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Recently, analytical solutions of a nonlinear Fokker-Planck equation describing anomalous diffusion with an external linear force were found using a non extensive thermostatistical Ansatz. We have extended these solutions to the case when…
We consider the initial/boundary value problem for the fractional diffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two "simple" fully discrete schemes based on the Galerkin finite element…
Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by…
We present a model of diffusion in heterogeneous environment, which qualitatively reflects the transport properties of a polymeric membrane with carbon nanotubes. We derived Fokker-Planck equation from system of stochastic equations,…
We address the problem of determining the stationary distribution of the multi-allelic, neutral-evolution Wright-Fisher model in the diffusion limit. A full solution to this problem for an arbitrary K x K mutation rate matrix involves…
Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…
We associate a coupled nonlinear Fokker-Planck equation on $\R^d$, i.e. with solution paths in $\scr P$, to a linear Fokker-Planck equation for probability measures on the product space $\R^d\times \scr P$, i.e. with solution paths in $\scr…
The Fokker-Planck equations (FPEs) for stochastic systems driven by additive symmetric $\alpha$-stable noises may not adequately describe the time evolution for the probability densities of solution paths in some practical applications,…
In this paper, we introduce and analyze a numerical scheme for solving the Cauchy-Dirichlet problem associated with fractional nonlinear diffusion equations. These equations generalize the porous medium equation and the fast diffusion…
Diffusion models have achieved huge empirical success in data generation tasks. Recently, some efforts have been made to adapt the framework of diffusion models to discrete state space, providing a more natural approach for modeling…
One proves the existence and uniqueness of a generalized (mild) solution for the nonlinear Fokker-Planck equation (FPE) \begin{align*} &u_t-\Delta (\beta(u))+{\mathrm{ div}}(D(x)b(u)u)=0, \quad t\geq0,\ x\in\mathbb{R}^d,\ d\ne2, \\…
This paper derives the Fokker-Planck (FP) equation for a particle moving in potential by a randomly modulated dipole. The FP equation describes the anomalous diffusion observed in the companion paper [1] and breaks the conservation of the…
Of primary interest in this paper is the numerical approximation of a time dependent fractional, in space, diffusion equation where the domain is assumed to be nonhomogeneous, having different axial diffusion coefficients. This work is…
We solve a physically significant extension of a classic problem in the theory of diffusion, namely the Ornstein-Uhlenbeck process [G. E. Ornstein and L. S. Uhlenbeck, Phys. Rev. 36, 823, (1930)]. Our generalised Ornstein-Uhlenbeck systems…
In this paper, the Feynman path integral formulation of the continuous-continuous filtering problem, a fundamental problem of applied science, is investigated for the case when the noise in the signal and measurement model is additive. It…
We show how the steady-state solution of the Smoluchowski (Fokker-Planck) equation for a color reaction-counterdiffusion problem, together with equilibrium trajectory information (e.g., from molecular simulations or confocal microscopy…
The Fokker-Planck (FP) equation is a linear partial differential equation which governs the temporal and spatial evolution of the probability density function (PDF) associated with the response of stochastic dynamical systems. An exact…
Current strategies for solving image-based inverse problems apply latent diffusion models to perform posterior sampling.However, almost all approaches make no explicit attempt to explore the solution space, instead drawing only a single…
We establish the existence theory of several commonly used finite element (FE) nonlinear fully discrete solutions, and the convergence theory of a linearized iteration. First, it is shown for standard FE, SUPG and edge-averaged method…
This paper is concerned with the problem of continuous-time nonlinear filtering for stochastic processes on a connected matrix Lie group. The main contribution of this paper is to derive the feedback particle filter (FPF) algorithm for this…