Anomalous subdiffusion characterizes transport in diverse physical systems and is especially prevalent inside biological cells. In cell biology, the prevailing model for chemical activation rates has recently changed from the first passage time (FPT) of a single searcher to the FPT of the fastest searcher out of many searchers to reach a target, which is called an extreme statistic or extreme FPT. In this paper, we investigate extreme statistics of searchers which move by anomalous subdiffusion. We model subdiffusion by a fractional Fokker-Planck equation involving the Riemann-Liouville fractional derivative. We prove an explicit and very general formula for every moment of subdiffusive extreme FPTs and approximate their full probability distribution. While the mean FPT of a single subdiffusive searcher is infinite, the fastest subdiffusive searcher out of many subdiffusive searchers typically has a finite mean FPT. In fact, we prove the counterintuitive result that extreme FPTs of subdiffusion are faster than extreme FPTs of normal diffusion. Mathematically, we employ a stochastic representation involving a random time change of a standard Ito drift-diffusion according to the trajectory of the first crossing time inverse of a Levy subordinator. A key step in our analysis is generalizing Varadhan's formula from large deviation theory to the case of subdiffusion, which yields the short-time distribution of subdiffusion in terms of a certain geodesic distance.
In this paper we analyze fractional Fokker-Planck equation describing subdiffusion in the general infinitely divisible (ID) setting. We show that in the case of space-time-dependent drift and diffusion and time-dependent jump coefficient, the corresponding stochastic process can be obtained by subordinating two-dimensional system of Langevin equations driven by appropriate Brownian and Levy noises. Our result solves the problem of stochastic representation of subdiffusive Fokker-Planck dynamics in full generality.
This paper develops solutions of fractional Fokker-Planck equations describing subdiffusion of probability densities of stochastic dynamical systems driven by non-Gaussian Lévy processes, with space-time-dependent drift, diffusion and jump coefficients, thus significantly extends Magdziarz and Zorawik's result in "M. Magdziarz and T. Zorawik, Stochastic representation of fractional subdiffusion equation. The case of infinitely divisible waiting times, Lévy noise and space-time-dependent coefficients. Proc. Amer. Math. Soc., Accepted (2015)." Fractional Fokker-Planck equation describing subdiff…
By investigating McKean-Vlasov SDEs, the order preservation and positive correlation are characterized for nonlinear Fokker-Planck equations. The main results recover the corresponding criteria on these properties established in [3, 5] for diffusion processes or linear Fokker-Planck equations.
The work concerns the superposition between the Zakai equations and the Fokker-Planck equations on measure spaces. First, we prove a superposition principle for the Fokker-Planck equations on $\mR^\mN$ under the integrable condition. And then by means of it, we show two superposition principles for the weak solutions of the Zakai equations from the nonlinear filtering problems and the weak solutions of the Fokker-Planck equations on measure spaces. As a by-product, we give some weak conditions under which the Fokker-Planck equations can be solved in the weak sense.
One obtains a probabilistic representation for the entropic generalized solutions to a nonlinear Fokker-Planck equation in $\mathbb R^d$ with multivalued nonlinear diffusion term as density probabilities of solutions to a nonlinear stochastic differential equation. The case of a nonlinear Fokker-Planck equation with linear space dependent drift is also studied.
One proves the uniqueness of distributional solutions to nonlinear Fokker--Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean--Vlasov stochastic differential equation (SDE).
We develop a general technique to prove uniqueness of solutions for Fokker--Planck equations on infinite dimensional spaces. We illustrate this method by implementing it for Fokker--Planck equations in Hilbert spaces with Kolmogorov operators with irregular coefficients and both non-degenerate or degenerate second order part.
We prove a version of the Ambrosio-Figalli-Trevisan superposition principle for a restricted subclass of solutions to the Fokker-Planck-Kolmogorov equation, that is valid on separable infinite-dimensional Hilbert spaces. Furthermore, we transfer this restricted superposition principle into a nonlinear setting.
We consider a stochastic model of the two-dimensional chemostat as a diffusion process for the concentration of substrate and the concentration of biomass. The model allows for the washout phenomenon: the disappearance of the biomass inside the chemostat. We establish the Fokker-Planck associated with this diffusion process, in particular we describe the boundary conditions that modelize the washout. We propose an adapted finite difference scheme for the approximation of the solution of the Fokker-Planck equation.
We prove the superposition principle for probability measure-valued solutions to non-local Fokker-Planck equations, which in turn yields the equivalence between martingale problems for SDEs with jumps and such non-local PDEs with rough coefficients. As an application, we obtain a probabilistic representation for weak solutions of fractional porous media equations.
We present new $Φ$-entropy inequalities for diffusion semigroups under the curvature-dimension criterion. They include the isoperimetric function of the Gaussian measure. Applications to the long time behaviour of solutions to Fokker-Planck equations are given.
By constructing successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker-Planck equations on $\R^m\times\R^{d}$. The main results are also applied to the study of gradient estimate, entropy/transportation-cost inequality and heat kernel inequalities.
We consider the Fokker--Planck equations with irregular coefficients. Two different cases are treated: in the degenerate case, the coefficients are assumed to be weakly differentiable, while in the non-degenerate case the drift satisfies only the Ladyzhenskaya--Prodi--Serrin condition. Using Trevisan's superposition principle which represents the solution as the marginal of the solution to the martingale problem of the diffusion operator, we establish quantitative stability estimates for the solutions of Fokker--Planck equations.
We classify (1+3)-dimensional Fokker-Planck equations with a constant diagonal diffusion matrix that are solvable by the method of separation of variables. As a result, we get possible forms of the drift coefficients $B_1(\vec x),B_2(\vec x),B_3(\vec x)$ providing separability of the corresponding Fokker-Planck equations and carry out variable separation in the latter. It is established, in particular, that the necessary condition for the Fokker-Planck equation to be separable is that the drift coefficients $\vec B(\vec x)$ must be linear. We also find the necessary condition for R-separabilit…
The object of this paper is the uniqueness for a $d$-dimensional Fokker-Planck type equation with non-homogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so called Barenblatt solution of the fast diffusion equation which is the partial differential equation $\partial_t u = \partial^2_{xx} u^m$ with $m\in(0,1)$. Together with the mentioned Fokker-Planck equation, we make use of small time density estimates uniformly with respect to the initial condition
The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.
We prove a superposition principle for nonlinear Fokker-Planck-Kolmogorov equations on Euclidean spaces and their corresponding linearized first-order continuity equation over the space of Borel (sub-)probability measures. As a consequence, we obtain equivalence of existence and uniqueness results for these equations. Moreover, we prove an analogous result for stochastically perturbed Fokker-Planck-Kolmogorov equations. To do so, we particularly show that such stochastic equations for measures are, similarly to the deterministic case, intrinsically related to linearized second-order equations …
This paper establishes Fokker-Planck-Kolmogorov type equations for time-changed Gaussian processes. Examples include those equations for a time-changed fractional Brownian motion with time-dependent Hurst parameter and for a time-changed Ornstein-Uhlenbeck process. The time-change process considered is the inverse of either a stable subordinator or a mixture of independent stable subordinators.
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 Levy 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 differential equations with Levy motion arise the mathematical models for various phenomenon in geophysical and biochemical sciences. The Fokker Planck equation for such a stochastic differential equations is a nonlocal partial differential equations. We prove the existence and uniqueness of the weak solution for this equation.
We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness of generalized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutions for second order integro-differential equation of Fokker-Planck type.
Convergence of a system of particles, interacting with a fluid, to Navier-Stokes-Vlasov-Fokker-Planck system is studied. The interaction between particles and fluid is described by Stokes drag force. The empirical measure of particles is proved to converge to the Vlasov-Fokker-Planck component of the system and the velocity of the fluid coupled with the particles converges in the uniform topology to the the Navier-Stokes component. A new uniqueness result for the PDE system is added.
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.
In this article, we establish a general formula for higher order linear functional derivatives for the composition of an arbitrary smooth functional on the 1-Wasserstein space with the solution of a Fokker-Planck PDE. This formula has important links with the theory of propagation of chaos and mean-field games.
Reaction-diffusion equations are one of the most common mathematical models in the natural sciences and are used to model systems that combine reactions with diffusive motion. However, rather than normal diffusion, anomalous subdiffusion is observed in many systems and is especially prevalent in cell biology. What are the reaction-subdiffusion equations describing a system that involves first-order reactions and subdiffusive motion? In this paper, we answer this question. We derive fractional reaction-subdiffusion equations describing an arbitrary number of molecular species which react at fir…
We introduce a stochastic particle system that corresponds to the Fokker-Planck equation with decay in the many-particles limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Gamma-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is again a variational formulation for the Fokker-Planck equation with decay.
By using the probability approach (the Malliavin calculus), we prove the existence of smooth fundamental solutions for degenerate kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity, where the dissipative term is the generator of an anisotropic Lévy process and the drift term is allowed to be cubic growth.
We consider Fokker-Planck type differential operators associated with general Langevin processes admitting a Gibbs stationary distribution. Under assumptions insuring suitable resolvent estimates, we prove Eyring-Kramers formulas for the bottom of the spectrum of these operators in the low temperature regime. Our approach is based on the construction of sharp Gaussian quasimodes which avoids supersymmetry or PT-symmetry assumptions.
The Fokker-Planck equations describe time evolution of probability densities of stochastic dynamical systems and are thus widely used to quantify random phenomena such as uncertainty propagation. For dynamical systems driven by non-Gaussian Lévy processes, however, it is difficult to obtain explicit forms of Fokker-Planck equations because the adjoint operators of the associated infinitesimal generators usually do not have exact formulation. In the present paper, Fokker- Planck equations are derived in terms of infinite series for nonlinear stochastic differential equations with non-Gaussian L…
In this paper, we study the long-time behaviour of solutions to the Vlasov-Fokker-Planck equation where the confining potential is non-convex. This is a nonlocal nonlinear partial differential equation describing the time evolution of the probability distribution of a particle moving under the influence of a non-convex potential, an interaction potential, a friction force and a stochastic force. Using the free-energy approach, we show that under suitable assumptions solutions of the Vlasov-Fokker-Planck equation converge to an invariant probability.
