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A forward-backward SDE from the 2D nonlinear stochastic heat equation. (English) Zbl 1487.35463

Summary: 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 \(\varepsilon > 0\) but divided by a factor of \(\sqrt{\log{\varepsilon^{-1}}} \). 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 \(\varepsilon \downarrow 0\), 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 F. Caravenna et al. [Ann. Appl. Probab. 27, No. 5, 3050–3112 (2017; Zbl 1387.82032)].

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35K15 Initial value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)

Citations:

Zbl 1387.82032
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References:

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