Analysis Of Some Deterministic and Stochastic Evolution Equations With Solutions Taking Values In An Infinite Dimensional Hilbert Manifold

The objective of this thesis is threefold:

Firstly, to deal with the deterministic problem consisting of non-linear heat equation of gradient type. It comes out as projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearly of degree 2n-1, onto the tangent space of a sphere M in a Hilbert space H. We are going to deal with questions of the existence and the uniqueness of a global solution, and the invariance of manifold M i.e. if the suitable initial data lives on M then all trajectories of solutions also belong to M.

Secondly, to generalize the deterministic model to its stochastic version i.e. stochastic non-linear heat equation driven by the noise of Stratonovich type. We are going to show that if the suitable initial data belongs to manifold $M$ then M-valued unique global solution to the generalized stochastic model exists.

Thirdly, to investigate the small noise asymptotics of the stochastic model. A Freidlin-Wentzell large deviation principle is established for the laws of solutions of stochastic heat equation on Hilbert manifold.