Extended resolvent of the heat operator with a multisoliton potential

We consider the heat operator with a general multisoliton potential and derive its extended resolvent depending on a parameter q ?? ?². We show that it is bounded in all variables and find its singularities in q. We introduce the Green??s functions and study their properties in detail.     

Averaging evolutionary equations perturbed by random processes with jumps

Weak convergence of measures generated by solutions of an evolutionary equation dependent on a small parameter to the unique solution of the martingale problem corresponding to the stochastic evolutionary equation is proved. The coefficients of the initial equation depend on random Markov processes with jumps.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 197–207, February, 1992.     

An Ablowitz-Ladik system with a discrete potential: I. Extended resolvent

An Ablowitz-Ladik linear system with the potential taking the values 0 or 1 is considered. The extended resolvent of this system is constructed, and the singularities of this operator are analyzed in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 20–33, April, 1999.     

The Carleman property of the resolvent of a one-dimensional Dirac operator with an integrable potential

It is proved that the resolvent of a Dirac operator with a potential thai is integrable on the entire axis is a Carleman operator.     

Homogenization of the Stokes equations with a random potential

Homogenization of the Stokes equations in a random porous medium is considered. Instead of the homogeneous Dirichlet condition on the boundaries of numerous small pores, used in the existing work on the subject, we insert a term with a positive rapidly oscillating potential into the equations. Physically, this corresponds to porous media whose rigid matrix is slightly permeable to fluid. This relaxation of the boundary value problem permits one to study the asymptotics of the solutions and to justify the Darcy law for the limit functions under much fewer restrictions. Specifically, homogenization becomes possible without any connectedness conditions for the porous domain, whose verification would lead to problems of percolation theory that are insufficiently studied. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 504–520, April, 1996. The work of the first author was supported by the INTAS under grant No. 93-2716.     

Averaging of random walks and shift-invariant measures on a Hilbert space

We study random walks in a Hilbert space H and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on H. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure λ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in H. We define the Hilbert space H of equivalence classes of complex-valued functions on H that are square integrable with respect to a shift-invariant measure λ. Using averaging of the shift operator in H over random vectors in H with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on H, we define a one-parameter semigroup of contracting self-adjoint transformations on H, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.     

Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients

In this paper we consider the parabolic equation with random coefficients: