arXiv subjects

Ivan Remizov

Publications and source records attributed to Ivan Remizov.

2 recordsLinked to original sources

arXiv2024

Open problems in one-parameter operator semigroups theory

One-parameter strongly continuous semigroups of linear bounded operators on Banach spaces (also known as $C_0$-semigroups) are a fundamental operator-theoretic tool used in the study of linear and non-linear evolution PDEs arising in physics, probability, control theory and other areas of science and technology, including quantum theory, transportation problems and finance. Since 2021 the annual online conference on one-parameter semigroups of operators (OPSO) offers an opportunity for researchers worldwide to discuss the current state of the art and exchange knowledge. On every OPSO conference open problems are proposed, collected and discussed, some of them are found to be solved and hence get crossed out from the list. In this paper we provide a collection of open problems of semigroup theory that were proposed by participants of OPSO 2021--2024 conferences and have not been solved, yet. For each problem some comments on the relevance and and history of the problem are provided.

math.FA↗

arXiv2021

Chernoff approximations of Feller semigroups in Riemannian manifolds

Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.

math.FA↗