论文标题
来自2D非线性随机热方程的前向后SDE
A forward-backward SDE from the 2D nonlinear stochastic heat equation
论文作者
论文摘要
我们考虑空间尺寸中的非线性随机热方程$ d = 2 $，由空间相关长度$ \ VAREPSILON> 0 $强迫，但除以$ \ sqrt {\ log log \ log \ \ varepsilon^{ - 1}} $。我们对非线性的Lipschitz常数施加条件，以使问题处于“弱噪声”状态。我们表明，作为$ \ varepsilon \ downarrow0 $，解决方案的单点分布收敛，其限制以溶液的特征为前向后返回的随机微分方程（FBSDE）。我们还表征了解决方案的限制多点统计，当以相似的术语选择适当的尺度上选择点时。即使对于线性案例，我们的方法也是新的，在线性情况下，可以明确解决FBSDE，并恢复CARAVENNA，SUN和ZYGOURAS的结果（Ann。Appl。prob。27（5）：3050---3112，2017）。
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 $\varepsilon\downarrow0$, 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).
