, Alexander; Gu, Yu
• 署名单位: New York University; University System of Maryland; University of Maryland College Park
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1563
• 发表日期: 2022
• 页码: 1204-1253
• 关键词: kpz equation; dimensions 3; moments; limit

摘要

We consider a nonlinear stochastic heat equation in spatial dimension d = 2, forced by a white-in-time multiplicative Gaussian noise with spatial correlation lengths epsilon > 0 but divided by a factor of root log epsilon(-1). We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the weak noise regime. We show that, as epsilon down arrow 0, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras