Searcharxiv⌕ Search

arXiv subjects

Ante Mimica

Publications and source records attributed to Ante Mimica.

16 recordsLinked to original sources
arXiv2018

Estimates of Dirichlet heat kernels for subordinate Brownian motions

In this paper, we discuss estimates of transition densities of subordinate Brownian motions in open subsets of Euclidean space. When $D$ is a $C^{1,1}$ domain, we establish sharp two-sided estimates for the transition densities of a large class of subordinate Brownian motions in $D$ whose scaling order is not necessarily strictly below $2$. Our estimates are explicit and written in terms of the dimension, the Euclidean distance between two points, the distance to the boundary and Laplace exponent of the corresponding subordinator only.

math.PR↗
arXiv2017

Asymptotical properties of distributions of isotropic L\' evy processes

In this paper, we establish the precise asymptotic behaviors of the tail probability and the transition density of a large class of isotropic Lévy processes when the scaling order is between 0 and 2 including 2. We also obtain the precise asymptotic behaviors of the tail probability of subordinators when the scaling order is between 0 and 1 including 1. The asymptotic expressions are given in terms of the radial part of characteristic exponent $ψ$ and its derivative. In particular, when $ψ(λ)-\fracλ{2}ψ'(λ)$ varies regularly, as $\frac{tψ(r^{-1})^2}{ψ(r^{-1})-(2r)^{-1}ψ'(r^{-1})} \to 0$ the tail probability $\mathbb{P}(|X_t|\geq r)$ is asymptotically equal to a constant times $ t( ψ(r^{-1})-(2r)^{-1}ψ'(r^{-1})).$

math.PR↗
arXiv2017

Harnack inequality for subordinate random walks

In this paper, we consider a large class of subordinate random walks $X$ on integer lattice $\mathbb{Z}^d$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions.

math.PR↗
arXiv2016

Markov Chain Approximation of Pure Jump Processes

In this paper we discuss weak convergence of continuous-time Markov chains to a non-symmetric pure jump process. We approach this problem using Dirichlet forms as well as semimartingales. As an application, we discuss how to approximate a given Markov process by Markov chains.

math.PR↗
arXiv2016

Heat kernel estimates for subordinate Brownian motions

In this article we study transition probabilities of a class of subordinate Brownian motions. Under mild assumptions on the Laplace exponent of the corresponding subordinator, sharp two sided estimates of the transition probability are established. This approach, in particular, covers subordinators with Laplace exponents that vary regularly at infinity with index one, e.g. \[ ϕ(λ)=\fracλ{\log(1+λ)}-1 \quad \text{ or }\quad ϕ(λ)=\fracλ{\log(1+λ^{β/2})},\ β\in (0,2)\, \] that correspond to subordinate Brownian motions with scaling order that is not necessarily strictly between 0 and 2. These estimates are applied to estimate Green function (potential) of subordinate Brownian motion. We also prove the equivalence of the lower scaling condition of the Laplace exponent and the near diagonal upper estimate of the transition estimate.

math.PR↗
arXiv2016

On subordinate random walks

In this article subordination of random walks in $R^d$ is considered. We prove that subordination of random walks in the sense of [BSC12] yields the same process as subordination of Lévy processes (in the sense of Bochner). Furthermore, we prove that appropriately scaled subordinate random walk converges to a multiple of a rotationally $2α$-stable process if and only if the Laplace exponent of the corresponding subordinator varies regularly at zero with index $α\in (0,1]$.

math.PR↗
arXiv2016

Exponential decay of measures and Tauberian theorems

We study behavior of a measure on $[0,\infty)$ by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the measure occurs and under certain assumptions we show that the rate of the decay is given by the so called abscissa of convergence and extend the result of Nakagawa from [Nak05]. Under stronger assumptions we give behavior of density of the measure by considering its Laplace transform. In situations when there is no exponential decay we study occurrence of heavy tails and give an application in the theory of non-local equations.

math.CA↗
arXiv2015

Intrinsic scaling properties for nonlocal operators II

We study integrodifferential operators and regularity estimates for solutions to integrodifferential equations. Our emphasis is on kernels with a critically low singularity which does not allow for standard scaling. For example, we treat operators that have a logarithmic order of differentiability. For corresponding equations we prove a growth lemma and derive a priori estimates. We derive these estimates by classical methods developed for partial differential operators. Since the integrodifferential operators under consideration generate Markov jump processes, we are able to offer an alternative approach using probabilistic techniques.

math.AP↗
arXiv2013

Intrinsic scaling properties for nonlocal operators

We study growth lemmas and questions of regularity for generators of Markov processes. The generators are allowed to have an arbitrary order of differentiability less than 2. In general, this order is represented by a function and not by a number. The approach enables a careful study of regularity issues up to the phase boundary between integro-differential (positive order of differentiability) and integral operators (nonnegative order of differentiability). The proof is based on intrinsic scaling properties of the underlying operators and stochastic processes.

math.AP↗
arXiv2013

Unavodiable collections of balls for isotropic Lévy processes

A collection $\{\bar{B}(x_n,r_n)\}_{n\ge 1}$ of pairwise disjoint balls in the Euclidean space $\R^d$ is said to be avoidable with respect to a transient process $X$ if the process with positive probability escapes to infinity without hitting any ball. In this paper we study sufficient and necessary conditions for avoidability with respect to unimodal isotropic Lévy processes satisfying a certain scaling hypothesis. These conditions are expressed in terms of the characteristic exponent of the process, or alternatively, in terms of the corresponding Green function. We also discuss the same problem for a random collection of balls. The results are generalization of several recent results for the case of Brownian motion.

math.PR↗
arXiv2013

Unavoidable collections of balls for censored stable processes

We study avoidability of collections of balls in bounded $C^{1,1}$ opens sets for censored $α$-stable processes, $α\in (1,2)$. The results are analog to the ones obtained for Brownian motion in S. J. Gardiner, M. Ghergu, Champagne subregions of the unit ball with unavoidable bubbles, Ann. Acad. Sci. Fenn. Math. 35 (2010) 321-329. On the way we derive a Wiener-Aikawa-type criterion for minimal thinness with respect to the censored stable processes.

math.PR↗
arXiv2013

Green function estimates for subordinate Brownian motions : stable and beyond

A subordinate Brownian motion $X$ is a Lévy process which can be obtained by replacing the time of the Brownian motion by an independent subordinator. In this paper, when the Laplace exponent $ϕ$ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for $X$ on arbitrary open sets. Then we give an explicit form of sharp two-sided estimates on the Green functions of these subordinate Brownian motions in any bounded $C^{1,1}$ open set. As a consequence, we prove the boundary Harnack inequality for $X$ on any $C^{1,1}$ open set with explicit decay rate. Unlike {KSV2, KSV4}, our results cover geometric stable processes and relativistic geometric stable process, i.e. the cases when the subordinator has the Laplace exponent $$ϕ(λ)=\log(1+λ^{α/2}) (0<α\leq 2, d > α)$$ and $$ϕ(λ)=\log(1+(λ+m^{α/2})^{2/α}-m) (0<α<2,\, m>0, d >2) .$$

math.PR↗
arXiv2012

Harnack Inequalities for Subordinate Brownian Motions

In this paper, we consider transient subordinate Brownian motion X in R^d, d \geq 1, where the Laplace exponent ϕof the corresponding subordinator satisfies some mild conditions. The scaleinvariant Harnack inequality is proved for X. We first give new forms of asymptotical properties of the Levy and potential density of the subordinator near zero. Using these results we find asymptotics of the Levy density and potential density of X near the origin, which is essential to our approach. The examples which are covered by our results include geometric stable processes and relativistic geometric stable processes, i.e. the cases when the subordinator has the Laplace exponent ϕ(λ)=\log(1+λ^{α/2}) (0<α\leq 2, d > α) and ϕ(λ)=\log(1+(λ+m^{α/2})^{2/α}-m) (0<α<2,\,m>0,d >2).

math.PR↗
arXiv2012

On harmonic functions of symmetric Levy processes

We consider some classes of Levy processes for which the estimate of Krylov and Safonov (as in [BL02]) fails and thus it is not possible to use the standard iteration technique to obtain a-priori Holder continuity estimates of harmonic functions. Despite the faliure of this method, we obtain some a-priori regularity estimates of harmonic functions for these processes. Moreover, we extend results from [SSV06] and obtain asymptotic behavior of the Green function and the Levy density for a large class of subordinate Brownian motions, where the Laplace exponent of the corresponding subordinator is a slowly varying function.

math.PR↗
arXiv2011

Analysis of jump processes with nondegenerate jumping kernels

We prove regularity estimates for functions which are harmonic with respect to certain jump processes. The aim of this article is to extend the method of Bass-Levin[BL02] and Bogdan-Sztonyk[BS05] to more general processes. Furthermore, we establish a new version of the Harnack inequality that implies regularity estimates for corresponding harmonic functions.

math.PR↗