A stochastic solution of a high order parabolic equation (Q1091043)

The paper is concerned with a stochastic solution of a parabolic partial differential equation of the form \(\partial W/\partial t=(A+B)W\) for \(t>0\) and \(x\in {\mathbb{R}}^ d\), subject to the initial condition \(W(0,x)=f(x)\). Here A and B are suitable partial differential operators (with respect to x) of order 2q for some natural number q, satisfying a condition under which \(A+B\) is a strongly parabolic operator. The stochastic solution is based on the notion of A-process, a ''Markov process'' (in a suitable sense) associated to the equation \(\partial u/\partial t=Au\) for \(t>0\) and \(x\in {\mathbb{R}}^ d.\) The author defines singular stochastic integrals of A-processes, which are related to differential operators of orders up to 2q. Using singular stochastic integrals, a Girsanov type formula is obtained, yielding a stochastic solution of the original parabolic initial value problem. Uniqueness and regularity of the stochastic solution is finally proved.