Stochastic PDES in S'

Abstract: In the usual literature, the drift coefficient of an SPDE contain second and first order differential operators, whereas the diffusion coefficient contains only first order differential operator. In this paper we try to present a general model for an SPDE which contains zero’th order terms in its drift as well as in diffusion coefficients. We aim to prove the existence and uniqueness of the following SPDE in S' (the space of tempered distributions):

\(dY_t\)= \(L(t,Y_t)dt\) + \(A(t,Y_t)\cdot dB_{t},\)

\(Y_0=\xi;\)

Where \(\xi\) is \(F_0\)-measurable, \({B_{t}}\) is standard \(d\)-dimensional \(F_t\)- Brownian motion. And we assume that non-linear operators \((L,A)\)satisfy certain `Monotonicity type inequalities'.

This is a joint work with Prof. B. Rajeev (ISI Bangalore).