Feller Semigroups Research Articles

QUASI-ERGODICITY OF COMPACT STRONG FELLER SEMIGROUPS ON $L^2$

Abstract We study the quasi-ergodicity of compact strong Feller semigroups $U_t$ , $t> 0$ , on $L^2(M,\mu )$ ; we assume that M is a locally compact Polish space equipped with a locally finite Borel measue $\mu $ . The operators $U_t$ are ultracontractive and positivity preserving, but not necessarily self-adjoint or normal. We are mainly interested in those cases where the measure $\mu $ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on $L^p(M,\mu )$ and uniqueness of the quasi-stationary measure with the finiteness of the heat content of the semigroup (for large values of t) and with the progressive uniform ground state domination property. The latter property is equivalent to a variant of quasi-ergodicity which progressively propagates in space as $t \uparrow \infty $ ; the propagation rate is determined by the decay of . We discuss several applications and illustrate our results with examples. This includes a complete description of quasi-ergodicity for a large class of semigroups corresponding to non-local Schrödinger operators with confining potentials.

New classes of p-adic pseudo-differential operators with negative definite symbols and their applications

This paper introduces new classes of p-adic operators representing m-dissipative pseudo-differential operators with negative definite symbols under certain conditions. We will study new types of semilinear problems and martingale problems associated with these operators, and we will prove that these pseudo-differential operators are the infinitesimal generators of strongly continuous contraction semigroups on L2(Qpn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2({\\mathbb {Q}}_p^n)$$\\end{document}. Also, this article introduces new families of measures, resolvent of measures, positive definite measures, Feller semigroups, and Markov processes.

P-adic Bessel α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-potentials and some of their applications

In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-potentials. These operators are denoted and defined in the form (Eϕ,αf)(x)=-Fζ→x-1max{1,|ϕ(||ζ||p)|}-αf^(ζ),x∈Qpn,α∈R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (\\mathcal {E}_{\\varvec{\\phi },\\alpha }f)(x)=-\\mathcal {F}^{-1}_{\\zeta \\rightarrow x}\\left( \\left[ \\max \\{1,|\\varvec{\\phi }(||\\zeta ||_{p})|\\} \\right] ^{-\\alpha }\\widehat{f}(\\zeta )\\right) , \ ext { } x\\in {\\mathbb {Q}}_{p}^{n}, \\ \\ \\alpha \\in \\mathbb {R}, \\end{aligned}$$\\end{document}where f is a p-adic distribution and max{1,|ϕ(||ζ||p)|}-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left[ \\max \\{1,|\\varvec{\\phi }(||\\zeta ||_{p})|\\}\\right] ^{-\\alpha }$$\\end{document} is the symbol of the operator. We will study some properties of the convolution kernel (denoted as Kα\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_{\\alpha }$$\\end{document}) of the pseudo-differential operator Eϕ,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {E}_{\\varvec{\\phi },\\alpha }$$\\end{document}, α∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in \\mathbb {R}$$\\end{document}; and demonstrate that the family (Kα)α>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(K_{\\alpha })_{\\alpha >0}$$\\end{document} determines a convolution semigroup on Qpn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_{p}^{n}$$\\end{document}. Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.

Diffusion in media with membranes and some nonlocal parabolic problems

UDC 519.21 We establish the classical solvability of a certain conjugation problem for one-dimensional (with respect to a spatial variable) Kolmogorov backward equation with discontinuous coefficients and some variants of the general nonlocal Feller–Wentzell boundary condition given on nonsmooth boundaries of considered curvilinear domains. In addition, we prove, that the two-parameter Feller semigroup defined by the solution of this problem describes some inhomogeneous diffusion process with moving membranes on the given region of the real line. We also show the relationship between the constructed process and the generalized diffusion in the sense of M. I. Portenko.

New classes of parabolic pseudo-differential equations, Feller semigroups, contraction semigroups and stochastic process on the p-adic numbers

Two types of p-adic pseudo-differential operators (denoted, respectively, by Tf1,f2l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}_{{\\varvec{f}}_{1},{\\varvec{f}}_{2}}^{l}$$\\end{document} and Jf1,f2α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {J}}_{{\\varvec{f}}_{1},{\\varvec{f}}_{2}}^{\\alpha }$$\\end{document}) are introduced in this article. We will show that the operator Tf1,f2l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}_{{\\varvec{f}}_{1},{\\varvec{f}}_{2}}^{l}$$\\end{document} determines certain Feller semigroups and stochastic processes with state space the p-adic numbers. The second type of these operators (defined on a new class of p-adic Sobolev space) are connected with contraction semigroups and parabolic pseudo-differential equations.

Correction: Singular integrals and Feller semigroups with jump phenomena

Correction: Singular integrals and Feller semigroups with jump phenomena

Singular integrals and Feller semigroups with jump phenomena

Singular integrals and Feller semigroups with jump phenomena

Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions

Lévy walks represent important modeling tools for a variety of real-life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far, these scaling limits have been derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here, we derive the corresponding limiting equations in the case of position-depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo–Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalizations are analyzed.

Some further classes of pseudo-differential operators in the p-adic context and their applications.

The purpose of this article is to study new non-Archimedean pseudo-differential operators whose symbols are determined from the behavior of two functions defined on the p-adic numbers. Thanks to the characteristics of our symbols, we can find connections between these operators and new types of non-homogeneous differential equations, Feller semigroups, contraction semigroups and strong Markov processes.

New classes of p-adic evolution equations and their applications

In this article, we study new classes of evolution equations in the p-adic context. We establish rigorously that the fundamental solutions of the homogeneous Cauchy problem, naturally associated to these equations, are transition density functions of some strong Markov processes {mathfrak {X}} with state space the n-dimensional p-adic unit ball ({mathbb {Z}}_{p}^{n}). We introduce a family of operators {T_{t}}_{tge 0} (obtained explicitly) that determine a Feller semigroup on C_{0}({mathbb {Z}}_{p}^{n}). Also, we study the asymptotic behavior of the survival probability of a strong Markov processes {mathfrak {X}} on a ball B_{-m}^{n}subset {mathbb {Z}}_{p}^{n}, min {mathbb {N}}. Moreover, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the mentioned above Feller semigroup.

A monotone convergence theorem for strong Feller semigroups

For an increasing sequence (T_n) of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps mathbb {1} to a continuous function. This is instrumental in the study of elliptic operators on mathbb {R}^d with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and to show that the semigroup is strongly Feller under very mild regularity assumptions on the coefficients. We also provide counterexamples to demonstrate that the assumptions in our main result are close to optimal.

Chernoff approximations of Feller semigroups in Riemannian manifolds

AbstractChernoff 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.

On admissible singular drifts of symmetric α‐stable process

AbstractWe consider the problem of existence of a (unique) weak solution to the SDE describing symmetric α‐stable process with a locally unbounded drift , , . In this paper, b belongs to the class of weakly form‐bounded vector fields, the class providing the L2 theory of the non‐local operator behind the SDE, i.e. . It contains as proper sub‐classes other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, the weak class and the Campanato–Morrey class (in general, such b makes invalid the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that the operator with weakly form‐bounded b admits a realization as (minus) Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of in , .

Affine pure-jump processes on positive Hilbert–Schmidt operators

We show the existence of a broad class of affine Markov processes on the cone of positive self-adjoint Hilbert–Schmidt operators. Such processes are well-suited as infinite-dimensional stochastic covariance models. The class of processes we consider is an infinite-dimensional analogue of the affine processes on the cone of positive semi-definite and symmetric matrices studied in Cuchiero et al. (2011).As in the finite-dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well as jumps governed by a jump measure that depends affine linearly on the state. The fact that the cone of positive self-adjoint Hilbert–Schmidt operators has empty interior calls for a new approach to proving existence: instead of using standard localization techniques, we employ the theory on generalized Feller semigroups introduced in Dörsek and Teichmann (2010) and further developed in Cuchiero and Teichmann (2020). Our approach requires a second moment condition on the jump measures involved, consequently, we obtain explicit formulas for the first and second moments of the affine process.

On some generalizations of non-archimedean pseudo-differential operators and their applications

In this article we define and study various generalizations non-archimedean pseudo-differential operators and its connections with probability theory. We provide explicit and sufficient conditions to ensure that these operators can satisfy the positive maximum principle, the heat Kernel or fundamental solution of the Cauchy problem (or heat equation) naturally associated with these operators determine a Feller semigroup and a transition function of some strong Markov processes X with state space the p-adic numbers. Also, we will study the property of Conservation of mass, the Comparison principle and some aspects of these processes including the first passage time problem and the probability of survival. These equations are proposed as mathematical models (connected with energy landscapes) to study the spread of an infectious or contagious disease that takes into account social clusters in a situation of extreme social isolation.

THE NONLOCAL CONJUGATION PROBLEM FOR A LINEAR SECOND ORDER PARABOLIC EQUATION OF KOLMOGOROV'S TYPE WITH DISCONTINUOUS COEFFICIENTS

In this paper, we construct the two-parameter Feller semigroup associated with a certain one-dimensional inhomogeneous Markov process. This process may be described as follows. At the interior points of the finite number of intervals $(-\infty,r_1(s)),\,(r_1(s),r_2(s)),\ldots,\,(r_{n}(s),\infty)$ separated by points $r_i(s)\,(i=1,\ldots,n)$, the positions of which depend on the time variable, this process coincides with the ordinary diffusions given there by their generating differential operators, and its behavior on the common boundaries of these intervals is determined by the Feller-Wentzell conjugation conditions of the integral type, each of which corresponds to the inward jump phenomenon from the boundary. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding nonlocal conjugation problem for a second order linear parabolic equation of Kolmogorov’s type with discontinuous coefficients. The main part of the paper consists in the investigation of this parabolic conjugation problem, the peculiarity of which is that the domains on the plane, where the equations are given, are curvilinear and have non-smooth boundaries: the functions $r_i(s)\,(i=1,\ldots,n)$, which determine the boundaries of these domains satisfy only the Hölder condition with exponent greater than $\frac{1}{2}$. Its classical solvability in the space of continuous functions is established by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials. It is also proved that the solution of this problem has a semigroup property. The availability of the integral representation for the constructed semigroup allows us to prove relatively easily that this semigroup yields the Markov process.

Upper Envelopes of Families of Feller Semigroups and Viscosity Solutions to a Class of Nonlinear Cauchy Problems

In this paper, we consider the (upper) semigroup envelope, i.e., the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, ...

Feller semigroups and degenerate elliptic operators III

AbstractThis paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups in the characteristic case via the Fichera function. Probabilistically, our result may be stated as follows: We construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the interior of the state space, without reaching the boundary. We make use of the Hille–Yosida–Ray theorem that is a Feller semigroup version of the classical Hille–Yosida theorem in terms of the positive maximum principle. Our proof is based on a method of elliptic regularizations essentially due to Oleĭnik and Radkevič. The weak convergence of approximate solutions follows from the local sequential weak compactness of Hilbert spaces and Mazur's theorem in normed linear spaces.

On a nonclassical problem for the heat equation and the Feller semigroup generated by it

The initial boundary value problem for the equation of heat conductivity with the Wenzel conjugation condition is studied. It does not fit into the general theory of parabolic initial boundary value problems and belongs to the class of conditionally correct ones. In space of bounded continuous functions by the method of boundary integral equations its classical solvability under some conditions is established. In addition, it is proved that the obtained solution is a Feller semigroup, which represents some homogeneous generalized diffusion process in the area considered here.

Logistic Neumann problems with discontinuous coefficients

The purpose of this paper is for the first time to provide a careful and accessible exposition of the study of the existence of positive solutions of semilinear Neumann problems for diffusive logistic equations with discontinuous coefficients, which models population dynamics in environments with spatial heterogeneity. A biological interpretation of our main result is that when the environment has an impassable boundary and is on the average unfavorable, then high diffusion rates have the same effect (that is, the ultimate extinction of the population) as they always have when the boundary is deadly; but if the boundary is impassable and the environment is on the average neutral or favorable, then the population can persist, no matter what its rate of diffusion. The approach here is based on explicit representation formulas for the solutions of the Neumann problem and also on the $$L^{p}$$ boundedness of Calderon–Zygmund singular integral operators appearing in those representation formulas. That is why we consider the case where the space dimension is greater than 2. Moreover, we make use of an $$L^{p}$$ variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups.