Nonlocal Fokker-Planck Equation Research Articles

Fokker–Planck equation and Feynman–Kac formula for multidimensional stochastic dynamical systems with Lévy noises and time-dependent coefficients

The aim of this paper is to establish a version of the Feynman–Kac formula for the time-dependent multidimensional nonlocal Fokker–Planck equation corresponding to a class of time-dependent stochastic differential equations driven by multiplicative symmetric (or asymmetric) α-stable Lévy noise. First the forward nonlocal Fokker–Planck equation is derived by the adjoint operator method, overcoming the challenges posed by time-dependent multidimensional nonlinear symmetric α-stable Lévy noise. Subsequently, the Feynman–Kac formula for the forward multidimensional time-dependent nonlocal Fokker–Planck equation is established by applying techniques for the backward nonlocal Fokker–Planck equations, which is associated with the backward stochastic differential equation driven by the multiplicative symmetric α-stable Lévy noise. Notably, in the case of asymmetric α-stable Lévy noise case, the presence of the characteristic function in the nonlocal operator adds complexity to the analysis. Using the Feynman–Kac formula, it is demonstrated that the solution of the forward nonlocal Fokker–Planck equation can be readily simulated through Monte Carlo approximation, especially in scenarios involving long-time simulation settings with large steps. These concepts are illustrated with intriguing examples, and the dynamic evolution of the probability density function corresponding to the stochastic SIS model and the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, ComK and ComS) are investigated over an extended period.

Quantifying random collisions between particles inside and outside a circle

Random collisions of particles occur in various biophysical and physical systems. Inspired by the binding of receptor and ligand on the cell membrane, we devised a method based on stochastic dynamical modeling to quantify the likelihood of two random particles colliding on a circle. We consider the dynamics of a receptor binding to a ligand on the cell membrane, where the receptor and ligand perform different motions and are thus modeled by stochastic differential equations with non-Gaussian noise. We use neural networks based on the Onsager–Machlup function to compute the probability P1 of an unbounded receptor diffusing to the cell membrane. Meanwhile, we compute the probability P2 of the extracellular ligand arriving at the cell membrane by solving the associated nonlocal Fokker–Planck equation. We can then calculate the most probable binding probability by combining P1 and P2. In this way, we conclude with some indication of how the receptors could distribute on the membrane, as well as where the ligand will most probably encounter the receptor, contributing to a better understanding of the cell’s response to external stimuli and communication with other cells.

Existence and uniqueness of solutions for forward and backward nonlocal Fokker-Planck equations with time-dependent coefficients

Based on the Feynman-Kac formula, we develop a new approach to show the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations, which are associated with a class of time-dependent stochastic differential equations driven by symmetric (or asymmetric) non-Gaussian Lévy process. First, after deriving the forward time-dependent multidimensional nonlocal Fokker-Planck equation, the Feynman-Kac formula for the forward nonlocal Fokker-Planck equation is established by using Itô's formula and techniques for the backward nonlocal Fokker-Planck equation corresponding to the backward stochastic differential equation driven by Lévy process. Then, from the Feynman-Kac formula, we prove the global existence and uniqueness of the solutions for the forward and backward time-dependent multidimensional nonlocal Fokker-Planck equations by using techniques from stochastic analysis. Finally, the dynamic evolution of the probability density function corresponding to the stochastic model for the MeKS network (reflecting the interactions among the MecA complex, comK, comS) is investigated over an extended period by Feynman-Kac formula and Monte Carlo simulations. In particular, we reveal the influence of symmetric and asymmetric stable noises as well as tempered stable noise on the probability density function corresponding to the stochastic MeKS network model.

Most Probable Dynamics of the Single-Species with Allee Effect under Jump-Diffusion Noise

We explore the most probable phase portrait (MPPP) of a stochastic single-species model incorporating the Allee effect by utilizing the nonlocal Fokker–Planck equation (FPE). This stochastic model incorporates both non-Gaussian and Gaussian noise sources. It has three fixed points in the deterministic case. One is the unstable state, which lies between the two stable equilibria. Our primary focus is on elucidating the transition pathways from extinction to the upper stable state in this single-species model, particularly under the influence of jump-diffusion noise. This helps us to study the biological behavior of species. The identification of the most probable path relies on solving the nonlocal FPE tailored to the population dynamics of the single-species model. This enables us to pinpoint the corresponding maximum possible stable equilibrium state. Additionally, we derive the Onsager–Machlup function for the stochastic model and employ it to determine the corresponding most probable paths. Numerical simulations manifest three key insights: (i) when non-Gaussian noise is present in the system, the peak of the stationary density function aligns with the most probable stable equilibrium state; (ii) if the initial value rises from extinction to the upper stable state, then the most probable trajectory converges towards the maximally probable equilibrium state, situated approximately between 9 and 10; and (iii) the most probable paths exhibit a rapid ascent towards the stable state, then maintain a sustained near-constant level, gradually approaching the upper stable equilibrium as time goes on. These numerical findings pave the way for further experimental investigations aiming to deepen our comprehension of dynamical systems within the context of biological modeling.

On nonlocal Fokker–Planck equations with nonlinear force fields and perturbations

On nonlocal Fokker–Planck equations with nonlinear force fields and perturbations

Well-posedness of density dependent SDE driven by [formula omitted]-stable process with Hölder drifts

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by α-stable processes with α∈(1,2). The existence part is based on Euler’s approximation as Hao et al. (2021), while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker–Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being β0-order Hölder continuous in the spatial variable, where β0∈(1−α/2,1).

Trends to equilibrium for a nonlocal Fokker–Planck equation

We obtain equilibration rates for a one-dimensional nonlocal Fokker–Planck equation with time-dependent diffusion coefficient and drift, modeling the relaxation of a large swarm of robots, feeling each other in terms of their distance, towards the steady profile characterized by uniform spreading over a finite interval of the line. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution, with the properties of the quasi-stationary profile.

Solving the non-local Fokker-Planck equations by deep learning.

Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations (PDEs) under various initial and boundary conditions. In this paper, we propose trapz-PiNNs, physics-informed neural networks incorporated with a modified trapezoidal rule recently developed for accurately evaluating fractional Laplacian and solve the space-fractional Fokker-Planck equations in 2D and 3D. We describe the modified trapezoidal rule in detail and verify the second-order accuracy. We demonstrate that trapz-PiNNs have high expressive power through predicting the solution with low L 2 relative error by a variety of numerical examples. We also use local metrics, such as point-wise absolute and relative errors, to analyze where it could be further improved. We present an effective method for improving the performance of trapz-PiNN on local metrics, provided that physical observations or high-fidelity simulation of the true solution are available. The trapz-PiNN is able to solve PDEs with fractional Laplacian with arbitrary α ∈ ( 0 , 2 ) and on rectangular domains. It also has the potential to be generalized into higher dimensions or other bounded domains.

A Non-local Fokker-Planck Equation with Application to Probabilistic Evaluation of Sediment Replenishment Projects

A Non-local Fokker-Planck Equation with Application to Probabilistic Evaluation of Sediment Replenishment Projects

Data-driven approximation for extracting the transition dynamics of a genetic regulatory network with non-Gaussian Lévy noise

In the study of biological systems, several methods based on statistical physics or machine learning have been developed for inference or prediction in the presence of complicated nonlinear interactions and random noise perturbations. However, there have been few studies dealing with the stochastic non-Gaussian perturbation case, which is more natural and universal than Gaussian white noise. In this manuscript, for a two-dimensional biological model (the MeKS network) perturbed by non-Gaussian stable Lévy noise, we use a data-driven approach with theoretical probabilistic foundation to extract the rare transition dynamics representing gene expression. This involves theories of non-local Kramers–Moyal formulas and the non-local Fokker–Planck equation, as well as the corresponding numerical algorithms, aimed at extracting the maximum likelihood transition path. The feasibility and accuracy of the method are checked. Furthermore, several dynamical behaviors and indicators are investigated. In detail, the investigation shows a bistable transition probability state of the ComK protein concentration and bifurcations in the learned transition paths from vegetative state to competence state. Analysis of the tipping time illustrates the difficulty of the gene expression. This method will serve as an example in the study of stochastic systems with non-Gaussian perturbations from biological data, and provides some insights into the extraction of other dynamical indicators, such as the mean first exit time and the first escape probability with respect to their own biological interpretations.

Bifurcation and chaotic behavior in stochastic Rosenzweig–MacArthur prey–predator model with non-Gaussian stable Lévy noise

We perform dynamical analysis on a stochastic Rosenzweig–MacArthur model driven by α-stable Lévy motion. We analyze the existence of the equilibrium points, and provide a clear illustration of their stability. It is shown that the nonlinear model has at most three equilibrium points. If the coexistence equilibrium exists, it is asymptotically stable attracting all nearby trajectories. The phase portraits are drawn to gain useful insights into the dynamical underpinnings of prey–predator interaction. Specifically, we present a transcritical bifurcation curve at which system bifurcates. The stationary probability density is characterized by the non-local Fokker–Planck equation and confirmed by some numerical simulations. By applying Monte Carlo method and using statistical data, we plot a substantial number of simulated trajectories for stochastic system as parameter varies. For initial conditions that are arbitrarily close to the origin, parameter changes in noise terms can lead to significantly different future paths or trajectories with variations, which reflect chaotic behavior in mutualistically interacting two-species prey–predator system subject to stochastic influence.

Extracting stochastic governing laws by non-local Kramers-Moyal formulae.

With the rapid development of computational techniques and scientific tools, great progress of data-driven analysis has been made to extract governing laws of dynamical systems from data. Despite the wide occurrences of non-Gaussian fluctuations, the effective data-driven methods to identify stochastic differential equations with non-Gaussian Lévy noise are relatively few so far. In this work, we propose a data-driven approach to extract stochastic governing laws with both (Gaussian) Brownian motion and (non-Gaussian) Lévy motion, from short bursts of simulation data. Specifically, we use the normalizing flows technology to estimate the transition probability density function (solution of non-local Fokker-Planck equations) from data, and then substitute it into the recently proposed non-local Kramers-Moyal formulae to approximate Lévy jump measure, drift coefficient and diffusion coefficient. We demonstrate that this approach can learn the stochastic differential equation with Lévy motion. We present examples with one- and two-dimensional decoupled and coupled systems to illustrate our method. This approach will become an effective tool for discovering stochastic governing laws and understanding complex dynamical behaviours. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Arbitrarily high-order trapezoidal rules for functions with fractional singularities in two dimensions

In this paper, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the formsIi,j=∫R2ϕ(x)xixj|x|2+αdx,0<α<2where i,j∈{1,2} and ϕ∈CcN 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.

Learning the temporal evolution of multivariate densities via normalizing flows.

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning assisted construction of a time-dependent mapping that takes a reference distribution (say, a Gaussian) to each and every instance of our evolving distribution. If the reference distribution is the initial condition of a Fokker-Planck equation, what we learn is the time-T map of the corresponding solution. Specifically, the learned map is a multivariate normalizing flow that deforms the support of the reference density to the support of each and every density snapshot in time. We demonstrate that this approach can approximate probability density function evolutions in time from observed sampled data for systems driven by both Brownian and Lévy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.

Fokker–Planck equation for Feynman–Kac transform of anomalous processes

In this paper, we develop a novel and rigorous approach to the Fokker–Planck equation, or Kolmogorov forward equation, for the Feynman–Kac transform of non-Markov anomalous processes. The equation describes the evolution of the density of the anomalous process Yt=XEt under the influence of potentials, where X is a strong Markov process on a Lusin space X that is in weak duality with another strong Markov process X̂ on X and {Et,t≥0} is the inverse of a driftless subordinator S that is independent of X and has infinite Lévy measure. We derive a probabilistic representation of the density of the anomalous process under the Feynman–Kac transform by the dual Feynman–Kac transform in terms of the weak dual process X̂t and the inverse subordinator {Et;t≥0}. We then establish the regularity of the density function, and show that it is the unique mild solution as well as the unique weak solution of a non-local Fokker–Planck equation that involves the dual generator of X and the potential measure of the subordinator S. During the course of the study, we are naturally led to extend the notation of Riemann–Liouville integral to measures that are locally finite on [0,∞).

Stochastic bifurcations and tipping phenomena of insect outbreak systems driven by α-stable Lévy processes

In this work, we mainly characterize stochastic bifurcations and tipping phenomena of insect outbreak dynamical systems driven by α-stable Lévy processes. In one-dimensional insect outbreak model, we find the fixed points representing refuge and outbreak from the bifurcation curves, and carry out a sensitivity analysis with respect to small changes in the model parameters, the stability index and the noise intensity. We perform the numerical simulations of dynamical trajectories using Monte Carlo methods, which contribute to looking at stochastic hysteresis phenomenon. As for two-dimensional insect outbreak system, we identify the global stability properties of fixed points and express the probability density function by the stationary solution of the nonlocal Fokker-Planck equation. Through numerical modelling, the phase portrait makes it possible to detect critical transitions among the stable states. It is then worth describing stochastic bifurcation associated with tipping points induced by noise through a careful examination of the dynamical paths of the insect outbreak system with external forcing. The results give new insight into the sensitivity of ecosystems to realistic environmental changes represented by stochastic perturbations.

Kantorovich–Rubinstein distance and approximation for non-local Fokker–Planck equations

This work is devoted to studying complex dynamical systems under non-Gaussian fluctuations. We first estimate the Kantorovich–Rubinstein distance for solutions of non-local Fokker–Planck equations associated with stochastic differential equations with non-Gaussian Lévy noise. This is then applied to establish weak convergence of the corresponding probability distributions. Furthermore, this leads to smooth approximation for non-local Fokker–Planck equations, as illustrated in an example.

Stochastic bifurcation in single-species model induced by α-stable Lévy noise

Bifurcation analysis has many applications in different scientific fields, such as electronics, biology, ecology, and economics. In population biology, deterministic methods of bifurcation are commonly used. In contrast, stochastic bifurcation techniques are infrequently employed. Here we establish stochastic P-bifurcation behavior of (i) a growth model with state dependent birth rate and constant death rate, and (ii) a logistic growth model with state dependent carrying capacity, both of which are driven by multiplicative symmetric stable Lévy noise. Transcritical bifurcation occurs in the deterministic counterpart of the first model, while saddle-node bifurcation takes place in the logistic growth model. We focus on the impact of the variations of the growth rate, the per capita daily adult mortality rate, the stability index, and the noise intensity on the stationary probability density functions of the associated non-local Fokker–Planck equation. Implications of these bifurcations in population dynamics are discussed. In the first model the bifurcation parameter is the ratio of the population birth rate to the population death rate. In the second model the bifurcation parameter corresponds to the sensitivity of carrying capacity to change in the size of the population near equilibrium. In each case we show that as the value of the bifurcation parameter increases, the shape of the steady-state probability density function changes and that both stochastic models exhibit stochastic P-bifurcation. The unimodal density functions become more peaked around deterministic equilibrium points as the stability index increases, while an increase in any one of the other parameter has an effect on the stationary probability density function. That means the geometry of the density function changes from unimodal to flat, and its peak appears in the middle of the domain, which means a transition occurs.

Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise.

We characterize a stochastic dynamical system with tempered stable noise, by examining its probability density evolution. This probability density function satisfies a nonlocal Fokker-Planck equation. First, we prove a superposition principle that the probability measure-valued solution to this nonlocal Fokker-Planck equation is equivalent to the martingale solution composed with the inverse stochastic flow. This result together with a Schauder estimate leads to the existence and uniqueness of strong solution for the nonlocal Fokker-Planck equation. Second, we devise a convergent finite difference method to simulate the probability density function by solving the nonlocal Fokker-Planck equation. Finally, we apply our aforementioned theoretical and numerical results to a nonlinear filtering system by simulating a nonlocal Zakai equation.

Quantifying model uncertainty for the observed non-Gaussian data by the Hellinger distance

Mathematical models for complex systems under random fluctuations often certain uncertain parameters. However, quantifying model uncertainty for a stochastic differential equation with an α-stable Lévy process is still lacking. Here, we propose an approach to infer all the uncertain non-Gaussian parameters and other system parameters by minimizing the Hellinger distance over the parameter space. The Hellinger distance measures the similarity between an empirical probability density of non-Gaussian observations and a solution (as a probability density) of the associated nonlocal Fokker-Planck equation. Numerical experiments verify that our method is feasible for estimating single and multiple parameters. Meanwhile, we find an optimal estimation interval of the estimated parameters. This method is beneficial for extracting governing dynamical system models under non-Gaussian fluctuations, as in the study of abrupt climate changes in the Dansgaard-Oeschger events.