A forward-backward SDE from the 2D nonlinear stochastic heat equation

Abstract

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $\epsilon >0$ but divided by a factor of $\sqrt{\log {\epsilon }^{-1}}$. 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 $\epsilon \downarrow 0$, 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 (Ann. Appl. Probab. 27(5):3050--3112, 2017).

arXiv comment49 pages; accepted version, to appear in Annals of Probability

References in corpus (10)

• A theory of regularity structures
• On well-posedness of forward-backward SDEs-A unified approach
• The intermediate disorder regime for directed polymers in dimension $1+1$
• The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
• A Wong-Zakai theorem for stochastic PDEs
• The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher
• On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window
• Moments of the 2D SHE at criticality
• Subsequential scaling limits for Liouville graph distance
• The random heat equation in dimensions three and higher: the homogenization viewpoint

Cited by in corpus (5)

• A quenched local limit theorem for stochastic flows
• Gaussian Limits for Subcritical Chaos
• The Allen-Cahn equation with weakly critical random initial datum
• Mesoscopic averaging of the two-dimensional KPZ equation
• Equivalence of fluctuations of discretized SHE and KPZ equations in the subcritical weak disorder regime