2018/02/21 by Barbu, Viorel, Röckner, Michael
#47J05 #FOS: Mathematics #Primary 60H15 #Probability (math.PR) #Secondary 47H05
paper · doi:10.48550/arxiv.1802.07533
One introduces a new variational concept of solution for the stochastic differential equation dX+A(t)X dt+λX dt=X dW, t∈(0,T); X(0)=x in a real Hilbert space where A(t)=∂φ(t), t∈(0,T), is a maximal monotone subpotential operator in H while W is a Wiener process in H on a probability space \Ω,F,ℙ\. In this new context, the solution X=X(t,x) exists for each x∈ H, is unique, and depends continuously on x. This functional scheme applies to a general class of stochastic PDE not covered by the classical variational existence theory ([15], [16], [17]) and, in particular, to stochastic variational inequalities and parabolic stochastic equations with general monotone nonlinearities with low or superfast growth to +∞.