S. E. Kuznetsov

Branching measure-valued processes

SummaryThe objective of this paper is to investigate the structure of a general subcritical branching measure-valued processX subject to the usual regularity conditions. We prove that, if the second moments of the total massXt(E) are finite, thenX is a superprocess and we give an explicit expression of the branching characteristicsQ andl in terms of the continuous martingale component of the total massXt(E) and the Lévy measure (jumps compensator) ofX.

Nonhomogeneous Markov processes

We present the foundations of the theory of nonhomogeneous Markov processes in general state spaces and we give a survey of the fundamental papers in this topic. We consider the following questions:1.The existence of transition functions for a Markov process.2.The construction of regularization of processes.3.The properties of right and left processes: the strict Markov property, the behavior of excessive functions, etc.4.The relation of right and left processes with dual homogeneous processes and the application of the results of the nonhomogeneous theory to dual homogeneous processes, etc.

Linear additive functionals of superdiffusions and related nonlinear P.D.E.

Let L be a second order elliptic differential operator in a bounded smooth domain D in Rd and let 1 < α ≤ 2. We get necessary and sufficient conditions on measures η, ν under which there exists a positive solution of the boundary value problem (*) −Lv + v = η in D, v = ν on ∂D. The conditions are stated both analytically (in terms of capacities related to the Green’s and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called (L, α)-superdiffusions). We also investigate a closely related subject — linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain E in Rd, we establish a 1-1 correspondence between a class of such functionals and a class of L-excessive functions h (which we describe in terms of their Martin integral representation). The Laplace transform of A satisfies an integral equation which can be considered as a substitute for (*).

Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary

Let L be a second order elliptic differential operator on a Riemannian manifold E with no zero order terms. We say that a function h is L-harmonic if Lh 0. Every positive L-harmonic function has a unique representation

Markov snakes and superprocesses

SummaryWe suggest the name Markov snakes for a class of path-valued Markov processes introduced recently by J.-F. Le Gall in connection with the theory of branching measure-valued processes. Le Gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator Δv−v2. We establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of Markov snakes and vice versa. By using this theorem, we get limit theorems for discrete Markov snakes.

Solutions ofLu=u α dominated byL-harmonic functions

AbstractLetL be a second order elliptic differential operator on a differentiable manifoldM and let 1 <α≤2. We investigate connections bewween classU of all positive solutions of the equationLu=uα and classH of all positiveL-harmonic functions (i.e., solutions of the equationsLh=0). Putu∈U0 ifu∈U and ifu≤h for someh∈H. To everyu∈U0 there corresponds the minimalL-harmonic functionhu which dominatesu andu→hu is a 1–1 mapping fromU0 onto a subsetH0 ofH. The inverse mapping associates with everyh∈H0 the maximal element ofU dominated byh.Supposeg(x, dy) is Greens kernel,k(x, y) is the Martin kernel and ϖM is the Martin boundary associated withL. A subset Γ of ϖM is calledR-polar if it is not hit by the rangeR of the (L, α)-superdiffusion. It is calledM-polar if

σ-Moderate solutions of Lu = uα and fine trace on the boundary

\int\limits_M {g\left( {c,dx} \right)[\int\limits_\Gamma {k(x,y)v(dy)]^\alpha } }

On removable lateral singularities for quasilinear parabolic PDE's

is equal to 0 or ∞ for everyc∈M and every measure ρ.Everyh∈H has a unique representation