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Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs
Probability Theory and Related Fields ( IF 2 ) Pub Date : 2020-10-26 , DOI: 10.1007/s00440-020-01012-6
Nils Dabrock , Martina Hofmanová , Matthias Röger

We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. We establish existence of martingale solutions which are strong in the PDE sense and study their large-time behavior. Our analysis is based on a viscous approximation and new global bounds, namely, an $L^{\infty}_{\omega,x,t}$ estimate for the gradient and an $L^{2}_{\omega,x,t}$ bound for the Hessian. The proof makes essential use of the delicate interplay between the deterministic mean curvature part and the stochastic perturbation, which permits to show that certain gradient-dependent energies are supermartingales. Our energy bounds in particular imply that solutions become asymptotically spatially homogeneous and approach a Brownian motion perturbed by a random constant.

中文翻译:

图的随机平均曲率流的鞅解和大时间行为的存在性

我们关心的是在任何空间维度的周期域上的图的随机平均曲率流。我们建立了 PDE 意义上很强的鞅解的存在性,并研究了它们的大时间行为。我们的分析基于粘性近似和新的全局边界,即梯度的 $L^{\infty}_{\omega,x,t}$ 估计值和 $L^{2}_{\omega, x,t}$ 绑定到 Hessian。该证明充分利用了确定性平均曲率部分和随机扰动之间微妙的相互作用,这允许证明某些依赖于梯度的能量是超鞅。我们的能量界限特别意味着解决方案在空间上渐近均匀,并接近受随机常数扰动的布朗运动。
更新日期:2020-10-26
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