A. Yu. Pilipenko

NONLINEAR TRANSFORMATIONS OF SMOOTH MEASURES ON INFINITE-DIMENSIONAL SPACES

We investigate the properties of the image of a differentiable measure on an infinitely-dimensional Banach space under nonlinear transformations of the space. We prove a general result concerning the absolute continuity of this image with respect to the initial measure and obtain a formula for density similar to the Ramer–Kusuoka formula for the transformations of the Gaussian measure. We prove the absolute continuity of the image for classes of transformations that possess additional structural properties, namely, for adapted and monotone transformations, as well as for transformations generated by a differential flow. The latter are used for the realization of the method of characteristics for the solution of infinite-dimensional first-order partial differential equations and linear equations with an extended stochastic integral with respect to the given measure.

On the properties of an operator of stochastic differentiation constructed on a group

We construct a differential operator by an admissible group in the space L2 (ℝm,P)and study its properties.

Non uniform averagings in the ergodic theorem for stochastic flows

We study convergence of non uniform averagings of the Kozlov-Treschev type for ergodic diffusion processes generated by stochastic differential equations.

Stroock–Varadhan Theorem for Flows Generated by Stochastic Differential Equations with Interaction

We prove a theorem that characterizes the support of a flow generated by a system of stochastic differential equations with interaction.

On the existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process

We study the problem of existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process. For the conditional mathematical expectation of a solution, we obtain a partial differential equation.

On local operators diagonal with respect to the system of Hermitian polynomials

We present necessary conditions for operators diagonal with respect to the system of Hermitian polynomials to be local.