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, Ẇ 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 soluti...[ Read more ]

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, Ẇ 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.