On the probabilistic solution of the Cauchy problem for parabolic equations

Abstract

The questions about finding (conditional) mathematical expectations, the joint and marginal distribution of different functionals from the trajectories of random processes, expressed through the process itself, the ordinary stochastic integral and the stochastic Ito integral (stochastic integrals are understood as integrals in the mean square sense) are among the important issues of both the theory itself random processes and its numerous applications. But it is not always possible to find (joint) distributions of the indicated functionals by direct computations, therefore, they usually resort to some methods of finding the required characteristics. One of such methods is the so-called method of differential equations, which reduces the problem of finding joint distributions of functionals from random processes to solving (connected with these functionals) partial differential equations. The aim of the present paper is to find a joint distribution of the above-mentioned types of functionals and the integrands present in the definitions of these functionals depend both on the time and on the spatial coordinates. To do this, we first derive an equation for the joint characteristic function of the functionals under consideration and show that in some special cases the determination of the Laplace transformation of the solution of this equation can be reduced to the solution of an ordinary differential equation with constant coefficients. As an application, explicit distributions of some functionals of the Wiener process are found and some of their possible applications are discussed.