An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
In this talk, we develop a new general approach to the existence and unique-ness theory of infinite dimensional stochastic equations of the form $dX(t)+A(t,X(t))dt=X(t)dW(t)$ in $(0,T)\times H$, where $A$ is a time-dependent nonlinear monotone and demicontinuous operator from $V$ to $V'$, coercive and with poly-nomial growth. Here, $V$ is a reflexive Banach space continuously and densely embedded in a Hilbert space $H$ of (generalized) functions on a domain $\Omega \subset \mathbb R^d$ 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 stochastic partial differential equations, as e.g. stochastic transport equations. http://cvgmt.sns.it/seminar/384/
When
Tue Mar 11, 2014 2pm – 3pm Coordinated Universal Time
