Nonlocal Fokker-Planck Equation Research Articles

Mathematical modelling of collagen fibres rearrangement during the tendon healing process

Tendon injuries present a significant clinical challenge to modern medicine as they heal slowly and rarely recover the structure and mechanical strength of a healthy tendon. Moreover, tendon represents a highly under-researched tissue with the process of healing not fully elucidated. To improve the understanding of tendon function and healing process we propose a new model of collagen fibers rearrangement during tendon 1 healing. The model consists of integro-differential equation describing the dynamics of collagen fibers distribution. We further reduce the model in a suitable asymptotic regime leading to a nonlinear non-local Fokker-Planck type equation for the spatial and orientation distribution of col-lagen fiber bundles. The reduced model allows for possible parameter estimation based on data due to its simplicity. We showcase some of the qualitative properties of this model simulating its long time asymptotic behavior and the total time for tendon fibers to align in terms of the model parameters. A possible biological interpretation of the numerical experiments performed leads us to the working hypothesis of the importance of the tendon cell size in patients' recovery.

The lattice Fokker–Planck equation for models of wealth distribution

Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker-Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the 'inverse problem' of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the 'forward problem' of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Numerical analysis and applications of Fokker-Planck equations for stochastic dynamical systems with multiplicative α-stable noises

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.

The tipping times in an Arctic sea ice system under influence of extreme events.

In light of the rapid recent retreat of Arctic sea ice, the extreme weather events triggering the variability in Arctic ice cover has drawn increasing attention. A non-Gaussian α-stable Lévy process is thought to be an appropriate model to describe such extreme events. The maximal likely trajectory, based on the nonlocal Fokker-Planck equation, is applied to a nonautonomous Arctic sea ice system under α-stable Lévy noise. Two types of tipping times, the early-warning tipping time and the disaster-happening tipping time, are used to predict the critical time for the maximal likely transition from a perennially ice-covered state to a seasonally ice-free one and from a seasonally ice-free state to a perennially ice-free one, respectively. We find that the increased intensity of extreme events results in shorter warning time for sea ice melting and that an enhanced greenhouse effect will intensify this influence, making the arrival of warning time significantly earlier. Meanwhile, for the enhanced greenhouse effect, we discover that increased intensity and frequency of extreme events will advance the disaster-happening tipping time, in which an ice-free state is maintained throughout the year in the Arctic Ocean. Finally, we identify values of the Lévy index α and the noise intensity ϵ in the αϵ-space that can trigger a transition between the Arctic sea ice state. These results provide an effective theoretical framework for studying Arctic sea ice variations under the influence of extreme events.

The maximum likelihood climate change for global warming under the influence of greenhouse effect and Lévy noise.

An abrupt climatic transition could be triggered by a single extreme event, and an α-stable non-Gaussian Lévy noise is regarded as a type of noise to generate such extreme events. In contrast with the classic Gaussian noise, a comprehensive approach of the most probable transition path for systems under α-stable Lévy noise is still lacking. We develop here a probabilistic framework, based on the nonlocal Fokker-Planck equation, to investigate the maximum likelihood climate change for an energy balance system under the influence of greenhouse effect and Lévy fluctuations. We find that a period of the cold climate state can be interrupted by a sharp shift to the warmer one due to larger noise jumps with low frequency. Additionally, the climate change for warming 1.5°C under an enhanced greenhouse effect generates a steplike growth process. These results provide important insights into the underlying mechanisms of abrupt climate transitions triggered by a Lévy process.

Transitions between metastable states in a simplified model for the thermohaline circulation under random fluctuations

In this work we study the impact of non-Gaussian α-stable Lévy motion on transitions between metastable equilibrium states (or attractors) in a stochastic Stommel two-compartment model for the thermohaline circulation. By maximizing probability density of the solution process associated with a nonlocal Fokker-Planck equation, we compute maximal likely pathways and identify corresponding maximal likely stable equilibrium states. Our numerical results indicate that random fluctuations with small intensity induces a weakened thermohaline circulation when the Lévy noise stability index is from 0.1 to 0.7.

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

We study the most probable transition pathways and maximal likely trajectories in a genetic regulation model of the transcription factor activator’s concentration evolution, with Gaussian noise and non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account, respectively. We compute the most probable transition pathways by the Onsager–Machlup least action principle, and calculate the maximal likely trajectories by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker–Planck equation. We have observed the rare most probable transition pathways in the case of Gaussian noise, for certain noise intensity, evolution time scale and system parameters. We have especially studied the maximal likely trajectories starting at the low concentration metastable state, and examined whether they evolve to or near the high concentration metastable state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period); (ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including: (a) A smaller noise intensity may trigger the transcription process, while a larger noise intensity cannot, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise; (b) The symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription.These findings provide insights for further experimental research, in order to achieve or to avoid specific gene transcriptions, with possible relevance for medical advances.

Stochastic nonlinear Fokker–Planck equations

The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker–Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with common noise. The uniqueness of solutions is obtained without any higher moment assumption on the solution by means of a duality argument to a backward stochastic PDE.

The role of a strong confining potential in a nonlinear Fokker–Planck equation

We show that solutions of nonlinear nonlocal Fokker–Planck equations in a bounded domain with no-flux boundary conditions can be approximated by Cauchy problems with increasingly strong confining potentials defined in the whole space. Two different approaches are analyzed, making crucial use of uniform estimates for L2 energy functionals and free energy (or entropy) functionals respectively. In both cases, we prove that the weak formulation of the problem in a bounded domain can be obtained as the weak formulation of a limit problem in the whole space involving a suitably chosen sequence of large confining potentials. The free energy approach extends to the case degenerate diffusion.

A conservative, free energy dissipating, and positivity preserving finite difference scheme for multi-dimensional nonlocal Fokker–Planck equation

In this work, we design and analyze a conservative, positivity preserving, and free energy dissipating finite difference method for the multi-dimensional nonlocal Fokker–Planck (FP) equation. Based on a non-logarithmic Landau transformation, a central-differencing spatial discretization using harmonic-mean approximations is developed. Both forward and backward Euler discretizations in time are employed to derive an explicit scheme and a linearized semi-implicit scheme, respectively. Three desired properties that are possessed by analytical solutions: i) mass conservation, ii) free-energy dissipation, and iii) positivity, are proved to be maintained at discrete level. Remarkably, numerical analysis demonstrates that the semi-implicit time discretization ensures the property of positivity preserving unconditionally. Due to the advantages brought by the harmonic-mean approximations, an estimate on the upper bound of the condition number of the resulting coefficient matrix is further established for the semi-implicit scheme. Extensive numerical tests are performed to validate aforementioned properties numerically.

Target search of a protein on DNA in the presence of position-dependent bias

We study the target search on DNA for proteins in the presence of non-constant drift. This search is realized by the facilitated diffusion process. Existing works on this problem focus on the case of constant drift. Starting from a non-local Fokker–Planck equation with the -order fractional Laplace operator and a ‘sink’ term, we obtain the possibility density function for a protein occurring at position x at time t. Based on this, we further compute the survival probability and the first arrival density in order to quantify the searching mechanisms. The numerical results show that in the linear drift case, there is an optimal index for the search to be most likely successful (searching reliability reaches its maximum). This optimal index depends on the initial position–target separation. It is also found that the diffusion intensity plays a positive role in improving the search success. The nonlinear double-well drift could drive the protein to reach the target with a larger possibility than the linear drag at the initial time period, but viewed over a long time duration, the linear drift is more beneficial for target search success. In contrast to the linear drift case, the search reliability and efficiency with nonlinear double-well drift have a monotonic relationship with the index, that is, the smaller the index is, the higher the likelihood of a protein finding its target.

Fokker-Planck equation driven by asymmetric Lévy motion

Non-Gaussian L\'evy 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 L\'evy 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.

Analysis of a Nonlocal and Nonlinear Fokker--Planck Model for Cell Crawling Migration

Cell movement has essential functions in development, immunity and cancer. Various cell migration patterns have been reported and a general rule has recently emerged, the so-called UCSP (Universal Coupling between cell Speed and cell Persistence), [30]. This rule says that cell persistence, which quantifies the straightness of trajectories, is robustly coupled to migration speed. In [30], the advection of polarity cues by a dynamic actin cytoskeleton undergoing flows at the cellular scale was proposed as a first explanation of this universal coupling. Here, following ideas proposed in [30], we present and study a simple model to describe motility initiation in crawling cells. It consists of a non-linear and non-local Fokker-Planck equation, with a coupling involving the trace value on the boundary. In the one-dimensional case we characterize the following behaviours: solutions are global if the mass is below the critical mass, and they can blow-up in finite time above the critical mass. In addition, we prove a quantitative convergence result using relative entropy techniques.

Existence and regularity of a linear nonlocal Fokker–Planck equation with growing drift

The nonlocal Fokker–Planck equations for a class of stochastic differential equations with non-Gaussian α-stable Lévy motion in Euclidean space are studied. The existence and uniqueness of weak solution are obtained with rough drift. The solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded. Moreover, the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity.

A kinetic equation for economic value estimation with irrationality and herding

A kinetic inhomogeneous Boltzmann-type equation is proposed to model the dynamics of the number of agents in a large market depending on the estimated value of an asset and the rationality of the agents. The interaction rules take into account the interplay of the agents with sources of public information, herding phenomena, and irrationality of the individuals. In the formal grazing collision limit, a nonlinear nonlocal Fokker-Planck equation with anisotropic (or incomplete) diffusion is derived. The existence of global-in-time weak solutions to the Fokker-Planck initial-boundary-value problem is proved. Numerical experiments for the Boltzmann equation highlight the importance of the reliability of public information in the formation of bubbles and crashes. The use of Bollinger bands in the simulations shows how herding may lead to strong trends with low volatility of the asset prices, but eventually also to abrupt corrections.

Smooth solution of a nonlocal Fokker–Planck equation associated with stochastic systems with Lévy noise

It is shown that the solution of a nonlocal Fokker–Planck equation is smooth with respect to both time and space variable whenever the divergence of the smooth drift has a lower bound.

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 on either a bounded or infinite domain. Under a specified condition, 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.

Local and global solution for a nonlocal Fokker–Planck equation related to the adaptive biasing force process

We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker–Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non-linear term is related to a conditional expectation, and is thus non-local. The proof uses tools from the theory of semigroups of linear operators for the local existence result, and an a priori estimate based on a supersolution for the global existence result.

A nonlocal Fokker–Planck equation for non-Gaussian stochastic dynamical systems

This work is devoted to the study of a nonlocal Fokker–Planck equation for a class of stochastic differential equations with α-stable Lévy motion (0<α<2). The existence and uniqueness of the weak solution is proved under suitable conditions, by a vanishing viscosity method.

Rate-Independent Dynamics and Kramers-Type Phase Transitions in Nonlocal Fokker–Planck Equations with Dynamical Control

The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker–Planck equations that involve two small parameters and are driven by a time-dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and characterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical stability of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.