Generalized solution of some parabolic equations with a random drift (Q1124211)

The author considers fundamental solutions of parabolic stochastic partial differential equations of the form \[ (1)\quad du=2^{-1}\nu (\partial^ 2u/\partial x^ 2)dt-(\partial u/\partial x)d\eta_ t(x),\quad u_ 0=\delta_ x, \] where \(\eta_ t(x):=\int^{t}_{0}\sigma (x,s)dB_ s\), \(B_.\) a scalar Wiener process and \(\sigma\) measurable and bounded. It is shown that (1) admits a unique solution in the space of generalized Brownian functionals (multiple Wiener integrals with singular kernels). In the spatially homogeneous case (\(\sigma\) (x,s)\(\equiv \sigma (s))\) the solution \(u^ 0\) of the Stratonovich version of (1) for \(\nu =0\) is given by Donskers delta function. If \(\nu >0\) the solution \(u^{\nu}\) is a regular \((L^ 2\)-) Brownian functional and it is shown that \(u^{\nu}\to u^ 0\) (as \(\nu\) \(\downarrow 0)\).