In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating th... more In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms Ii,j = R 2 φ(x) xixj |x| 2+α dx, 0 < α < 2 where i, j ∈ {1, 2} and φ ∈ C N c for N ≥ 2. This type of singular integrals and its quadrature rule appear in the numerical discretization of fractional Laplacian in non-local Fokker-Planck Equations in 2D. The quadrature rules are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is 2p + 4 − α, where p ∈ N0 is associated with total number of correction weights. Although we work in 2D setting, we formulate definitions and theorems in n ∈ N dimensions when appropriate for the sake of generality. We present numerical experiments to validate the order of convergence of the proposed modified quadrature rules.
This paper proposes a general symplectic Euler scheme for a class of Hamiltonian stochastic diffe... more This paper proposes a general symplectic Euler scheme for a class of Hamiltonian stochastic differential equations driven by Lévy noise in the sense of Marcus form. The convergence of the symplectic Euler scheme for this Hamiltonian stochastic differential equations is investigated. Realizable numerical implementation of this scheme is also provided in details. Numerical experiments are presented to illustrate the effectiveness and superiority of the proposed method by the simulations of its orbits, symplectic structure and Hamlitonian.
This work is about low dimensional reduction for a slow-fast data assimilation system with non-Ga... more This work is about low dimensional reduction for a slow-fast data assimilation system with non-Gaussian α−stable Lévy noise via stochastic averaging. When the observations are only available for slow components, we show that the averaged, low dimensional filter approximates the original filter, by examining the corresponding Zakai stochastic partial differential equations. Furthermore, we demonstrate that the low dimensional slow system approximates the slow dynamics of the original system, by examining parameter estimation and most probable paths.
Despite there are numerous theoretical studies of stochastic differential equations with a symmet... more 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].
Advances in Computational Mathematics, Oct 24, 2018
Non-Gaussian Lévy noises are present in many models for understanding underlining principles of p... more Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology and more. In this work, we consider the Fokker-Planck equation(FPE) due to one-dimensional asymmetric Lévy motion, which is a nonlocal partial differential equation. We present an accurate numerical quadrature for the singular integrals in the nonlocal FPE and develop a fast summation method to reduce the order of the complexity from O(J 2) to O(J log J) in one time-step, where J is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises and the domain size.
Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynam... more 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.
A class of Hamiltonian stochastic differential equations with multiplicative Lévy noise in the se... more A class of Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus, and the construction and numerical implementation methods of symplectic Euler scheme, are considered. A general symplectic Euler scheme for this kind of Hamiltonian stochastic differential equations is devised, and its convergence theorem is proved. The second part presents realizable numerical implementation methods for this scheme in details. Some numerical experiments are conducted to demonstrate the effectiveness and superiority of the proposed method by the simulations of its orbits, Hamlitonian,and convergence order over a long time interval.
Nonlinear random vibration under excitations of both Gaussian and Poisson white noises is conside... more 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.
Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinea... more Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems. However, these nonlinear systems are often subject to random fluctuations or noises. It is thus desirable to quantify the impact of noises on the invariant manifolds. When the noise intensity is small, this impact is estimated via asymptotic analysis in the context of Liapunov–Perron formulation. Namely, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold, with a well-defined bound for the deviation.
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 sy... more 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.
A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the sys... more A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the system induced by Gaussian noise has been intensively investigated, but Gaussian noise cannot incorporate large bursts typically occurring in real experiments. This paper aims to examine effects of variations from one protein imposed by a non-Gaussian Lévy noise, which is able to describe even large jumps, on the coherent switch and the on/off switch via the steady-state probability density, the joint steady-state probability density, and the mean first passage time. We find that a large burst of one protein due to the Lévy noises can induce coherent switches even with small noise intensities in contrast to the Gaussian case which requires large intensities for this. The influences of the stability index, skewness parameter and noise intensity on the on/off switch are analyzed, leading to an adjustment of the concentrations of both proteins and a decision which stable point to stay most. Th...
DESCRIPTION We are devoted to the study of nonlocal Fokker-Plank equations for a class of stochas... more DESCRIPTION We are devoted to the study of nonlocal Fokker-Plank equations for a class of stochastic differential equations with non-Gaussian $\alpha$-stable L\'evy motion in Euclidean space. More precisely, in cases of free L\'evy motion and L\'evy motion with Ornstein-Uhlenbeck drift, we prove that there exists a unique solution, and it is smooth in space for all $0<\alpha<2$.
42nd AIAA Aerospace Sciences Meeting and Exhibit, 2004
Public reporting burden for this collection of information is estimated to average 1 hour per res... more Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are
Intracellular sorting centers, including the trans-Golgi network, the endoplasmic reticulum, and ... more Intracellular sorting centers, including the trans-Golgi network, the endoplasmic reticulum, and the endocytic recycling compartment, all contain membrane tube elements with cylindrical curvatures. The question of how curvature and sorting are coupled is central to the understanding of the function of organelle homeostasis, membrane trafficking, and intracellular sorting. Of particular interest, but underexplored, are non-equilibrium phenomena fundamentally linked to membrane transport. Here we present a straightforward and well controlled model system that allows us to characterize lipid transport at steady state: ternary lipid mixture tubular membranes pulled from phase-separated giant unilamellar vesicles (GUVs) by means of optical tweezers. The tubule composition, when initially formed from the liquid-ordered (Lo) phase region of the vesicle, is dependent on the velocity at which it is pulled: fast extraction velocities create tethers of the Lo phase, while slow extraction velocities can generate tethers that are liquiddisordered (Ld) phase. Thus, the speed with which highly curved tubules are formed may possibly serve as a control variable in living cells to adjust tubule composition. Furthermore, in tubules which are initially Lo phase, we find curvature-induced nucleation of Ld domains at the neck between tubule and vesicle. These Ld domains display characteristic, curvature dependent parabolic growth behavior that can be understood via a straightforward analytical mass transfer model that we derived from linear irreversible thermodynamics. We have also developed numerical schemes that capture shape transitions of tubular membranes on the basis of measured biophysical parameters in support of our experimental findings.
We present a new numerical method for solving two-dimensional Stokes flow with deformable interfa... more We present a new numerical method for solving two-dimensional Stokes flow with deformable interfaces such as dynamics of suspended drops or bubbles. The method is based on a boundary integral formulation for the interfacial velocity and is spectrally accurate in space. We analyze the singular behavior of the integrals (single-layer and double-layer integrals) appearing in the equations. The interfaces are formulated in the tangent angle and arc-length coordinates and, to reduce the stiffness of the evolution equation, the marker points are evenly distributed in arc-length by choosing a proper tangential velocity along the interfaces. Examples of Stokes flow with bubbles are provided to demonstrate the accuracy and effectiveness of the numerical method.
The mean exit time and escape probability are deterministic quantities that can quantify dynamica... more The mean exit time and escape probability are deterministic quantities that can quantify dynamical behaviors of stochastic differential equations with non-Gaussian α-stable type Lévy motions. Both deterministic quantities are characterized by differential-integral equations (i.e., differential equations with nonlocal terms) but with different exterior conditions. A convergent numerical scheme is developed and validated for computing the mean exit time and escape probability for two-dimensional stochastic systems with rotationally symmetric α-stable type Lévy motions. The effects of drift, Gaussian noises, intensity of jump measure and domain sizes on the mean exit time are discussed. The difference between the one-dimensional and two-dimensional cases is also presented.
This work is devoted to investigating the evolution of concentration in a genetic regulation syst... more This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcati...
The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian α−stable symmetri... more 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.
In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating th... more In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms Ii,j = R 2 φ(x) xixj |x| 2+α dx, 0 < α < 2 where i, j ∈ {1, 2} and φ ∈ C N c for N ≥ 2. This type of singular integrals and its quadrature rule appear in the numerical discretization of fractional Laplacian in non-local Fokker-Planck Equations in 2D. The quadrature rules are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is 2p + 4 − α, where p ∈ N0 is associated with total number of correction weights. Although we work in 2D setting, we formulate definitions and theorems in n ∈ N dimensions when appropriate for the sake of generality. We present numerical experiments to validate the order of convergence of the proposed modified quadrature rules.
This paper proposes a general symplectic Euler scheme for a class of Hamiltonian stochastic diffe... more This paper proposes a general symplectic Euler scheme for a class of Hamiltonian stochastic differential equations driven by Lévy noise in the sense of Marcus form. The convergence of the symplectic Euler scheme for this Hamiltonian stochastic differential equations is investigated. Realizable numerical implementation of this scheme is also provided in details. Numerical experiments are presented to illustrate the effectiveness and superiority of the proposed method by the simulations of its orbits, symplectic structure and Hamlitonian.
This work is about low dimensional reduction for a slow-fast data assimilation system with non-Ga... more This work is about low dimensional reduction for a slow-fast data assimilation system with non-Gaussian α−stable Lévy noise via stochastic averaging. When the observations are only available for slow components, we show that the averaged, low dimensional filter approximates the original filter, by examining the corresponding Zakai stochastic partial differential equations. Furthermore, we demonstrate that the low dimensional slow system approximates the slow dynamics of the original system, by examining parameter estimation and most probable paths.
Despite there are numerous theoretical studies of stochastic differential equations with a symmet... more 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].
Advances in Computational Mathematics, Oct 24, 2018
Non-Gaussian Lévy noises are present in many models for understanding underlining principles of p... more Non-Gaussian Lévy noises are present in many models for understanding underlining principles of physics, finance, biology and more. In this work, we consider the Fokker-Planck equation(FPE) due to one-dimensional asymmetric Lévy motion, which is a nonlocal partial differential equation. We present an accurate numerical quadrature for the singular integrals in the nonlocal FPE and develop a fast summation method to reduce the order of the complexity from O(J 2) to O(J log J) in one time-step, where J is the number of unknowns. We also provide conditions under which the numerical schemes satisfy maximum principle. Our numerical method is validated by comparing with exact solutions for special cases. We also discuss the properties of the probability density functions and the effects of various factors on the solutions, including the stability index, the skewness parameter, the drift term, the Gaussian and non-Gaussian noises and the domain size.
Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynam... more 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.
A class of Hamiltonian stochastic differential equations with multiplicative Lévy noise in the se... more A class of Hamiltonian stochastic differential equations with multiplicative Lévy noise in the sense of Marcus, and the construction and numerical implementation methods of symplectic Euler scheme, are considered. A general symplectic Euler scheme for this kind of Hamiltonian stochastic differential equations is devised, and its convergence theorem is proved. The second part presents realizable numerical implementation methods for this scheme in details. Some numerical experiments are conducted to demonstrate the effectiveness and superiority of the proposed method by the simulations of its orbits, Hamlitonian,and convergence order over a long time interval.
Nonlinear random vibration under excitations of both Gaussian and Poisson white noises is conside... more 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.
Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinea... more Invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems. However, these nonlinear systems are often subject to random fluctuations or noises. It is thus desirable to quantify the impact of noises on the invariant manifolds. When the noise intensity is small, this impact is estimated via asymptotic analysis in the context of Liapunov–Perron formulation. Namely, the random invariant manifold is represented as a perturbation of the deterministic invariant manifold, with a well-defined bound for the deviation.
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 sy... more 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.
A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the sys... more A bistable toggle switch is a paradigmatic model in the field of biology. The dynamics of the system induced by Gaussian noise has been intensively investigated, but Gaussian noise cannot incorporate large bursts typically occurring in real experiments. This paper aims to examine effects of variations from one protein imposed by a non-Gaussian Lévy noise, which is able to describe even large jumps, on the coherent switch and the on/off switch via the steady-state probability density, the joint steady-state probability density, and the mean first passage time. We find that a large burst of one protein due to the Lévy noises can induce coherent switches even with small noise intensities in contrast to the Gaussian case which requires large intensities for this. The influences of the stability index, skewness parameter and noise intensity on the on/off switch are analyzed, leading to an adjustment of the concentrations of both proteins and a decision which stable point to stay most. Th...
DESCRIPTION We are devoted to the study of nonlocal Fokker-Plank equations for a class of stochas... more DESCRIPTION We are devoted to the study of nonlocal Fokker-Plank equations for a class of stochastic differential equations with non-Gaussian $\alpha$-stable L\'evy motion in Euclidean space. More precisely, in cases of free L\'evy motion and L\'evy motion with Ornstein-Uhlenbeck drift, we prove that there exists a unique solution, and it is smooth in space for all $0<\alpha<2$.
42nd AIAA Aerospace Sciences Meeting and Exhibit, 2004
Public reporting burden for this collection of information is estimated to average 1 hour per res... more Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to
This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are
Intracellular sorting centers, including the trans-Golgi network, the endoplasmic reticulum, and ... more Intracellular sorting centers, including the trans-Golgi network, the endoplasmic reticulum, and the endocytic recycling compartment, all contain membrane tube elements with cylindrical curvatures. The question of how curvature and sorting are coupled is central to the understanding of the function of organelle homeostasis, membrane trafficking, and intracellular sorting. Of particular interest, but underexplored, are non-equilibrium phenomena fundamentally linked to membrane transport. Here we present a straightforward and well controlled model system that allows us to characterize lipid transport at steady state: ternary lipid mixture tubular membranes pulled from phase-separated giant unilamellar vesicles (GUVs) by means of optical tweezers. The tubule composition, when initially formed from the liquid-ordered (Lo) phase region of the vesicle, is dependent on the velocity at which it is pulled: fast extraction velocities create tethers of the Lo phase, while slow extraction velocities can generate tethers that are liquiddisordered (Ld) phase. Thus, the speed with which highly curved tubules are formed may possibly serve as a control variable in living cells to adjust tubule composition. Furthermore, in tubules which are initially Lo phase, we find curvature-induced nucleation of Ld domains at the neck between tubule and vesicle. These Ld domains display characteristic, curvature dependent parabolic growth behavior that can be understood via a straightforward analytical mass transfer model that we derived from linear irreversible thermodynamics. We have also developed numerical schemes that capture shape transitions of tubular membranes on the basis of measured biophysical parameters in support of our experimental findings.
We present a new numerical method for solving two-dimensional Stokes flow with deformable interfa... more We present a new numerical method for solving two-dimensional Stokes flow with deformable interfaces such as dynamics of suspended drops or bubbles. The method is based on a boundary integral formulation for the interfacial velocity and is spectrally accurate in space. We analyze the singular behavior of the integrals (single-layer and double-layer integrals) appearing in the equations. The interfaces are formulated in the tangent angle and arc-length coordinates and, to reduce the stiffness of the evolution equation, the marker points are evenly distributed in arc-length by choosing a proper tangential velocity along the interfaces. Examples of Stokes flow with bubbles are provided to demonstrate the accuracy and effectiveness of the numerical method.
The mean exit time and escape probability are deterministic quantities that can quantify dynamica... more The mean exit time and escape probability are deterministic quantities that can quantify dynamical behaviors of stochastic differential equations with non-Gaussian α-stable type Lévy motions. Both deterministic quantities are characterized by differential-integral equations (i.e., differential equations with nonlocal terms) but with different exterior conditions. A convergent numerical scheme is developed and validated for computing the mean exit time and escape probability for two-dimensional stochastic systems with rotationally symmetric α-stable type Lévy motions. The effects of drift, Gaussian noises, intensity of jump measure and domain sizes on the mean exit time are discussed. The difference between the one-dimensional and two-dimensional cases is also presented.
This work is devoted to investigating the evolution of concentration in a genetic regulation syst... more This work is devoted to investigating the evolution of concentration in a genetic regulation system, when the synthesis reaction rate is under additive and multiplicative asymmetric stable Lévy fluctuations. By focusing on the impact of skewness (i.e., non-symmetry) in the probability distributions of noise, we find that via examining the mean first exit time (MFET) and the first escape probability (FEP), the asymmetric fluctuations, interacting with nonlinearity in the system, lead to peculiar likelihood for transcription. This includes, in the additive noise case, realizing higher likelihood of transcription for larger positive skewness (i.e., asymmetry) index β, causing a stochastic bifurcation at the non-Gaussianity index value α = 1 (i.e., it is a separating point or line for the likelihood for transcription), and achieving a turning point at the threshold value β≈-0.5 (i.e., beyond which the likelihood for transcription suddenly reversed for α values). The stochastic bifurcati...
The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian α−stable symmetri... more 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.
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