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Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-05-05 , DOI: 10.1137/18m1211076
Michael Röckner , Bo Wu , Rongchan Zhu , Xiangchan Zhu

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2237-2274, January 2020.
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits the Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson--Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bound of the Ricci curvature are presented related to the stochastic heat equation.


中文翻译:

通过Dirichlet形式在流形中具有值的随机热方程

SIAM数学分析杂志,第52卷,第3期,第2237-2274页,2020
年1月。在本文中,我们证明了随机热方程的mar解的存在性,该方程采用黎曼流形中的值,该流形接受了维纳(布朗桥) )使用适当的Dirichlet形式在黎曼路径(回路)空间上进行度量作为不变度量。使用Andersson-Driver逼近法,我们启发式地推导出由Dirichlet形式给出的过程求解的方程式。此外,我们在路径空间中建立了Dirichlet形式的log-Sobolev不等式。此外,还提出了有关Ricci曲率下界的一些与随机热方程有关的特征。
更新日期:2020-06-30
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