Stochastic geometric heat equations

Abstract:

I will show that an approach from the paper Brzezniak and Ondrejat (2007) can be applied to the stochastic heat flow equation in the case when the domain is one dimensional. The one dimensionality of the domain allows us to work with the energy space, i.e. the Hilbert space H1,2(S1,Rd) as a state space since in this case the embedding of the energy space into the Banach space C(S1,Rd) of continuous functions holds. Some techniques that have been developed by the speaker in collaboration with Goldys and Jegaraj (2010) are essential. Let us point out a difference between our proof of the global existence and the one in the deterministic case by Eells-Sampson (1964) and Hamilton (1975). While in the latter papers the crucial step is to prove that the energy density solves certain scalar parabolic equation, in our case the crucial step is to prove an inequality for the L2-norm of the gradient of the solution which is based on certain geometric property of the target manifold M. Based on a joint work of the speaker with B. Goldys (UNSW) and M. Ondrejat (Mathematical Institute AS CR, Prague, Czech Republic). 

About the speaker: Zdzislaw Brzezniak is Professor at the Department of Mathematics of the University of York (UK). His main research interests are Stochastic Partial Derivative Equations, Feynman path integrals and Mathematical Finance.