We analyse a Fokker-Planck like equation, driven by a scalar parameter in order to reach an integral constraint. We exhibit criteria guaranteeing existence-uniqueness of a solution. We also provide counter-examples. This problem is motivated by an application to the immune control of tumor growth.
In this article, we obtain hessian estimates for Kolmogorov-Fokker-Planck operators in non-divergence form in several Banach function spaces. Our approach relies on a representation formula and newly developed sparse domination techniques in Harmonic Analysis. Our result when restricted to weighted Lebesgue spaces yields sharp quantitative hessian estimates for the Kolmogorov-Fokker-Planck operators.
This paper is devoted to $ϕ$-entropies applied to Fokker-Planck and kinetic Fokker-Planck equations in the whole space, with confinement. The so-called $ϕ$-entropies are Lyapunov functionals which typically interpolate between Gibbs entropies and L2 estimates. We review some of their properties in the case of diffusion equations of Fokker-Planck type, give new and simplified proofs, and then adapt these methods to a kinetic Fokker-Planck equation acting on a phase space with positions and velocities. At kinetic level, since the diffusion only acts on the velocity variable, the transport operat…
We consider the Vlasov-Manev-Fokker-Planck (VMFP) system in three dimensions, which differs from the Vlasov-Poisson-Fokker-Planck in that it has the gravitational potential of the form $-1/r - 1/r^2$ instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions. The proof extends to several related kinetic systems.
The present article is concerned with global subelliptic estimates for Kramers-Fokker-Planck operators with polynomials of degree less than or equal to two. The constants appearing in those estimates are accurately formulated in terms of the coefficients, especially when those are large.
We consider the kinetic Fokker-Planck equation with weak confinement force. We proved some (polynomial and sub-exponential) rate of convergence to the equilibrium (depending on the space to which the initial datum belongs). Our results generalized the result in [4, 5, 21, 11, 10, 9, 1, 14] to weak confinement case.
In this paper, we investigate the spectral analysis (from the point of view of semi-groups) of discrete, fractional and classical Fokker-Planck equations. Discrete and fractional Fokker-Planck equations converge in some sense to the classical one. As a consequence, we first deal with discrete and classical Fokker-Planck equations in a same framework, proving uniform spectral estimates using a perturbation argument and an enlargement argument. Then, we do a similar analysis for fractional and classical Fokker-Planck equations using an argument of enlargement of the space in which the semigroup …
We consider a Fokker-Planck operator with electric potential and electromagnetic fields. We establish the sharp weighted and subelliptic estimates, involving the control of the derivatives of electric potential and electromagnetic fields. Our proof relies on a localization argument as well as a careful calculation on commutators.
Noise or fluctuations play an important role in the modeling and understanding of the behavior of various complex systems in nature. Fokker-Planck equations are powerful mathematical tool to study behavior of such systems subjected to fluctuations. In this paper we establish local well-posedness result of a new nonlinear Fokker-Planck equation. Such equations appear in the modeling of the grain boundary dynamics during microstructure evolution in the polycrystalline materials and obey special energy laws.
Solutions of stationary Fokker-Planck equations in the narrow beam regime are commonly approximated by either ballistic linear transport or by a Fermi pencil-beam equation. We present a rigorous approximation analysis of these three models in a half-space geometry. Error estimates are obtained in a 1-Wasserstein sense, which is an adapted metric to quantify beam spreading. The required well-posedness and regularity results for the stationary Fokker-Planck equation with singular internal and boundary sources are also presented in detail.
In this article we study some Kramers-Fokker-Planck operators with a polynomial potential $V(q)$ of degree greater than two having quadratic limiting behavior. This work provides an accurate global subelliptic estimate for KFP operators under some conditions imposed on the potential.
In this work, we prove an optimal global-in-time L p --L q estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as t $\rightarrow$ +$\infty$ is the same as the heat equation in x-variables and the divergence rate as t $\rightarrow$ 0 + is related to the sub-ellipticity with loss of 1/3 derivatives of the Kramers-Fokker-Planck operator.
We consider the modeling of light beams propagating in highly forward-peaked turbulent media by fractional Fokker-Planck equations and their approximations by fractional Fermi pencil-beam models. We obtain an error estimate in a 1-Wasserstein distance for the latter model showing that beam spreading is well captured by the Fermi pencil-beam approximation in the small diffusion limit.
This paper is concerned with finding Fokker-Planck equations in $\mathbb{R}^d$ with the fastest exponential decay towards a given equilibrium. For a prescribed, anisotropic Gaussian we determine a non-symmetric Fokker-Planck equation with linear drift that shows the highest exponential decay rate for the convergence of its solutions towards equilibrium. At the same time it has to allow for a decay estimate with a multiplicative constant arbitrarily close to its infimum. Such an optimal Fokker-Planck equation is constructed explicitly with a diffusion matrix of rank one, hence being hypocoerciv…
We prove the exponential convergence to the equilibrium, quantified by Rényi divergence, of the solution of the Fokker-Planck equation with drift given by the gradient of a strictly convex potential. This extends the classical exponential decay result on the relative entropy for the same equation.
We obtain local Holder continuity estimates up to the boundary for a kinetic Fokker-Planck equation with rough coefficients, with the prescribed influx boundary condition. Our result extends some recent developments that incorporate De Giorgi methods to kinetic Fokker-Planck equations. We also obtain higher order asymptotic estimates near the incoming part of the boundary. In particular, when the equation has a zero boundary conditions and no source term, we prove that the solution vanishes at infinite order on the incoming part of the boundary.
This paper is devoted to the rigorous derivation of the macroscopic limit of a Vlasov-Fokker-Planck equation in which the Laplacian is replaced by a fractional Laplacian. The evolution of the density is governed by a fractional heat equation with the addition of a convective term coming from the external force. The analysis is performed by a modified test function method and by obtaining a priori estimates from quadratic entropy bounds. In addition, we give the proof of existence and uniqueness of solutions to the Vlasov-fractional-Fokker-Planck equation.
This paper is devoted to the construction of an \emph{eigen-solution} for the Fokker-Planck operator with heavy tail equilibrium. We propose an \textit{alternative} method in dimension 1, which will be generalizable in higher dimension. The later method is inspired by the work of H. Koch on non-linear KdV equation \cite{Koch}. As a consequence of this construction, we recover the result of G. Lebeau and M. Puel \cite{LebPu} on the fractional diffusion limit for the Fokker-Planck equation.
We consider the compressible Vlasov-Poisson-Fokker-Planck-Navier-Stokes system in a three dimensional bounded domain with nonhomogeneous Dirichlet boundary conditions. The system describes the evolution of charged particles ensemble dispersed in an isentropic fluid. For the adiabatic coefficient $γ>3/2$, we establish the global existence of weak solutions to this system with arbitrary large initial and boundary data.
This paper is devoted to the linearized Vlasov-Poisson-Fokker-Planck system in presence of an external potential of confinement. We investigate the large time behaviour of the solutions using hypocoercivity methods and a notion of scalar product adapted to the presence of a Poisson coupling. Our framework provides estimates which are uniform in the diffusion limit. As an application in a simple case, we study the one-dimensional case and prove the exponential convergence of the nonlinear Vlasov-Poisson-Fokker-Planck system without any small mass assumption.
We present global Schauder type estimates in all variables and unique solvability results in kinetic Hölder spaces for kinetic Kolmogorov-Fokker-Planck (KFP) equations. The leading coefficients are Hölder continuous in the $x, v$ variables and are merely measurable in the temporal variable. Our proof is inspired by Campanato's approach to Schauder estimates and does not rely on the estimates of the fundamental solution of the KFP operator.
In the present paper, we study the diffusion limit of the classical solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system with initial data near a global Maxwellian. We prove the convergence and establish the optimal convergence rate of the global strong solution to the VPFP system towards the solution to the drift-diffusion-Poisson system based on the spectral analysis with precise estimation on the initial layer.
We provide a quantitative asymptotic analysis for the nonlinear Vlasov--Poisson--Fokker--Planck system with a large linear friction force and high force-fields. The limiting system is a diffusive model with nonlocal velocity fields often referred to as aggregation-diffusion equations. We show that a weak solution to the Vlasov--Poisson--Fokker--Planck system strongly converges to a strong solution to the diffusive model. Our proof relies on the modulated macroscopic kinetic energy estimate based on the weak-strong uniqueness principle together with a careful analysis of the Poisson equation.
In this paper we show that Markov uniqueness for symmetric pre-Dirichlet operators $L$ follows from the uniqueness of the corresponding Fokker-Planck-Kolmogorov equation (FPKE). Since in recent years a considerable number of uniqueness results for FPKE's have been achieved, we obtain new Markov uniqueness results in concrete cases. A selection of such will be presented in this paper. They include cases with killing and with degenerate diffusion coefficients.
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.
In this paper we study the spectral property of a Fokker-Planck operator with potential. By virtue of a multiplier method inspired by Nicolas Lerner, we obtain new compactness criteria for its resolvent, involving the control of the positive eigenvalues of the Hessian matrix of the potential.
For quickly decreasing potentials with one position variable, the first threshold zero is always a resonance of the Kramers-Fokker-Planck operator. In this article we study low-energy spectral properties of the operator and calculate large time asymptotics of solutions in terms of the Maxwellian.
J. Feng and T. Nguyen have defined a cost on curves of measures which is finite exactly on the curves which solve a Fokker-Planck equation with $L^2$ drift. In this paper, using ideas of D. Gomes and E. Valdinoci, we give a different construction of the cost of Feng and Nguyen.
We obtain estimates for the Kantorovich functionals between solutions to different Fokker -- Planck -- Kolmogorov equations for measures with same diffusion part but different drifts and different initial conditions. We show possible applications of such estimates to the study of the well-posedness for nonlinear equations.
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 work, we study the bilinear optimal stabilization of a non-homogeneous Fokker-Planck equation. We first study the problem of optimal control in a finite-time interval and then focus on the case of the infinite time horizon. We further show that the obtained optimal control guarantees the strong stability of the system at hand. An illustrating numerical example is given.
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.
