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 Rieman...

Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical Bochner formula on M is related to lower bounds on Ricci curvature. Using this formalism, we obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and streamlined proofs of the previous characterizations of bounded Ricci curvature of the second author (arXiv:1306.6512).

Ricci Curvature and Bochner Formulas for Martingales

In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, $$L^p$$
 -inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can be used in particular to characterize general geometric flow and Ricci flow by functional inequalities.

https://orbilu.uni.lu/bitstream/10993/28890/1/Boundness-curvature_JGEA-D-16-00266R1.pdf

Characterization of Pinched Ricci Curvature by Functional Inequalities

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ with values in a general Riemannian maifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper \cite{RWZZ17} on finite volume case. 
Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution if the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative.

/pdf/stochastic-heat-equations-for-infinite-strings-with-values-19feqz55l4.pdf

Stochastic heat equations for infinite strings with values in a manifold

Let $M$ be a complete Riemannian manifold possibly with a boundary $\partial M$. For any $C^1$-vector field $Z$, by using gradient/functional inequalities of the (reflecting) diffusion process generated by $L:=\Delta+Z$, pointwise characterizations are presented for the Bakry-Emery curvature of $L$ and the second fundamental form of $\partial M$ if exists. These extend and strengthen the recent results derived by A. Naber for the uniform norm $\|{\mathbf{Ric}}_Z\|_\infty$ on manifolds without boundary. A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first named author, such that the proofs are significantly simplified.

Pointwise Characterizations of Curvature and Second Fundamental Form on Riemannian Manifolds

In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces.

Remarks on spectral gaps on the Riemannian path space

/pdf/remarks-on-spectral-gaps-on-the-riemannian-path-space-4120ih6757.pdf

Shizan Fang

Papers

Remarks on spectral gaps on the Riemannian path space

Remarks on spectral gaps on the Riemannian path space