arXiv subjects

Ilya Pavlyukevich

Publications and source records attributed to Ilya Pavlyukevich.

34 records · Page 2Linked to original sources

arXiv2019

Stratonovich SDE with irregular coefficients: Girsanov's example revisited

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 of anomalous diffusions in heterogeneous media. We determine its weak and strong solutions, which are homogeneous strong Markov processes \chng{spending zero time at $0$: for $α\in (0,1)$, these solutions have the form $$ X_t^θ=\bigl((1-α)B_t^θ\bigr)^{1/(1-α)}, $$ where $B^θ$ is the $θ$-skew Brownian motion driven by $B$ and starting at $\frac{1}{1-α}(X_0)^{1-α}$, $θ\in [-1,1]$,} and $(x)^γ=|x|^γ\operatorname{sign} x$; for $α\in(-1,0]$, only the case $θ=0$ is possible. The central part of the paper consists in the proof of the existence of a quadratic covariation $[f(B^θ),B]$ for a locally square integrable function $f$ and is based on the time-reversion technique for Markovian diffusions.

math.PR↗

arXiv2019

The first passage problem for stable linear delay equations perturbed by power law Lévy noise

This article studies a linear scalar delay differential equation subject to small multiplicative power tail Lévy noise. We solve the first passage (the Kramers) problem with probabilistic methods and discover an asymptotic loss of memory in this non-Markovian system. Furthermore, the mean exit time increases as the power of the small noise amplitude, whereas the pre-factor accounts for memory effects. In particular, we discover a non-linear delay-induced exit acceleration due to a non-normal growth phenomenon. Our results are illustrated in the well-known linear delay oscillator driven by $α$-stable Lévy flights.

math.PR↗

arXiv2018

Non-Gaussian Limit Theorem for Non-Linear Langevin Equations Driven by Lévy Noise

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-Gaussian Lévy processes $Z^\varepsilon$. This equation describes the dynamics of a one-degree-of-freedom mechanical system subject to non-linear friction and noisy vibrations. For a compound Poisson noise, the process $x^\varepsilon$ on the macroscopic time scale $t/\varepsilon$ has a natural interpretation as a non-linear filter which responds to each single jump of the driving process. We prove that a system driven by a general symmetric Lévy noise exhibits essentially the same asymptotic behaviour under the principal condition $α+2β<4$, where $α\in [0,2]$ is the ``uniform'' Blumenthal--Getoor index of the family $\{Z^\varepsilon\}_{\varepsilon>0}$.

math.PR↗

arXiv2017

Random Perturbations of a Periodically Driven Nonlinear Oscillator: Escape from a resonance zone

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, as $\varepsilon \to 0$, the measure of the set of initial conditions which lead to 'capture in a resonance zone' goes to zero. In this paper we study the effect of weak noise on the escape from a resonance zone.

math.PR↗

arXiv2015

Spectral Analysis of a Discrete Metastable System Driven by Lévy Flights

In this paper we consider a finite state time discrete Markov chain that mimics the behaviour of solutions of the stochastic differential equation $dX=-U'(X)dt+εdL$, where $U$ is a multi-well potential with $n\geq 2$ local minima and L is a symmetric α-stable Lévy process (Lévy flights process). We investigate the spectrum of the generator of this Markov chain in the limit $ε\to 0$ and localize the top n eigenvalues $λ^ε_1,\dots, λ^ε_n$. These eigenvalues turn out to be of the same algebraic order $O(ε^α)$ and are well separated from the rest of the spectrum by a spectral gap. We also determine the limits $\lim_{ε\to 0}ε^{-α} λ^ε_i$, $1\leq i\leq n$, and show that the corresponding eigenvectors are approximately constant over the domains which correspond to the potential wells of $U$.

math.PR↗

arXiv2014

Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces

We study the convergence in probability in the non-standard $M_1$ Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type $\int_0^t F_γ(t-s)\,d L(s)$ to a process $\int_0^t F(t-s)\, d L(s)$ driven by a Lévy process $L$. In Banach spaces we introduce strong, weak and product modes of $M_1$-convergence, prove a criterion for the $M_1$-convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behaviour of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein--Uhlenbeck processes with diagonalisable generators.

math.PR↗

arXiv2014

Marcus versus Stratonovich for Systems with Jump Noise

The famous Itô-Stratonovich dilemma arises when one examines a dynamical system with a multiplicative white noise. In physics literature, this dilemma is often resolved in favour of the Stratonovich prescription because of its two characteristic properties valid for systems driven by Brownian motion: (i) it allows physicists to treat stochastic integrals in the same way as conventional integrals, and (ii) it appears naturally as a result of a small correlation time limit procedure. On the other hand, the Marcus prescription [IEEE Trans. Inform. Theory 24, 164 (1978); Stochastics 4, 223 (1981)] should be used to retain (i) and (ii) for systems driven by a Poisson process, Lévy flights or more general jump processes. In present communication we present an in-depth comparison of the Itô, Stratonovich, and Marcus equations for systems with multiplicative jump noise. By the examples of areal-valued linear system and a complex oscillator with noisy frequency (the Kubo-Anderson oscillator) we compare solutions obtained with the three prescriptions.

cond-mat.stat-mech↗

arXiv2014

Metastability of Morse-Smale dynamical systems perturbed by heavy-tailed Lévy type noise

We consider a general class of finite dimensional deterministic dynamical systems with finitely many local attractors $K^i$ each of which supports a unique ergodic probability measure $P^i$, which includes in particular the class of Morse-Smale systems in any finite dimension. The dynamical system is perturbed by a multiplicative non-Gaussian heavy-tailed Lévy type noise of small amplitude $\varepsilon>0$. Specifically we consider perturbations leading to a Itô, Stratonovich and canonical (Marcus) stochastic differential equation. The respective asymptotic first exit time and location problem from each of the domains of attractions $D^i$ in case of inward pointing vector fields in the limit of $\varepsilon \to 0$ was solved by the authors in [J. Stoch. An. Appl. 32(1), 163-190, 2014]. We extend these results to domains with characteristic boundaries and show that the perturbed system exhibits a metastable behavior in the sense that there exits a unique $\varepsilon$-dependent time scale on which the random system converges to a continuous time Markov chain switching between the invariant measures $P^i$. As examples we consider $α$-stable perturbations of the Duffing equation and a chemical system exhibiting a birhythmic behavior.

math.PR↗

arXiv2014

Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by $α$-stable Lévy processes

In this paper, we study the asymptotic behaviour of one-dimensional integrated Ornstein-Uhlenbeck processes driven by $α$-stable Lévy processes of small amplitude. We prove that the integrated Ornstein-Uhlenbeck process converges weakly to the underlying $α$-stable Lévy process in the Skorokhod $M_1$-topology which secures the weak convergence of first passage times. This result follows from a more general result about approximations of an arbitrary Lévy process by continuous integrated Ornstein-Uhlenbeck processes in the $M_1$-topology.

math.PR↗

arXiv2013

The exit problem from the neighborhood of a global attractor for heavy-tailed Lévy diffusions

We consider a finite dimensional deterministic dynamical system with a global attractor A with a unique ergodic measure P concentrated on it, which is uniformly parametrized by the mean of the trajectories in a bounded set D containing A. We perturbe this dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ε>0 and solve the asymptotic first exit time and location problem from a bounded domain D around the attractor A in the limit of ε to 0. In contrast to the case of Gaussian perturbations, the exit time has the asymptotically algebraic exit rate as a function of ε, just as in the case when A is a stable fixed point. In the small noise limit, we determine the joint law of the first time and the exit location on the complement of D. As an example, we study the first exit problem from a neighbourhood of a stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise.

math.PR↗

arXiv2012

First exit times of solutions of stochastic differential equations driven by multiplicative Levy noise with heavy tails

In this paper we study first exit times from a bounded domain of a gradient dynamical system $\dot Y_t=-\nabla U(Y_t)$ perturbed by a small multiplicative Lévy noise with heavy tails. A special attention is paid to the way the multiplicative noise is introduced. In particular we determine the asymptotics of the first exit time of solutions of Itô, Stratonovich and Marcus canonical SDEs.

math.PR↗

arXiv2010

Lévy Ratchet in a Weak Noise Limit: Theory and Simulation

We study the motion of a particle embedded in a time independent periodic potential with broken mirror symmetry and subjected to a Lévy noise possessing Lévy stable probability law (Lévy ratchet). We develop analytical approach to the problem based on the asymptotic probabilistic method of decomposition proposed by P. Imkeller and I. Pavlyukevich [J. Phys. A {\bf39}, L237 (2006); Stoch. Proc. Appl. {\bf116}, 611 (2006)]. We derive analytical expressions for the quantities characterizing the particle motion, namely the splitting probabilities of first escape from a single well, the transition probabilities and the particle current. A particular attention is devoted to the interplay between the asymmetry of the ratchet potential and the asymmetry (skewness) of the Lévy noise. Intensive numerical simulations demonstrate a good agreement with the analytical predictions for sufficiently small intensities of the Lévy noise driving the particle.

cond-mat.stat-mech↗

arXiv2009

First exit times for Lévy-driven diffusions with exponentially light jumps

We consider a dynamical system described by the differential equation $\dot{Y}_t=-U'(Y_t)$ with a unique stable point at the origin. We perturb the system by the Lévy noise of intensity $\varepsilon$ to obtain the stochastic differential equation $dX^{\varepsilon}_t=-U'(X^{\varepsilon}_{t-}) dt+\varepsilon dL_t.$ The process $L$ is a symmetric Lévy process whose jump measure $ν$ has exponentially light tails, $ν([u,\infty))\sim\exp(-u^α)$, $α>0$, $u\to \infty$. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval $(-1,1)$. In the small noise limit $\varepsilon\to0$, the law of the first exit time $σ_x$, $x\in(-1,1)$, has exponential tail and the mean value exhibiting an intriguing phase transition at the critical index $α=1$, namely, $\ln\mathbf{E}σ\sim\varepsilon^{-α}$ for $0<α<1$, whereas $\ln\mathbf{E}σ\sim\varepsilon^{- 1}|\ln\varepsilon|^{1-{1}/α}$ for $α>1$.

math.PR↗

arXiv2006

Metastable Behaviour of Small Noise Levy-Driven Diffusions

We consider a dynamical system in R driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Levy noise of small intensity and such that the heaviest tail of its Levy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature of the random perturbation, the results differ strongly from the well studied purely Gaussian case.

math.PR↗

arXiv2006

First exit times of solutions of non-linear stochastic differential equations driven by symmetric Levy processes with alpha-stable components

We study the exit problem of solutions of the stochastic differential equation dX(t)=-U'(X(t))dt+epsilon dL(t) from bounded or unbounded intervals which contain the unique asymptotically stable critical point of the deterministic dynamical system dY=-U'(Y) dt. The process L is composed of a standard Brownian motion and a symmetric alpha-stable Levy process. Using probabilistic estimates we show that in the small noise limit epsilon->0, the exit time of X from an interval is an exponentially distributed random variable and determine its expected value. Due to the heavy-tail nature of the alpha-stable component of L, the results differ strongly from the well known case in which the deterministic dynamical system undergoes purely Gaussian perturbations.

math.PR↗

arXiv2003

Stochastic Resonance in Two-State Markov Chains

In this paper we introduce a model which provides a new approach to the phenomenon of stochastic resonance. It is based on the study of the properties of the stationary distribution of the underlying stochastic process. We derive the formula for the spectral power amplification coefficient, study its asymptotic properties and dependence on parameters.

math.PR↗