May 2022 A forward-backward SDE from the 2D nonlinear stochastic heat equation
Alexander Dunlap, Yu Gu
Author Affiliations +
Ann. Probab. 50(3): 1204-1253 (May 2022). DOI: 10.1214/21-AOP1563


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 ε>0 but divided by a factor of logε1. 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 ε0, 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 (2017) 3050–3112).

Funding Statement

The first author was partially supported by the NSF Mathematical Sciences Postdoctoral Fellowship program under Grant no. DMS-2002118, and by NSF Grant no. DMS-1910023 and BSF Grant no. 2014302 (both awarded to Lenya Ryzhik). The second author was partially supported by the NSF through Grants no. DMS-1907928 and CAREER-2042384.


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Alexander Dunlap. Yu Gu. "A forward-backward SDE from the 2D nonlinear stochastic heat equation." Ann. Probab. 50 (3) 1204 - 1253, May 2022.


Received: 1 October 2020; Revised: 1 October 2021; Published: May 2022
First available in Project Euclid: 27 April 2022

MathSciNet: MR4413215
zbMATH: 1487.35463
Digital Object Identifier: 10.1214/21-AOP1563

Primary: 60H15
Secondary: 35R60 , 60H10

Keywords: forward-backward stochastic differential equation , Stochastic heat equation

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • May 2022
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