Evgeni B Dynkin

Learn More
where L is an elliptic differential operator of the second order, E is a bounded smooth 1 domain in R and ψ is a continuously differentiable positive function. Our goal is to describe the set U of all positive solutions of this equation. We say that an element u of U is moderate if u ≤ h for some h such that Lh = 0 in E. We say that u is σ-moderate if it is(More)
Semilinear equations Lu = ψ(u) where L is an elliptic differential operator and ψ is a positive function can be investigated by using (L, ψ)superdiffusions. In a special case ∆u = u2 a powerful probabilistic tool – the Brownian snake – introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation.(More)
All positive harmonic functions in an arbitrary domain of a Euclidean space can be described in terms of the so-called exit boundary. This was established in 1941 by R. S. Martin. A probabilistic approach to the Martin theory is due to Doob and Hunt. It was extended later to harmonic functions associated with a wide class of Markov processes. The subject of(More)
Let T(k) (epsilon)(lambda; t(1),..., t(k)) = rho(X(t) (1))q(epsilon)(X(t) (2) - X(t) (1))... q(epsilon)(X(t(k) ) - X(t(k) )-1), where X(t) is a Brownian motion in R(2), lambda(dx) = rho(x)dx, and q(epsilon) converges to Dirac's delta function as epsilon downward arrow 0. The self-intersection local times of order k are described by a generalized random(More)
The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of Σ is, in general, not large enough to contain the diagrams of all subsystems of Σ, the answer to this(More)
Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation Lu = uα with α > 1 in a bounded smooth domain E ⊂ Rd. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called σ-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved(More)