Existence, uniqueness and regularity for stochastic evolution equations with irregular initial values
Preprint, 2014
We consider stochastic evolution equations (SEEs) of parabolic type in Hilbert space
with smooth coefficients, driven by multiplicative, not necessarily trace class, Gaussian noise.
We present the notion of extended transition semigroups for such equations
and we show, under suitable assumptions, that the extended transition semigroup is a solution
to the Kolmogorov equation in infinite dimensions.
In addition, Fr\'{e}chet differentiability
of the extended
transition semigroup
in negative order spaces
is established.
The order of smoothness is the same as the order of
smoothes of the coefficients of the corresponding equation.
In order to define the extended transition semigroup, stochastic evolution equations with irregular
initial values and initial singularities in the coefficients are investigated
and an abstract existence and uniqueness result for such equations is presented.
Stochastic evolution equations
Kolmogorov equations in infinite dimensions