Ilya Pavlyukevich
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View article: Early warning signs for tipping points in systems with non-Gaussian $$\alpha$$-stable noise
Early warning signs for tipping points in systems with non-Gaussian $$\alpha$$-stable noise Open
Forecasting rapid, non-linear change or so-called tipping points is a major concern in ecology and environmental science. Statistical early warning signs, based on the theory of stochastic dynamical systems, are now regularly applied to ob…
View article: Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs
Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs Open
For a solution $X$ of a Lévy-driven $d$-dimensional Marcus (canonical) stochastic differential equation, we show that the Wong--Zakai type approximation scheme $X^h$ has a strong convergence of order $\frac12$: for each $T\in [0,\infty)$ a…
View article: Cutoff Ergodicity Bounds in Wasserstein Distance for a Viscous Energy Shell Model with Lévy Noise
Cutoff Ergodicity Bounds in Wasserstein Distance for a Viscous Energy Shell Model with Lévy Noise Open
This article establishes explicit non-asymptotic ergodic bounds in the renormalized Wasserstein–Kantorovich–Rubinstein (WKR) distance for a viscous energy shell lattice model of turbulence with random energy injection. The system under con…
View article: Homogenization of a Multivariate Diffusion with Semipermeable Interfaces
Homogenization of a Multivariate Diffusion with Semipermeable Interfaces Open
We study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that …
View article: Early warning signs of critical transitions -- The $α$-stable case
Early warning signs of critical transitions -- The $α$-stable case Open
Statistical early warning signs can be used to identify an approaching bifurcation in stochastic dynamical systems and are now regularly employed in applications concerned with the identification of potential rapid, non-linear change or ti…
View article: Stochastic energy-balance model with a moving ice line
Stochastic energy-balance model with a moving ice line Open
In [SIAM J. Appl. Dyn. Sys., 12(4):2068--2092, 2013], Widiasih proposed and analyzed a deterministic one-dimensional Budyko-Sellers energy-balance model with a moving ice-line. In this paper, we extend this model to the stochastic setting …
View article: Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks
Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks Open
In this paper we study Markov chains with the state space given by the coordinate axes of $\mathbb R^m$, $m \geq 2$, whose step sizes on each positive half-axis are distributed according to a centered probability distribution with variance…
View article: Stochastic selection problem for a Stratonovich SDE with power non-linearity
Stochastic selection problem for a Stratonovich SDE with power non-linearity Open
In our paper [Bernoulli 26(2), 2020, 1381-1409], we found all strong Markov solutions that spend zero time at $0$ of the Stratonovich stochastic differential equation $d X=|X|^α\circ dB$, $α\in (0,1)$. These solutions have the form $X_t^θ=…
View article: Homogenization of a multivariate diffusion with semipermeable interfaces
Homogenization of a multivariate diffusion with semipermeable interfaces Open
We study the homogenization problem for a system of stochastic differential equation with local time terms that models a multivariate diffusion in presence of semipermeable hyperplane interfaces with oblique penetration. We show that this …
View article: First Order Linear Marcus SPDEs
First Order Linear Marcus SPDEs Open
In this paper we solve a Lévy driven linear stochastic first order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic charact…
View article: Limit behaviour of random walks on ℤ<sup><i>m</i></sup>with two-sided membrane
Limit behaviour of random walks on ℤ<sup><i>m</i></sup>with two-sided membrane Open
We study Markov chains on ℤ m , m ≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane) H = {0} × ℤ m −1 . The exit probabilities from the membrane (penetration probabilities) H are periodic and also d…
View article: Limit behaviour of random walks on $\mathbb Z^m$ with two-sided membrane
Limit behaviour of random walks on $\mathbb Z^m$ with two-sided membrane Open
We study Markov chains on $\mathbb Z^m$, $m\geq 2$, that behave like a standard symmetric random walk outside of the hyperplane (membrane) $H=\{0\}\times \mathbb Z^{m-1}$. The transition probabilities on the membrane $H$ are periodic and a…
View article: Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds
Geodesic random walks, diffusion processes and Brownian motion on Finsler manifolds Open
We show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.
View article: Generalized selection problem with Lévy noise
Generalized selection problem with Lévy noise Open
Let $A_\pm>0$, $β\in(0,1)$, and let $Z^{(α)}$ be a strictly $α$-stable Lévy process with the jump measure $ν(\mathrm{d} z)=(C_+\mathbb{I}_{(0,\infty)}(z)+ C_-\mathbb{I}_{(-\infty,0)}(z))|z|^{-1-α}\,\mathrm{d} z$, $α\in (1,2)$, $C_\pm\geq 0…
View article: Drift estimation for a Lévy-driven Ornstein–Uhlenbeck process with heavy tails
Drift estimation for a Lévy-driven Ornstein–Uhlenbeck process with heavy tails Open
We consider the problem of estimation of the drift parameter of an ergodic Ornstein–Uhlenbeck type process driven by a Lévy process with heavy tails. The process is observed continuously on a long time interval [0, T ], $$T\rightarrow \inf…
View article: First order convergence of weak Wong--Zakai approximations of Lévy driven Marcus SDEs
First order convergence of weak Wong--Zakai approximations of Lévy driven Marcus SDEs Open
For solutions $X=(X_t)_{t\in[0,T]}$ of Lévy-driven Marcus stochastic differential equations we study the Wong--Zakai type time discrete approximations $\bar X=(\bar X_{kh})_{0\leq k\leq T/h}$, $h>0$, and establish the first order convergen…
View article: Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited
Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited Open
In this paper, we study the Stratonovich stochastic differential equation $\\mathrm{d}X=|X|^{\\alpha }\\circ \\mathrm{d}B$, $\\alpha \\in (-1,1)$, which has been introduced by Cherstvy et al. (New J. Phys. 15 (2013) 083039) in the context …
View article: Generalized Peano problem with Lévy noise
Generalized Peano problem with Lévy noise Open
We revisit the zero-noise Peano selection problem for Lévy-driven stochastic differential equation considered in [Pilipenko and Proske, Statist. Probab. Lett., 132:62–73, 2018] and show that the selection phenomenon pertains in the multipl…
View article: First-passage properties of asymmetric Lévy flights
First-passage properties of asymmetric Lévy flights Open
Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the ‘jump lengths’—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the v…
View article: Design, modeling and research of the new non-autonomous chaotic generator
Design, modeling and research of the new non-autonomous chaotic generator Open
тономного хаотичного генератора з нелiнiйнiстю “дiодоперацiйний пiдсилювач”. Це свого роду одна iз найпростiших схем, яка проявляє хаотичну поведiнку. Дана
\nсхема неавтономного хаотичного генератора мiстить чотири резистора, один конденса…
View article: Stratonovich SDE with irregular coefficients: Girsanov's example revisited
Stratonovich SDE with irregular coefficients: Girsanov's example revisited Open
In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^α\circ\mathrm{d} B$, $α\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context of analysis…
View article: Advection-diffusion equation on a half-line with boundary Lévy noise
Advection-diffusion equation on a half-line with boundary Lévy noise Open
In this paper we study a one-dimensional linear advection-diffusion equation on a half-line driven by a Lévy boundary noise. The problem is motivated by the study of contaminant transport models under random sources (P. P. Wang and C. Zhen…
View article: Non-Gaussian Limit Theorem for Non-Linear Langevin Equations Driven by Lévy Noise
Non-Gaussian Limit Theorem for Non-Linear Langevin Equations Driven by Lévy Noise Open
In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equation $\ddot x^\varepsilon_t +|\dot x^\varepsilon_t|^β=\dot Z^\varepsilon_{\varepsilon t}$, $β\in\mathbb R$, driven by symmetric non-Ga…
View article: Limit Theorem for Non-Linear Langevin Equations Driven by L\'evy Noise
Limit Theorem for Non-Linear Langevin Equations Driven by L\'evy Noise Open
In this paper, we study the small noise behaviour of solutions of a non-linear second order Langevin equation $\ddot x^\varepsilon_t +|\dot x^\varepsilon_t|^\beta=\dot Z^\varepsilon_{\varepsilon t}$, $\beta\in\mathbb R$, driven by symmetri…
View article: Random perturbations of a periodically driven nonlinear oscillator: escape from a resonance zone
Random perturbations of a periodically driven nonlinear oscillator: escape from a resonance zone Open
The phase space for a periodically driven nonlinear oscillator consists of many resonance zones. Let the strength of periodic excitation and the strength of the damping be indexed by a small parameter $\varepsilon$. It is well known that, …
View article: Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process
Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process Open
We study a bistable gradient systemperturbed by a stable-like additive process with a periodically varying stability index.Among a continuum of intrinsic time scales determined by the values of the stability indexwe single out the characte…
View article: Random Perturbations of Periodically Driven Nonlinear Oscillators
Random Perturbations of Periodically Driven Nonlinear Oscillators Open
This paper develops a unified approach to study the dynamics of nonlinear oscillators excited by both periodic and random per- turbations. This study is motivated by problems that range from nonlinear energy harvesting to ship capsizing in…