Well-posedness for singular drift SPDE with gradient dependent noise
A talk in the Bielefeld Stochastic Afternoon series by
Jonas M. Tölle
| Abstract: | We shall discuss well-posedness results for stochastic
nonlinear parabolic PDEs with singular drift and gradient Stratonovich
noise with coefficients depending on the spatial variable. The drift
term is given by a realization of a $p$-Laplace-type operator (for the
singular cases $1\le p \le 2$), including also the more general case of
non-homogeneous or multi-valued nonlinearities. For initial data in
$L^2$, we prove the unique existence of a continuous process solving the
SPDE in the sense of stochastic variational inequalities. The results
are based on geometric conditions on the spatial domain (and its
boundary symmetries) being related to the Itô-Stratonovich-corrector of
the gradient noise. By imposing a curvature-dimension condition as well
as a defective commutation condition, we obtain the higher order a
priori estimates that allow us to pass to the limit in the approximation
of the solution. The results are partially based on a joint work with
Ioana Ciotir (Normandie Université, INSA Rouen). |