Bounded perturbations of two-dimensional diffusion processes with nonlocal conditions near the boundary

We study the existence of Feller semigroups arising in the theory of multidimensional diffusion processes. We study bounded perturbations of elliptic operators with boundary conditions containing an integral over the closure of the domain with respect to a nonnegative Borel measure without assuming that the measure is small. We state sufficient conditions on the measure guaranteeing that the corresponding nonlocal operator is the generator of a Feller semigroup.     

Elliptic problems with nonlocal boundary conditions and Feller semigroups

This monograph is devoted to the following interrelated problems: the solvability and smoothness of elliptic linear equations with nonlocal boundary conditions and the existence of Feller semigroups that appear in the theory of multidimensional diffusion processes.     

On the Existence of Feller Semigroups with Discontinuous Coefficients

This paper is devoted to the functional analytic approach to the problem of the existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. In other words, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary.     

Path regularity for Feller semigroups via Gaussian kernel estimates and generalizations to arbitrary semigroups on <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

Given γ ∈ (−1,1), we present a dyadic growth condition on the finite dimensional distributions of operator semigroups on C₀(E which - for γ>0 and Feller semigroups - assures that the corresponding Feller process has paths in local Hölder spaces and in weighted Besov spaces of order γ. We show that, for operator semigroups satisfying Gaussian kernel estimates of order m>1, condition holds for all and even for all in the case of Feller semigroups. Such Gaussian kernel estimates are typical for Feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on ℝ^D of order m≥D.     

Pseudodifferential Operators with Rough Negative Definite Symbols

We consider state-space dependent continuous negative definite functions and use their associated pseudodifferential operators to construct Feller semigroups. Our method works with “rough” symbols ${p(x,\xi),\,{\rm i.e.}\,\xi \mapsto p(x,\xi)}We consider state-space dependent continuous negative definite functions and use their associated pseudodifferential operators to construct Feller semigroups. Our method works with “rough” symbols p(x,x), i.e. x? p(x,x){p(x,\xi),\,{\rm i.e.}\,\xi \mapsto p(x,\xi)} only needs to be continuous. The main part of this work concerns the development of an asymptotic expansion formula for the composition of two pseudodifferential operators with rough negative definite symbols. This presents an improvement over other symbolic calculi that typically require the symbols to be smooth. As an application we show how to adapt existing techniques to construct and approximate Feller semigroups to the case of rough symbols.     

Doubly Feller Property of Brownian Motions with Robin Boundary Condition

In this paper, we consider first order Sobolev spaces with Robin boundary condition on unbounded Lipschitz domains. Hunt processes are associated with these spaces. We prove that the semigroup of these processes are doubly Feller. As a corollary, we provide a condition for semigroups generated by these processes being compact.

     

Spectral Inclusions for Semigroups of Closed Operators

We show that several spectral inclusions known for C0-semigroups fail for semigroups of closed operators, even if they can be regularized. We introduce the notion of spectral completeness for the regularizing operator C which implies equality of the spectrum and the C-spectrum of the generator. We prove spectral inclusions under this additional assumption. We give a series of examples in which the regularizing operator is spectrally complete including generators of integrated semigroups, of distribution semigroups, and of some semigroups that are strongly continuous for t > 0.     

Cleavability in Semigroups

The notion of cleavability (splittability) is observed to apply not only to topological spaces, where it was first developed, but to semigroups also. Several results of the form 'if a semigroup D is cleavable over a class of semigroups each of which has property P, then D also has property P' are derived, and some suggestions for further investigations are put forward.     

Minimal Presentations and Efficiency of Semigroups

A finite semigroup S is said to be efficient if it can be defined by a presentation (A | R) with |R| -|A|=rank(H2(S)). In this paper we demonstrate certain infinite classes of both efficient and inefficient semigroups. Thus, finite abelian groups, dihedral groups D2n with n even, and finite rectangular bands are efficient semigroups. By way of contrast we show that finite zero semigroups and free semilattices are never efficient. These results are compared with some well-known results on the efficiency of groups.     

On the existence of Feller semigroups with discontinuous coefficients II

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with Dirichlet boundary condition, oblique derivative boundary condition and first-order Wentzell boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderon- Zygmund theory of singular integral operators with non-smooth kernels.     

Rational approximation schemes for bi-continuous semigroups

This paper extends the Hille-Phillips functional calculus and rational approximations results due to R. Hersh, T. Kato, P. Brenner, and V. Thomée to generators of bi-continuous semigroups. The method yields error estimates for rational time-discretization schemes for such semigroups, in particular for dual semigroups, Feller semigroups such as the Ornstein-Uhlenbeck semigroup, the heat semigroup, semigroups induced by nonlinear flows, implemented semigroups, and evolution semigroups. Furthermore, the results provide error estimates for a new class of inversion formulas for the Laplace transform.     

Trotter-Kato theorems for bi-continuous semigroups and applications to Feller semigroups

We give results on the convergence of bi-continuous semigroups introduced and studied by Kühnemund. As a consequence, we obtain a Lie-Trotter product formula and apply it to Feller semigroups generated by second order elliptic differential operators with unbounded coefficients in .     

Markov Transition Functions and Semigroups of Measures

The application of operator semigroups to Markov processes is extended to Markov transition functions which do not have the Feller property. Markov transition functions are characterized as solutions of forward and backward equations which involve the generators of integrated semigroups and are shown to induce integral semigroups on spaces of measures.     

On the existence of a Feller semigroup with atomic measure in a nonlocal boundary condition

The existence of Feller semigroups arising in the theory of multidimensional diffusion processes is studied. An elliptic operator of second order is considered on a plane bounded region G. Its domain of definition consists of continuous functions satisfying a nonlocal condition on the boundary of the region. In general, the nonlocal term is an integral of a function over the closure of the region G with respect to a nonnegative Borel measure μ(y, dη) ∈ ∂G. It is proved that the operator is a generator of a Feller semigroup in the case where the measure is atomic. The smallness of the measure is not assumed. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 164–179.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

Upper estimates of transition densities for stable-dominated semigroups

We derive upper estimates of transition densities for Feller semigroups with jump intensities lighter than that of the rotation invariant stable Lévy process.     

A history of topological and analytical semigroups: a personal view

A retrospective of the historical development of a topological and analytical theory of semigroups is given from a personal vantage point. It begins with SOPHUS LIE who from about 1880 onward dealt with semigroups by default, having no clear concept of a group at first. The algebraic theory of semigroups emerged in the first half of the 20th century, but its topological counterpart emancipated itself as late as in the second half. I shall comment on the genesis of a theory of compact topological semigroups in the fifties under the influence of A. D. WALLACE. These semigroups came into focus at about the same time E. S. LYAPIN raised the important issue of magnifying elements, thereby discovering the bicyclic semigroup wherever those exist. Compact topological semigroups, however, cannot contain bicyclic semigroups; this has interesting consequences. - Around 1970 D. S. SCOTT discovered what he called continuous lattices and what nowadays, in more general form, is called domains, whileJ. D. LAWSON drew semigroup theoreticians' attention to a very natural class of compact semilattices having enough homomorphisms into the unit interval semilattice. The class of continuous lattices agrees with the class of Lawson semilattices. It generates a network of applications in theoretical computer science under the name "domain theory". - A hundred years after SOPHUS LIE's differentiable groups and semigroups, attention returned back to semigroups and Lie theory. Lie semigroup theory, initiated by E. B. VINBERG, G. I. OLSHANSKY, J. D. LAWSON and the author among others, infused a strong geometric and analytical flavor into topological semigroup theory and generated a new lines of application of semigroup theory such as in geometric control theory, and in the area of unitary representation theory of Lie groups, particulary in the area of holomorphic extensions of unitary representations. A respectable number of mongraphs and collections have been and are being written in this field.     

Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

Journal of Theoretical Probability - We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $$Af=g$$ associated with...     

Asymptotic expansions of the representation formulae for (C 0) m-parameter operator semigroups 1 . 2

In this paper, we expand asymptotically the general representation formulae for (C _o) m-parameter operator semigroups. When we consider special semigroups, our results yield the asymptotic expansions for multivariate Feller operators. In particular, the asymptotic expansions for univariate and multivariate Bernstein operators are reobtained. See the related examples at the end.     

Approximation of Stable-dominated Semigroups

We consider Feller semigroups with jump intensity dominated by that of the rotation invariant stable Lévy process. Using an approximation scheme we obtain estimates of corresponding heat kernels.