Stochastic Characterization of Harmonic maps on Riemannian polyhedra

A stochastic approach to the harmonic map heat flow on manifolds with time-dependent Riemannian metric

We first prove stochastic representation formulae for space-time harmonic mappings defined on manifolds with evolving Riemannian metric. We then apply these formulae to derive Liouville type theorems under appropriate curvature conditions. Space-time harmonic mappings which are defined globally in time correspond to ancient solutions to the harmonic map heat flow. As corollaries, we establish triviality of such ancient solutions in a variety of different situations.

A probabilistic representation for heat flow of harmonic map on manifolds with time-dependent Riemannian metric

In this paper we will give a probabilistic representation for the heat flow of harmonic map with time-dependent Riemannian metric via a forward-backward stochastic differential equation on manifolds. Moreover, we can provide an alternative stochastic method for the proof of existence of a unique local solution for heat flow of harmonic map with time-dependent Riemannian metric.

Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds.

黎曼子流形上不连续鞅的随机雅可比场

Stochastic Jacobi fields along discontinuous martingales on Riemannian submanifolds

本文针对欧几里得空间子流形切丛上的不连续鞅，建立切丛上不连续半鞅的伊藤微积分，推导调和映射的导数相关性质及均值、导数公式，拓展了随机分析与几何的相关理论。

An equivalence between harmonic sections and sections that are harmonic maps

Let $π:(E,\nabla^{E}) \to (M,g)$ be an affine submersion with horizontal distribution, where $\nabla^{E}$ is a symmetric connection and $M$ is a Riemannian manifold. Let $σ$ be a section of $π$, namely, $π\circ σ= Id_{M}$. It is possible to study the harmonic property of section $σ$ in two ways. First, we see $σ$ as a harmonic map. Second, we see $σ$ as harmonic section. In the Riemannian context, it means that $σ$ is a critical point of the vertical functional energy. Our main goal is to find conditions to the assertion: $σ$ is a harmonic map if and only if $σ$ is a harmonic section.

非正$m$下非负$(m, V)$-Ricci曲率条件下的$V$-调和映射刘维尔定理

Liouville theorem for $V$-harmonic maps under non-negative $(m, V)$-Ricci curvature for non-positive $m$

本文证明了在非负$(m, V)$-Ricci曲率条件下多种增长$V$-调和映射的刘维尔定理，推广了Cheng、Hildebrandt-Jost-Wideman和Choi等人的经典结果，并建立了刘维尔性质与$Δ_V$-扩散过程常返性的联系。