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- Evgeni B Dynkin
- 2004

where L is an elliptic differential operator of the second order, E is a bounded smooth 1 domain in R and Ïˆ 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 â‰¤ h for some h such that Lh = 0 in E. We say that u is Ïƒ-moderate if it isâ€¦ (More)

Semilinear equations Lu = Ïˆ(u) where L is an elliptic differential operator and Ïˆ is a positive function can be investigated by using (L, Ïˆ)superdiffusions. In a special case âˆ†u = u2 a powerful probabilistic tool â€“ the Brownian snake â€“ introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation.â€¦ (More)

- Evgeni B Dynkin
- 1998

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

- Evgeni B Dynkin
- 2005

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 ofâ€¦ (More)

- Evgeni B Dynkin
- Proceedings of the National Academy of Sciencesâ€¦
- 1986

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 randomâ€¦ (More)

- Evgeni B Dynkin, A. N. Minchenko
- 2010

The root system Î£ 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 Î£ is, in general, not large enough to contain the diagrams of all subsystems of Î£, the answer to thisâ€¦ (More)

- Evgeni B Dynkin
- 2004

Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation Lu = uÎ± with Î± > 1 in a bounded smooth domain E âŠ‚ Rd. In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called Ïƒ-moderate. The question if all solutions belong to this class remained open. In 2002 Mselati provedâ€¦ (More)

Suppose that E is a bounded domain of class C2,Î» in Rd and L is a uniformly elliptic operator in E. The set U of all positive solutions of the equation Lu = Ïˆ(u) in E was investigated by a number of authors for various classes of functions Ïˆ. In [DK98] we defined, for every Borel subset Î“ of âˆ‚E, two such solutions uÎ“ â‰¤ wÎ“. We also introduced a class ofâ€¦ (More)