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.

中文翻译：

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具有流形值的无限弦的随机热方程

在本文中，我们构造了对应于 $\mathbb{R}^+$ 或 $\mathbb{R}$ 上的随机热方程的鞅解对应的保守马尔可夫过程，其值在一般黎曼梅折中，仅假设为是完整的和随机完整的。这项工作是之前关于有限体积情况的论文 \cite{RWZZ17} 的扩展。此外，我们还获得了一些与这些马尔可夫过程相关的函数不等式。这意味着在无限体积的情况下，如果 Ricci 曲率严格为正，则解的指数遍历性和如果截面曲率为负的过程的非遍历性。