Stochastic Fokker-Planck Equation Research Articles

Stochastic Fokker–Planck Equations for Conditional McKean–Vlasov Jump Diffusions and Applications to Optimal Control

The purpose of this paper is to study optimal control of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker–Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton–Jacobi–Bellman equation for the optimal control of conditional McKean–Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker–Planck equation reduces to a stochastic partial differential equation for the Radon–Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean–Vlasov jump diffusions, and optimal consumption from a cash flow.

Modeling of the Particle Build-Up Evolution on a Single-Wire Magnetic Capture from Axial Stream Flow

The kinetic equation of the accumulation of magnetic particles from axial flow on a magnetized ferromagnetic wire in an external homogeneous magnetic field has been developed in this study. A new differential equation of the evolution of the accumulation radius over time, which considers both the capture and the detachment of the particles in the accumulation profile on the wire, has been formulated. The evolution of the radius of the accumulation profile over time was obtained from both the differential kinetic equation based on population theory and from the stochastic Fokker–Planck equation. In the limit approach (t→∞), it was observed that the expressions of the saturation radius of the accumulation radius on the magnetized wire of the particles obtained from both models were the same. It is emphasized that the obtained results are valid for both the initial and steady-state build-up of the particle capture process. These results were compared with the experimental results from the literature, and it was observed that the theoretical and experimental results were in good agreement. The effects of both capture and detachment events on the accumulation of particles on the magnetized wire were evaluated.

Stochastic approach to gravitational waves from inflation

We propose a coarse-graining procedure for describing the superhorizon dynamics of inflationary tensor modes. Our aim is to formulate a stochastic description for the statistics of spin-2 modes which seed the background of gravitational waves from inflation. Using basic principles of quantum mechanics, we determine a probability density for coarse-grained tensor fields, which satisfies a stochastic Fokker-Planck equation at superhorizon scales. The corresponding noise and drift are computable, and depend on the cosmological system under consideration. Our general formulas are applied to a variety of cosmological scenarios, also considering cases seldom considered in the context of stochastic inflation, and which are important for their observational consequences. We start obtaining the expected expressions for noise and drift in pure de Sitter and power-law inflation, also including a discussion of effects of non-attractor phases. We then apply our methods to describe scenarios with a transition from inflation to standard cosmological eras of radiation and matter domination. We show how the interference between modes flowing through the cosmological horizon, and modes spontaneously produced at superhorizon scales, can affect the stochastic evolution of coarse-grained tensor quantities. In appropriate limits, we find that the corresponding spectrum of tensor modes at horizon crossing matches with the results of quantum field theory calculations, but we also highlight where differences can arise.

Modeling 2018 Ebola virus disease outbreak with Cholesky decomposition

In this paper, we analyze a modified Susceptible‐Exposed‐Infectious‐Dead‐Recovered (SEIDR) model in the literature of the Ebola disease with uncertainties. The model is constructed using a van Kampen expansion method to have an Ebola SEIDR stochastic Fokker–Planck equation model. This model has a deterministic equation and noise covariance matrix. The basic reproduction number of the deterministic equation is calculated using the next‐generation matrix method. We prove the uniqueness and existence of the deterministic model using Lipschitz conditions and also show that it is locally asymptotically stable at it endemic equilibrium states. We constructed two equivalent stochastic differential equations (SDEs) models, whereas the Weiner process is equal to (i) the number of model equations and (ii) the number of independent changes in the model. Our aim is to solve (i) computationally using Cholesky decomposition technique with the variance–covariance matrix. Our proposed Cholesky decomposition SEIRD‐SDEs model is compared with (ii) and also with the epidemic data of the 2018 Ebola outbreak in the Democratic Republic of Congo. We also use numerical analyses to show that the importance of post‐death transmission is difficult to identify with noise terms supported by statistical hypothesis testing.

Analytical solution for post-death transmission model of Ebola epidemics

In this paper, we shall introduce deceased human (D) transmission to the cycle phenomenon of disease modelling, which has a direct relationship with the infective compartment of the stochastic Susceptible-Infected-Recovered (SIR) disease model. Due to this, the noise covariance matrices of the standard stochastic SIR model will be modified. This will be done by using van Kampen's expansion method to approximate the master equation and the stochastic Fokker–Planck equation to analytically calculate a power spectral density (PSD) expression. A vector-valued process is used and shows that the absolute value of the real part of the principal diagonal of the PSD matrix solution is the spectral density of the system which is compared to the average PSD of the stochastic simulations. We aim to investigate the problem of identifiability when deceased humans act as an extended state of host infection using the SIR-D model during Ebola epidemics. Using our analytical solution model, we show that for an increasing degree of transmission parameter values the infected route cannot be identified, whereas the deceased human transmission shows enhancement for the persistence of Ebola virus disease using epidemiology data of the Democratic Republic of Congo.

Nonlinear Fokker–Planck equations driven by Gaussian linear multiplicative noise

Existence of a strong solution in H−1(Rd) is proved for the stochastic nonlinear Fokker–Planck equationdX−div(DX)dt−Δβ(X)dt=XdW in (0,T)×Rd,X(0)=x, respectively, for a corresponding random differential equation. Here d≥1, W is a Wiener process in H−1(Rd), D∈C1(Rd,Rd) and β is a continuous monotonically increasing function satisfying some appropriate sublinear growth conditions which are compatible with the physical models arising in statistical mechanics. The solution exists for x∈L1∩L∞ and preserves positivity. If β is locally Lipschitz, the solution is unique, pathwise Lipschitz continuous with respect to initial data in H−1(Rd). Stochastic Fokker–Planck equations with nonlinear drift of the form dX−div(a(X))dt−Δβ(X)dt=XdW are also considered for Lipschitzian continuous functions a:R→Rd.

3-D Diffusive-Drift Molecular Channel Characterization for Active and Passive Receivers

The analytical characterization of a 3-D diffusion-drift fluid channel with various molecular receivers is an open problem in the study of molecular communication (MC). In the absence of closed-form expressions of the molecular receptivities for different receivers, the performance evaluation of a receiver relies heavily on time-consuming simulations. In this paper, channel characterizations for active and passive receivers are provided. The diffusion-drift kernel and probability distribution function (PDF) of a particle’s location in a 3-D fluid environment are derived using the stochastic Fokker–Planck equation. Furthermore, the PDF of the absorbing point on an infinite absorbing wall in a 3-D geometry is obtained. Closed-form expressions for the molecular receptivities of the active and passive receivers are derived. Assuming interference at the molecular receiver, an error performance analysis of both types of receivers using a maximum-likelihood sequence detector with hard-decision Viterbi decoding is provided.

Boltzmann-type models with uncertain binary interactions

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods.

Diffusive transport in the presence of stochastically gated absorption.

We analyze a population of Brownian particles moving in a spatially uniform environment with stochastically gated absorption. The state of the environment at time t is represented by a discrete stochastic variable k(t)∈{0,1} such that the rate of absorption is γ[1-k(t)], with γ a positive constant. The variable k(t) evolves according to a two-state Markov chain. We focus on how stochastic gating affects the attenuation of particle absorption with distance from a localized source in a one-dimensional domain. In the static case (no gating), the steady-state attenuation is given by an exponential with length constant sqrt[D/γ], where D is the diffusivity. We show that gating leads to slower, nonexponential attenuation. We also explore statistical correlations between particles due to the fact that they all diffuse in the same switching environment. Such correlations can be determined in terms of moments of the solution to a corresponding stochastic Fokker-Planck equation.

Uniqueness for a class of stochastic Fokker–Planck and porous media equations

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.

Generalized Stochastic Fokker-Planck Equations

We consider a system of Brownian particles with long-range interactions. We go beyond the mean field approximation and take fluctuations into account. We introduce a new class of stochastic Fokker-Planck equations associated with a generalized thermodynamical formalism. Generalized thermodynamics arises in the case of complex systems experiencing small-scale constraints. In the limit of short-range interactions, we obtain a generalized class of stochastic Cahn-Hilliard equations. Our formalism has application for several systems of physical interest including self-gravitating Brownian particles, colloid particles at a fluid interface, superconductors of type II, nucleation, the chemotaxis of bacterial populations, and two-dimensional turbulence. We also introduce a new type of generalized entropy taking into account anomalous diffusion and exclusion or inclusion constraints.

Tuning structure and mobility of solvation shells surrounding tracer additives.

Molecular dynamics simulations and a stochastic Fokker-Planck equation based approach are used to illuminate how position-dependent solvent mobility near one or more tracer particle(s) is affected when tracer-solvent interactions are rationally modified to affect corresponding solvation structure. For tracers in a dense hard-sphere fluid, we compare two types of tracer-solvent interactions: (1) a hard-sphere-like interaction, and (2) a soft repulsion extending beyond the hard core designed via statistical mechanical theory to enhance tracer mobility at infinite dilution by suppressing coordination-shell structure [Carmer et al., Soft Matter 8, 4083-4089 (2012)]. For the latter case, we show that the mobility of surrounding solvent particles is also increased by addition of the soft repulsive interaction, which helps to rationalize the mechanism underlying the tracer's enhanced diffusivity. However, if multiple tracer surfaces are in closer proximity (as at higher tracer concentrations), similar interactions that disrupt local solvation structure instead suppress the position-dependent solvent dynamics.

How Local and Average Particle Diffusivities of Inhomogeneous Fluids Depend on Microscopic Dynamics.

Computer simulations and a stochastic Fokker-Planck equation based approach are used to compare the single-particle diffusion coefficients of equilibrium hard-sphere fluids exhibiting identical inhomogeneous static structure and governed by either Brownian (i.e., overdamped Langevin) or Newtonian microscopic dynamics. The physics of inhomogeneity is explored via the imposition of one-dimensional sinusoidal density profiles of different wavelengths and amplitudes. When imposed density variations are small in magnitude for distances on the scale of a particle diameter, bulk-like average correlations between local structure and mobility are observed. In contrast, when density variations are significant on that length scale, qualitatively different structure-mobility correlations emerge that are sensitive to the governing microscopic dynamics. Correspondingly, a previously proposed scaling between long-time diffusivities for bulk isotropic fluids of particles exhibiting Brownian versus Newtonian dynamics [Pond et al. Soft Matter 2011, 7, 9859-9862] cannot be generalized to describe the position-dependent behaviors of strongly inhomogeneous fluids. While average diffusivities in the inhomogeneous and homogeneous directions are coupled, their qualitative dependencies on inhomogeneity wavelength are sensitive to the details of the microscopic dynamics. Nonetheless, average diffusivities of the inhomogeneous fluids can be approximately predicted for either type of dynamics based on knowledge of bulk isotropic fluid behavior and how inhomogeneity modifies the distribution of available volume. Analogous predictions for average diffusivities of experimental, inhomogeneous colloidal dispersions (based on known bulk behavior) suggest that they will exhibit qualitatively different trends than those predicted by models governed by overdamped Langevin dynamics that do not account for hydrodynamic interactions.

A QUASI-LINEAR STOCHASTIC FOKKER–PLANCK EQUATION IN σ-FINITE MEASURES

Solutions of quasi-linear stochastic Fokker–Planck equations for the number density of a system of solute particles in suspension are derived. The initial values and the solutions take values in a class of σ-finite Borel measures over Rd where d ≥ 1. The stochastic driving noise is defined by Itô differentials. For the special case of semi-linear stochastic Fokker–Planck equations, the solutions can be represented as solutions of first-order stochastic transport equations driven by Stratonovich differentials.

Generalized Langevin Equations

A derivation is presented for a generalized Langevin equation of motion for a dynamical variable φ(R(t), P(t)) where R and P are the position and momentum of a single heavy particle in a bath of light particles. A detailed analysis is given for the conditions required for the validity of the equation. A stochastic Fokker–Planck equation is derived for the quantity D(t) = δ(R(t) − R̂)δ(P(t) − P̂) which may be used to obtain both the generalized Langevin equation for φ and a Fokker–Planck equation for the heavy particle distribution function. The correlation function for D(t) is computed and it is shown how this quantity may be used to obtain all other correlation functions of interest.