arXiv subjects

Nikola Sandrić

Publications and source records attributed to Nikola Sandrić.

36 records · Page 2Linked to original sources

arXiv2020

Periodic Homogenization of a Lévy-Type Process with Small Jumps

In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called Lévy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits "small jumps" only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.

math.PR↗

arXiv2020

On Sub-Geometric Ergodicity of Diffusion Processes

In this article, we discuss ergodicity properties of a diffusion process given through an Itô stochastic differential equation. We identify conditions on the drift and diffusion coefficients which result in sub-geometric ergodicity of the corresponding semigroup with respect to the total variation distance. We also prove sub-geometric contractivity and ergodicity of the semigroup under a class of Wasserstein distances. Finally, we discuss sub-geometric ergodicity of two classes of Markov processes with jumps.

math.PR↗

arXiv2019

Uniform polynomial rates of convergence for a class of Lévy-driven controlled SDEs arising in multiclass many-server queues

We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by $α$-stable processes which arise as the limiting equations of multiclass queueing models in the Halfin-Whitt regime that have heavy-tailed arrival processes. When the safety staffing parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence to the invariant probability measure in total variation, which is uniform over all stationary Markov controls resulting in a locally Lipschitz continuous drift. We also derive a matching lower bound on the rate of convergence (under no abandonment). On the other hand, when all abandonment rates are positive, we show that the SDEs are exponentially ergodic uniformly over the above-mentioned class of controls. Analogous results are obtained for Lévy-driven SDEs arising from multiclass many-server queues under asymptotically negligible service interruptions. For these equations, we show that the aforementioned ergodic properties are uniform over all stationary Markov controls. We also extend a key functional central limit theorem concerning diffusion approximations so as to make it applicable to the models studied here.

math.PR↗

arXiv2018

Ergodicity of Lévy-driven SDEs arising from multiclass many-server queues

We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anisotropic Lévy process as in (2) and a pure-jump Lévy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) $α$-stable Lévy process as a special case. We identify conditions on the parameters in the drift, the Lévy measure and/or covariance function which result in subexponential and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence is polynomial, and we provide a sharp quantitative characterization of the rate via matching upper and lower bounds.

math.PR↗

arXiv2018

Stability of the overdamped Langevin equation in double-well potential

In this article, we discuss stability of the one-dimensional overdamped Lange\-vin equation in double-well potential. We determine unstable and stable equilibria, and discuss the rate of convergence to stable ones. Also, we derive conditions for stability of general diffusion processes which generalize the classical and well-known results of Khasminskii 2012.

math.PR↗

arXiv2017

A Note on the Birkhoff Ergodic Theorem

The classical Birkhoff ergodic theorem states that for an ergodic Markov process the limiting behaviour of the time average of a function (having finite $p$-th moment, $p\ge1$, with respect to the invariant measure) along the trajectories of the process, starting from the invariant measure, is a.s. and in the $p$-th mean constant and equals to the space average of the function with respect to the invariant measure. The crucial assumption here is that the process starts from the invariant measure, which is not always the case. In this paper, under the assumptions that the underlying process is a Markov process on Polish space, that it admits an invariant probability measure and that its marginal distributions converge to the invariant measure in the $L^{1}$-Wasserstein metric, we show that the assertion of the Birkhoff ergodic theorem holds in the $p$-th mean, $p\geq1$, for any bounded Lipschitz function and any initial distribution of the process.

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

On Transience of Lévy-Type Processes

In this paper, we study weak and strong transience of a class of Feller processes associated with pseudo-differential operators, the so-called Lévy-type processes. As a main result, we derive Chung-Fuchs type conditions (in terms of the symbol of the corresponding pseudo-differential operator) for these properties, which are sharp for Lévy processes. Also, as a consequence, we discuss the weak and strong transience with respect to the dimension of the state space and Pruitt indices, thus generalizing some well-known results related to elliptic diffusion and stable Lévy processes. Finally, in the case when the symbol is radial (in the co-variable) we provide conditions for the weak and strong transience in terms of the Lévy measures.

math.PR↗

arXiv2016

Ergodicity of Lévy-Type Processes

In this paper, conditions for transience, recurrence, ergodicity and strong, subexponential (polynomial) and exponential ergodicity of a class of Feller processes are derived. The conditions are given in terms of the coefficients of the corresponding infinitesimal generator. As a consequence, mixing properties of these processes are also discussed.

math.PR↗

arXiv2015

Homogenization of periodic diffusion with small jumps

In this paper, we study the homogenization of a diffusion process with jumps, that is, Feller process generated by an integro-differential operator. This problem is closely related to the problem of homogenization of boundary value problems arising in studying the behavior of heterogeneous media. Under the assumptions that the corresponding generator has vanishing drift coefficient, rapidly periodically oscillating diffusion and jump coefficients, that it admits only "small jumps" (that is, the jump kernel has finite second moment) and under certain additional regularity conditions, we prove that the homogenized process is a Brownian motion. The presented results generalize the classical and well-known results related to the homogenization of a diffusion process.

math.PR↗

arXiv2015

On Recurrence and Transience of Two-Dimensional Lévy and Lévy-Type Processes

In this paper, we study recurrence and transience of Lévy-type processes, that is, Feller processes associated with pseudo-differential operators. Since the recurrence property of Lévy-type processes in dimensions greater than two is vacuous and the recurrence and transience of one-dimensional Lévy-type processes have been very well investigated, in this paper we are focused on the two-dimensional case only. In particular, we study perturbations of two-dimensional Lévy-type processes which do not affect their recurrence and transience properties, we derive sufficient conditions for their recurrence and transience in terms of the corresponding Lévy measure and we provide some comparison conditions for the recurrence and transience also in terms of the Lévy measures.

math.PR↗

arXiv2015

Ergodicity and fluctuations of a fluid particle driven by diffusions with jumps

In this paper, we study the long-time behavior of a fluid particle immersed in a turbulent fluid driven by a diffusion with jumps, that is, a Feller process associated with a non-local operator. We derive the law of large numbers and central limit theorem for the evolution process of the tracked fluid particle in the cases when the driving process: (i) has periodic coefficients, (ii) is ergodic or (iii) is a class of Lévy processes. The presented results generalize the classical and well-known results for fluid flows driven by elliptic diffusion processes.

math.PR↗

arXiv2014

Ergodic property of stable-like Markov chains

A stable-like Markov chain is a time-homogeneous Markov chain on the real line with the transition kernel $p(x,dy)=f_x(y-x)dy$, where the density functions $f_x(y)$, for large $|y|$, have a power-law decay with exponent $α(x)+1$, where $α(x)\in(0,2)$. In this paper, under a certain uniformity condition on the density functions $f_x(y)$ and additional mild drift conditions, we give sufficient conditions for recurrence in the case when $0<\liminf_{|x|\longrightarrow\infty}α(x)$, sufficient conditions for transience in the case when $\limsup_{|x|\longrightarrow\infty}α(x)<2$ and sufficient conditions for ergodicity in the case when $0<\inf\{α(x):x\in\mathbb{R}\}$. As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric $α$-stable random walk on $\mathbb{R}$ with the index of stability $α\neq1.$

math.PR↗

arXiv2014

Long-time behavior for a class of Feller processes

In this paper, as a main result, we derive a Chung-Fuchs type condition for the recurrence of Feller processes associated with pseudo-differential operators. In the Lévy process case, this condition reduces to the classical and well-known Chung-Fuchs condition. Further, we also discuss the recurrence and transience of Feller processes with respect to the dimension of the state space and Pruitt indices and the recurrence and transience of Feller-Dynkin diffusions and stable-like processes. Finally, in the one-dimensional symmetric case, we study perturbations of Feller processes which do not affect their recurrence and transience properties, and we derive sufficient conditions for their recurrence and transience in terms of the corresponding Lévy measure. In addition, some comparison conditions for recurrence and transience also in terms of the Lévy measures are obtained.

math.PR↗

arXiv2013

Recurrence and transience property for a class of Markov chains

We consider the recurrence and transience problem for a time-homogeneous Markov chain on the real line with transition kernel $p(x,\mathrm{d}y)=f_x(y-x)\,\mathrm{d}y$, where the density functions $f_x(y)$, for large $|y|$, have a power-law decay with exponent $α(x)+1$, where $α(x)\in(0,2)$. In this paper, under a uniformity condition on the density functions $f_x(y)$ and an additional mild drift condition, we prove that when $\lim\inf_{|x|\longrightarrow\infty}α(x)>1$, the chain is recurrent. Similarly, under the same uniformity condition on the density functions $f_x(y)$ and some mild technical conditions, we prove that when $\lim\sup_{|x|\longrightarrow\infty}α(x)<1$, the chain is transient. As a special case of these results, we give a new proof for the recurrence and transience property of a symmetric $α$-stable random walk on $\mathbb {R}$ with the index of stability $α\in(0,1)\cup(1,2).$

math.PR↗

arXiv2013

A transience condition for a class of one-dimensional symmetric Lévy processes

In this paper, we give a sufficient condition for transience for a class of one-dimensional symmetric Lévy processes. More precisely, we prove that a one-dimensional symmetric Lévy process with the Lévy measure $ν(dy)=f(y)dy$ or $ν(\{n\})=p_n$, where the density function $f(y)$ is such that $f(y)>0$ a.e. and the sequence $\{p_n\}_{n\geq1}$ is such that $p_n>0$ for all $n\geq1$, is transient if $$\int_1^{\infty}\frac{dy}{y^{3}f(y)}<\infty\quad\textrm{or}\quad \sum_{n=1}^{\infty}\frac{1}{n^{3}p_n}<\infty.$$ Similarly, we derive an analogous transience condition for one-dimensional symmetric random walks with continuous and discrete jumps.

math.PR↗

arXiv2012

Long-time behavior of stable-like processes

In this paper, we consider a long-time behavior of stable-like processes. A stable-like process is a Feller process given by the symbol $p(x,ξ)=-iβ(x)ξ+γ(x)|ξ|^{α(x)},$ where $α(x)\in(0,2)$, $β(x)\in\R$ and $γ(x)\in(0,\infty)$. More precisely, we give sufficient conditions for recurrence, transience and ergodicity of stable-like processes in terms of the stability function $α(x)$, the drift function $β(x)$ and the scaling function $γ(x)$. Further, as a special case of these results we give a new proof for the recurrence and transience property of one-dimensional symmetric stable Lévy processes with the index of stability $α\neq1.$

math.PR↗

arXiv2012

Recurrence and transience criteria for two cases of stable-like Markov chains

We give recurrence and transience criteria for two cases of time-homogeneous Markov chains on the real line with transition kernel $p(x,dy)=f_x(y-x)dy$, where $f_x(y)$ are probability densities of symmetric distributions and, for large $|y|$, have a power-law decay with exponent $α(x)+1$, with $α(x)\in(0,2)$. If $f_x(y)$ is the density of a symmetric $α$-stable distribution for negative $x$ and the density of a symmetric $β$-stable distribution for non-negative $x$, where $α,β\in(0,2)$, then the chain is recurrent if and only if $α+β\geq2.$ If the function $x\longmapsto f_x$ is periodic and if the set $\{x:α(x)=α_0:=\inf_{x\in\R}α(x)\}$ has positive Lebesgue measure, then, under a uniformity condition on the densities $f_x(y)$ and some mild technical conditions, the chain is recurrent if and only if $α_0\geq1.$

math.PR↗