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

Fractional Diffusion Equation for a Power-Law-Truncated Levy Process

Truncated Levy flights are stochastic processes which display a crossover from a heavy-tailed Levy behavior to a faster decaying probability distribution function (pdf). Putting less weight on long flights overcomes the divergence of the Levy distribution second moment. We introduce a fractional generalization of the diffusion equation, whose solution defines a process in which a Levy flight of exponent alpha is truncated by a power-law of exponent 5 - alpha. A closed form for the characteristic function of the process is derived. The pdf of the displacement slowly converges to a Gaussian in its central part showing however a power law far tail. Possible applications are discussed

Fluctuation-driven directed transport in the presence of Levy flights

Numerical evidence of directed transport driven by symmetric Levy noise in time-independent ratchet potentials in the absence of an external tilting force is presented. The results are based on the numerical solution of the fractional Fokker-Planck equation in a periodic potential and the corresponding Langevin equation with Levy noise. The Levy noise drives the system out of thermodynamic equilibrium and an up-hill net current is generated. For small values of the noise intensity there is an optimal value of the Levy noise index yielding the maximum current. The direction and magnitude of the current can be manipulated by changing the Levy noise asymmetry and the potential asymmetry

Generalized Elastic Model: thermal vs non-thermal initial conditions. Universal scaling, roughening, ageing and ergodicity

We study correlation properties of the generalized elastic model which accounts for the dynamics of polymers, membranes, surfaces and fluctuating interfaces, among others. We develop a theoretical framework which leads to the emergence of universal scaling laws for systems starting from thermal (equilibrium) or non-thermal (non-equilibrium) initial conditions. Our analysis incorporates and broadens previous results such as observables' double scaling regimes, (super)roughening and anomalous diffusion, and furnishes a new scaling behavior for correlation functions at small times (long distances). We discuss ageing and ergodic properties of the generalized elastic model in non-equilibrium conditions, providing a comparison with the situation occurring in continuous time random walk. Our analysis also allows to assess which observable is able to distinguish whether the system is in or far from equilibrium conditions in an experimental set-up

The BBGKY Hierarchy and Fokker-Planck Equation for Many-Body Dissipative Randomly Driven Systems

By generalizing Bogolyubov's reduced description method, we suggest a formalism to derive kinetic equations for many-body dissipative systems in external stochastic field. As a starting point, we use a stochastic Liouville equation obtained from Hamilton's equations taking dissipation and stochastic perturbations into account. The Liouville equation is then averaged over realizations of the stochastic field by an extension of the Furutsu-Novikov formula to the case of a non-Gaussian field. As the result, a generalization of the classical Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy is derived. In order to get a kinetic equation for the one-particle distribution function, we use a regular cut off procedure of the BBGKY hierarchy by assuming weak interaction between the particles and weak intensity of the field. Within this approximation we get the corresponding Fokker-Planck equation for the system in a non-Gaussian stochastic field. Two particular cases by assuming either Gaussian statistics of external perturbation or homogeneity of the system are discussed