Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

Michael Röckner; Bo Wu; Rongchan Zhu; Xiangchan Zhu

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arXiv · 1711.09570

Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

Michael Röckner·Bo Wu·Rongchan Zhu·Xiangchan Zhu

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Abstract

In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits 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 bounds of the Ricci curvature are presented related to the stochastic heat equation.

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BibTeX RIS

Michael Röckner, Bo Wu, Rongchan Zhu, Xiangchan Zhu. 2017-11-27. Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms. https://arxiv.org/abs/1711.09570

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