Studies on mild solutions of stochastic heat equations on Riemannian manifolds associated to higher-order elliptic operators

Abstract

In this thesis, we study an initial value problem of stochastic heat equation ｛_{u(0, ξ) = u0(ξ), ξ ∈ M.} ^{∂tu(t, ξ) = Hu(t, ξ) + b(ξ, u(t, ξ)) + σ(ξ, u(t, ξ)) ˙W (t, ξ), t > 0, ξ ∈ M} (1) where H is a certain 2m (m ∈ N) order elliptic operator b and σ are functions of ξ and u = {u(ξ)}_ξ∈M, W&#x0307 is formally a space-time white noise on M, and M is a compact, connected, and smooth Riemannian manifold of dimension N without boundary. We study a mild solution of stochastic heat equation on a higher dimensional Riemannian manifold. Specifically, we extend Funaki’s main theorem [4] to a higher dimensional Riemannian manifold based on Davies’s heat kernel estimate. We also show that the resulting mild solution obtained by this approach has ”nice” properties.

Date of Award	2022
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Yang XIANG (Supervisor)