arXiv subjects

Xiaofan Li

Publications and source records attributed to Xiaofan Li.

99 records · Page 6Linked to original sources

arXiv2016

Fokker-Planck equations for Marcus stochastic differential equations driven by Levy processes

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Levy 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 Levy 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 Levy processes.

math.DS↗

arXiv2016

Derivation of Fokker-Planck equations for stochastic dynamical systems under excitation of multiplicative non-Gaussian white noise

Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and play an important role in quantifying propagation and evolution of uncertainty. Although Fokker-Planck equations can be written explicitly for nonlinear dynamical systems excited by Gaussian white noise, they are not available in general for nonlinear dynamical systems excited by multiplicative non-Gaussian white noise. Marcus stochastic differential equations are often appropriate models in engineering and physics for stochastic dynamical systems excited by non-Gaussian white noise. In this paper, we derive explicit forms of Fokker-Planck equations for one dimensional systems modeled by Marcus stochastic differential equations under multiplicative non-Gaussian white noise. As examples to illustrate the theoretical results, the derived formula is used to obtain Fokker-Plank equations for nonlinear dynamical systems under excitation of (i) $α$-stable white noise; (ii) combined Gaussian and Poisson white noise, respectively.

math.DS↗

arXiv2015

A regularity result for the nonlocal Fokker-Planck equation with Ornstein-Uhlenbeck drift

Despite there are numerous theoretical studies of stochastic differential equations with a symmetric $α$-stable Lévy noise, very few regularity results exist in the case of $0<α\leq1$. In this paper, we study the fractional Fokker-Planck equation with Ornstein-Uhlenbeck drift, and prove that there exists a unique solution, which is $C^\infty$ in space for $t>0$ when $α\in (0, 2]$.

math.DS↗

arXiv2013

Fokker-Planck Equations for Stochastic Dynamical Systems with Symmetric Lévy Motions

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.

math.NA↗

arXiv2013

A Conservative Finite Difference Scheme for Poisson-Nernst-Planck Equations

A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson-Nernst-Planck(PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, which is second-order accurate in both space and time. We use the physical parameters specifically suited toward the modelling of ion channels. We present a simple iterative scheme to solve the system of nonlinear equations resulting from discretizing the equations implicitly in time, which is demonstrated to converge in a few iterations. We place emphasis on ensuring numerical methods to have the same physical properties that the PNP equations themselves also possess, namely conservation of total ions and correct rates of energy dissipation. We describe in detail an approach to derive a finite-difference method that preserves the total concentration of ions exactly in time. Further, we illustrate that, using realistic values of the physical parameters, the conservation property is critical in obtaining correct numerical solutions over long time scales.

math.NA↗

arXiv2013

State estimation under non-Gaussian Levy noise: A modified Kalman filtering method

The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance. A modified Kalman filter for linear systems with non-Gaussian Lévy noise is devised. It works effectively with reasonable computational cost. Simulation results are presented to illustrate this non-Gaussian filtering method.

math.DS↗

arXiv2012

Modeling nonlinear random vibration: Implication of the energy conservation law

Nonlinear random vibration under excitations of both Gaussian and Poisson white noises is considered. The model is based on stochastic differential equations, and the corresponding stochastic integrals are defined in such a way that the energy conservation law is satisfied. It is shown that Stratonovich integral and Di Paola-Falsone integral should be used for excitations of Gaussian and Poisson white noises, respectively, in order for the model to satisfy the underlining physical laws (e.g., energy conservation). Numerical examples are presented to illustrate the theoretical results.

math.DS↗

arXiv2012

An alternative expression of Di Paola and Falson's formula for stochastic dynamics

Di Paola and Falsone's formula is widely used in studying stochastic dynamics of nonlinear systems under Poisson white noise. In this short communication, an alternative expression is presented. Compared to Di Paola and Falsone's original expression, the alternative one is applicable under more general condition, and shows significantly improved performance in numerical implementation. The alternative expression turns out to be a special case of the Marcus integrals.

math.DS↗

arXiv2012

Mean exit time and escape probability for dynamical systems driven by Levy noise

The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian alpha-stable type Levy motions. Both deterministic quantities are characterized by differential-integral equations(i.e.,differential equations with non local terms) but with different exterior conditions. The non-Gaussianity of noises manifests as nonlocality at the level of mean exit time and escape probability. An objective of this paper is to make mean exit time and escape probability as efficient computational tools, to the applied probability community, for quantifying stochastic dynamics. An accurate numerical scheme is developed and validated for computing the mean exit time and escape probability. Asymptotic solution for the mean exit time is given when the pure jump measure in the Levy motion is small. From both the analytical and numerical results, it is observed that the mean exit time depends strongly on the domain size and the value of alpha in the alpha-stable Levy jump measure. The mean exit time can measure which of the two competing factors in alpha-stable Levy motion, i.e. the jump frequency or the jump size, is dominant in helping a process exit a bounded domain. The escape probability is shown to vary with the underlying vector field(i.e.,drift). The mean exit time and escape probability could become discontinuous at the boundary of the domain, when the process is subject to certain deterministic potential and the value of alpha is in (0,1).

math.NA↗