THESIS
2022
1 online resource (vii, 31 pages) : illustrations
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, 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...[
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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.
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