Research Profile and Principal Results

General theory of inhomogeneous Markov processes was the main subject of my earlier research. A construction of a Markov process with random birth and death times given in this context is widely used in modern theory of Markov processes under the name the Kuznetsov measure (see, e.g., R. K. Getoor, Excessive measures, Academic Press, New York, 1991 or C. Dellacherie and P.-A. Meyer, Probabilites et Potentiel, vol 5, Hermann, Paris, 1993). Among other results, it is worth to mention

Proof of the existence of a transition function for an arbitrary Markov process with values in a Luzin space (this result had filled a gap between the two definitions of a Markov process given by J. L. Doob and by E. B. Dynkin)
Construction of dual semigroups for Markov processes
Necessary and sufficient conditions for a unique decomposition of excessive functions into extremes
Construction of specifications for random fields

Since early 80s, I've concentrated on applications in statistics and economics. One of the results which I especially like is a construction (in terms of hyper-complex numbers) of an unbiased estimate for Aⁿ where A is a square matrix observed with Gaussian errors (established jointly with V. I. Orlov). Among other results,

Necessary conditions for the existence of consistent estimates for trend parameters
Stochastic analog of turnpike theorems (with I. V. Evstigneev)

Since 87, I became involved in some sort of software business. As a result of these activities, a time series analysis package MESOSAUR has been developed under my supervision. It combines traditional and non-traditional methods including expert systems approach. In particular, it includes methods of smoothing, forecasting (trends, exponential smoothing, the Box-Jenkins model), regression, spectral analysis, frequency filtration, disorder detection. The package was successfully distributed in fSU and in USA (the English version was prepared jointly with SYSTAT, Inc). It won the status of National Nominee at the yearly CEBIT exhibition in Hanover, Germany, in 1992.

Since 92, I've returned to the theory of Markov processes and I've concentrated on the theory of branching measure-valued processes or superprocesses, the processes which describe the evolution of a random cloud of infinitesimally small branching particles. Such processes are closely related to a semilinear operator Lu - u^a, where a belongs to (1,2] and L is a second order elliptic operator. Some results obtained in this contest could be reformulated in a purely analytic way and are new to specialists in P.D.E.'s. A substantial part of the results was obtained in a close collaboration with E. B. Dynkin. Among them,

a characterization of the structure of general subcritical branching measure-valued processes (with E.B.Dynkin and A.V.Skorohod)
regularity criteria for supercritical superprocesses
a characterization of removable boundary singularities for semilinear elliptic P.D.E.'s as boundary polar sets for the corresponding superprocesses or as null sets for a certain capacity (with E. B. Dynkin)
a similar characterization of removable lateral singularities for semilinear parabolic P.D.E.'s
a description of a class of natural linear additive functionals of superdiffusions and its relations with boundary value problems for semilinear P.D.E.'s (with E. B. Dynkin)
a characterization of a class of positive solutions of a semilinear elliptic P.D.E.'s in terms of their traces on the boundary (partly with E.Dynkin)

A recent breakthrough related with the last problem has put into consideration some new concepts in the theory of semilinear P.D.E. - singular points of a solution, fine topology on the boundary and a class of sigma-moderate solutions, which could be uniquely characterized by their fine traces. The principal question is to prove or disprove the existence of a non-sigma-moderate solution.

Created 9/27/97, Last updated 9/22/98

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