Stochastic heat equations for infinite strings with values in a manifold. (English) Zbl 1475.37088

Chen, Xin; Wu, Bo; Zhu, Rongchan; Zhu, Xiangchan

Trans. Am. Math. Soc. 374, No. 1, 407-452 (2021).

The authors construct a stochastic process representing the evolution of a one- or two-sided infinite string, forced by space-time white noise, in a complete and stochastically complete manifold. Formally, this is a stochastic heat equation corresponding to the Dirichlet energy defined using the metric on the manifold. The paper continues the program begun in [M. Röckner et al., SIAM J. Math. Anal. 52, No. 3, 2237–2274 (2020; Zbl 1445.60048)], which considers the case of a compact string. The infinite volume of the domain necessitates constructing the process in weighted \(L^{2}\) spaces on the half-line or the full line. In addition to the construction, the authors establish functional inequalities providing a partial characterization of the ergodic properties of the string process in terms of the curvature of the manifold.

Reviewer: Alexander Dunlap (New York)

MSC:

37L55	Infinite-dimensional random dynamical systems; stochastic equations
37A50	Dynamical systems and their relations with probability theory and stochastic processes
35K05	Heat equation
35R60	PDEs with randomness, stochastic partial differential equations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
58J35	Heat and other parabolic equation methods for PDEs on manifolds
58J60	Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)

Keywords:

stochastic heat equation; Ricci curvature; functional inequality; quasi-regular Dirichlet form; infinite volume

Citations:

Zbl 1445.60048

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