Fokker-Planck equation driven by asymmetric Lévy motion

Applied Mathematics and Computation, 2016

The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian α−stable symmetric Lévy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations, under either absorbing or natural conditions. The scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.

2001

The Fokker-Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. In this paper, we therefore derive a fractional Fokker-Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevintype equation, which is driven by a Lévy stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker-Planck equation.

arXiv (Cornell University), 2016

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker-Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker-Plank equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker-Planck equations for Marcus SDEs driven by Lévy processes.

Journal of Mathematical Physics, 2016

Physica A: Statistical Mechanics and its Applications, 2000

We demonstrate that the Fokker-Planck equation can be generalized into a 'Fractional Fokker-Planck' equation, i.e. an equation which includes fractional space differentiations, in order to encompass the wide class of anomalous diffusions due to a Lévy stable stochastic forcing. A precise determination of this equation is obtained by substituting a Lévy stable source to the classical gaussian one in the Langevin equation. This yields not only the anomalous diffusion coefficient, but a non trivial fractional operator which corresponds to the possible asymmetry of the Lévy stable source. Both of them cannot be obtained by scaling arguments. The (mono-) scaling behaviors of the Fractional Fokker-Planck equation and of its solutions are analysed and a generalization of the Einstein relation for the anomalous diffusion coefficient is obtained. This generalization yields a straightforward physical interpretation of the parameters of Lévy stable distributions. Furthermore, with the help of important examples, we show the applicability of the Fractional Fokker-Planck equation in physics.

arXiv (Cornell University), 2024

We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which respects the local limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris's theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to solutions of the usual Fokker-Planck equation uniformly in time-hence we show the approximation is asymptotic-preserving in this sense. The associated equilibrium has some interesting tail and regularity properties, which we also study.

2000

In this paper, we study a Fokker-Planck equation of the form ut = I(u) + div(xu) where the operator I, which is usually the Laplacian, is replaced here with a general Levy operator. We prove by the entropy production method the exponential decay in time of the solution to the only steady state of the associated stationnary equation.

arXiv (Cornell University), 2015

Despite there are numerous theoretical studies of stochastic differential equations with a symmetric αstable Lévy noise, very few regularity results exist in the case of 0 < α ≤ 1. In this paper, we study the fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift, and prove that there exists a unique solution, which is C ∞ in space for t > 0 when α ∈ (0, 2].

Entropy, 2018

The numerical solutions to a non-linear Fractional Fokker–Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.

Journal of Physics A: Mathematical and Theoretical

Complex systems are known to display anomalous diffusion, whose signature is a space/time scaling x ∼ t δ with δ = 1/2 in the Probability Density Function (PDF). Anomalous diffusion can emerge jointly with both Gaussian, e.g., fractional Brownian motion, and power-law decaying distributions, e.g., Lévy Flights (LFs) or Lévy Walks (LWs). LFs get anomalous scaling, but also infinite position variance and, being jumps of any size allowed even at short times, also infinite energy and discontinuous velocity. LWs are based on random trapping events, resemble a Lévy-type power-law distribution that is truncated in the large displacement range and have finite moments, finite energy and discontinuous velocity. However, both LFs and LWs cannot describe friction-diffusion processes and do not take into account the role of strong heterogeneity in many complex systems, such as biological transport in the crowded cell environment. We propose and discuss a model describing a Heterogeneous Ensemble of Brownian Particles (HEBP) based on a linear Langevin equation. We show that, for proper distributions of relaxation time and velocity diffusivity, the HEBP displays features similar to LWs, in particular power-law decaying PDF, longrange correlations and anomalous diffusion, at the same time keeping finite position moments and finite energy. The main differences between the HEBP model and two LWs are investigated, finding that, even if the PDFs are similar, they differ in three main aspects: (i) LWs are biscaling, while HEBP is monoscaling; (ii) a transition from anomalous (δ = 1/2) to normal (δ = 1/2) diffusion in the long-time regime; (iii) the power-law index of the position PDF and the space/time diffusion scaling are independent in the HEBP, while they both depend on the scaling of the inter-event time PDF in LWs. The HEBP model is derived from a friction-diffusion process, it has finite energy and it satisfies the fluctuation-dissipation theorem.