Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Levy processes

The Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and are thus widely used to quantify random phenomena such as uncertainty propagation. For dynamical systems driven by non-Gaussian L\'evy processes, however, it is difficult to obtain explicit forms of Fokker-Planck equations because the adjoint operators of the associated infinitesimal generators usually do not have exact formulation. In the present paper, Fokker- Planck equations are derived in terms of infinite series for nonlinear stochastic differential equations with non-Gaussian L\'evy processes. A few examples are presented to illustrate the method.Comment: 14 page

Marcus versus Stratonovich for Systems with Jump Noise

The famous It\^o-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, L\'evy flights or more general jump processes. In present communication we present an in-depth comparison of the It\^o, 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.Comment: 14 pages, 4 figure

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 L\'evy 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 It\^o, Stratonovich and Marcus canonical SDEs.Comment: 19 pages, 2 figure