close this window
An operatorial approach to SPDE with linear multiplicative noise
first name:
An operatorial approach to SPDE with
linear multiplicative noise

Michael Röckner (University of Bielefeld)

joint work with:
Viorel Barbu (Romanian Academy, Iasi)

In this talk, we develop a new general approach to the existence and uniqueness theory of in finite dimensional stochastic equations of the form

dX(t) + A(t,X(t))dt = X(t)dW(t) in (0,T)xH,

where A is a (random) time-dependent nonlinear monotone and demicontinuous operator from V to V', coercive and with polynomial growth. V is a reflexive Banach space continuously and densely embedded in a Hilbert space H of (generalized) functions on a Euclidean domain and V' is the dual of V in the duality induced by H as pivot space. Furthermore, W is a Wiener process in H. The new approach is based on an operatorial reformulation of the stochastic equation which is quite robust under perturbation of A. This leads to new existence and uniqueness results for a larger class of equations with linear multiplicative noise than the one treatable by the known approaches. In addition, we obtain regularity results for the solutions with respect to both the time and spatial variable which are sharper than the classical
ones. New applications include SPDE, as eg. stochastic transport equations.