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Mateusz Kwaśnicki

Publications and source records attributed to Mateusz Kwaśnicki.

50 records · Page 3Linked to original sources
arXiv2015

Martin kernels for Markov processes with jumps

We prove existence of boundary limits of ratios of positive harmonic functions for a wide class of Markov processes with jumps and irregular domains, in the context of general metric measure spaces. As a corollary, we prove uniqueness of the Martin kernel at each boundary point, that is, we identify the Martin boundary with the topological boundary. We also prove a Martin representation theorem for harmonic functions. Examples covered by our results include: strictly stable Lévy processes in R^d with positive continuous density of the Lévy measure; stable-like processes in R^d and in domains; and stable-like subordinate diffusions in metric measure spaces.

math.PR↗
arXiv2015

Ten equivalent definitions of the fractional Laplace operator

This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.

math.AP↗
arXiv2015

Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval

We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator psi(-Delta) in an interval, with zero exterior condition, for complete Bernstein functions psi such that x psi'(x) converges to infinity as x goes to infinity. This extends previous results obtained by the authors for the fractional Laplace operator (psi(x) = x^{alpha/2}) and for the Klein-Gordon square root operator (psi(x) = (1+x^2)^{1/2} - 1). The formula for the eigenvalues in (-a,a) is of the form lambda_n = psi(mu_n^2) + O(1/n), where mu_n is the solution of mu_n = (n pi)/(2 a) - theta(mu_n)/a, and theta(mu) is given as an integral involving psi.

math.SP↗
arXiv2015

Hitting times of points for symmetric Lévy processes with completely monotone jumps

Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Lévy processes are studied. The Lévy measure is assumed to have completely monotone density function, and a scaling-type condition inf z Psi"(z) / Psi'(z) > 0 is imposed on the Lévy-Khintchine exponent Psi. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin.

math.PR↗
arXiv2014

On the shape of the fundamental sloshing mode in axisymmetric containers

In the paper we numerically study positions of high spots (extrema) of the fundamental sloshing mode of liquid in an axisymmetric tank. Our approach is based on a linear model reducing the problem to appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method of making images of oscillating fluid. We also describe the relation of the high spot problem to the celebrated hot spots conjecture.

physics.flu-dyn↗
arXiv2014

New families of subordinators with explicit transition probability semigroup

There exist only a few known examples of subordinators for which the transition probability density can be computed explicitly along side an expression for its Lévy measure and Laplace exponent. Such examples are useful in several areas of applied probability, for example, they are used in mathematical finance for modeling stochastic time change, they appear in combinatorial probability to construct sampling formulae, which in turn is related to a variety of issues in the theory of coalescence models, moreover, they have also been extensively used in the potential analysis of subordinated Brownian motion in dimension greater than or equal to 2. In this paper, we show that Kendall's classic identity for spectrally negative Lévy processes can be used to construct new families of subordinators with explicit transition probability semigroups. We describe the properties of these new subordinators and emphasise some interesting connections with explicit and previously unknown Laplace transform identities and with complete monotonicity properties of certain special functions.

math.PR↗
arXiv2013

Rogers functions and fluctuation theory

Extending earlier work by Rogers, Wiener-Hopf factorisation is studied for a class of functions closely related to Nevanlinna-Pick functions and complete Bernstein functions. The name 'Rogers functions' is proposed for this class. Under mild additional condition, for a Rogers function f, the Wiener--Hopf factors of f(z)+q, as well as their ratios, are proved to be complete Bernstein functions in both z and q. This result has a natural interpretation in fluctuation theory of Lévy processes: for a Lévy process X_t with completely monotone jumps, under mild additional condition, the Laplace exponents kappa(q;z), kappa*(q;z) of ladder processes are complete Bernstein functions of both z and q. Integral representation for these Wiener--Hopf factors is studied, and a semi-explicit expression for the space-only Laplace transform of the supremum and the infimum of X_t follows.

math.CV↗
arXiv2013

Suprema of Lévy processes

In this paper we study the supremum functional $M_t=\sup_{0\le s\le t}X_s$, where $X_t$, $t\ge0$, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of $M_t$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_t$ if the Lévy-Khintchin exponent of the process increases on $(0,\infty)$.

math.PR↗
arXiv2013

Boundary Harnack inequality for Markov processes with jumps

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e.g., to many subordinate Brownian motions, Lévy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schrödinger, drift and jump perturbations of such processes.

math.PR↗
arXiv2011

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W in R^3. We study the case when W is an axially symmetric, convex, bounded domain satisfying the John condition. The Cartesian coordinates (x,y,z) are chosen so that the mean free surface of the liquid lies in (x,z)-plane and y-axis is directed upwards (y-axis is the axis of symmetry). Our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction phi there is a change of x,z coordinates by a rotation around y-axis so that phi is odd in x-variable. The second result of the paper gives the following monotonicity property of the fundamental eigenfunction phi. If phi is odd in x-variable then it is strictly monotonic in x-variable. This property has the following hydrodynamical meaning. If liquid oscillates freely with fundamental frequency according to phi then the free surface elevation of liquid is increasing along each line parallel to x-axis during one period of time and decreasing during the other half period. The proof of the second result is based on the method developed by D. Jerison and N. Nadirashvili for the hot spots problem for Neumann Laplacian.

math.AP↗
arXiv2010

Eigenvalues of the fractional Laplace operator in the interval

Two-term Weyl-type asymptotic law for the eigenvalues of one-dimensional fractional Laplace operator (-d^2/dx^2)^(alpha/2) (0 < alpha < 2) in the interval (-1,1) is given: the n-th eigenvalue is equal to (n pi/2 - (2 - alpha) pi/8)^alpha + O(1/n). Simplicity of eigenvalues is proved for alpha in [1, 2). L^2 and L^infinity properties of eigenfunctions are studied. We also give precise numerical bounds for the first few eigenvalues.

math.SP↗
arXiv2009

Spectral properties of the Cauchy process

We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0,infty) and the interval (-1,1). This process is related to the square root of one-dimensional Laplacian A = -sqrt(-d^2/dx^2) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions psi_lambda of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the half-line (or the heat kernel of A in (0,infty)), and for the distribution of the first exit time from the half-line follow. The formula for psi_lambda is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues lambda_n of A in the interval the asymptotic formula lambda_n = n pi/2 - pi/8 + O(1/n) is derived, and all eigenvalues lambda_n are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues lambda_n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point.

math.SP↗
arXiv2007

Estimates and structure of $α$-harmonic functions

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. This yields a unique representation of such functions as integrals against measures on $D^c\cup \{\infty\}$ satisfying an integrability condition. The corresponding Martin boundary of $D$ is a subset of the Euclidean boundary determined by an integral test.

math.PR↗