Integrability and continuity of solutions to Fokker–Planck–Kolmogorov equations

New results concerning the local integrability of any order and continuity of solution densities of Fokker–Planck–Kolmogorov equations with nondifferentiable coefficients are obtained.     

Convergence to Stationary Measures in Nonlinear Fokker–Planck–Kolmogorov Equations

The convergence of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary solutions is studied. Broad sufficient conditions for convergence in variation with an exponential bound are obtained.     

Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations

We describe conditions on non-gradient drift diffusion Fokker–Planck equations for its solutions to converge to equilibrium with a uniform exponential rate in Wasserstein distance. This asymptotic behaviour is related to a functional inequality, which links the distance with its dissipation and ensures a spectral gap in Wasserstein distance. We give practical criteria for this inequality and compare it to classical ones. The key point is to quantify the contribution of the diffusion term to the rate of convergence, in any dimension, which to our knowledge is a novelty.     

On uniqueness of solutions to the Cauchy problem for degenerate Fokker–Planck–Kolmogorov equations

We prove a new uniqueness result for highly degenerate second-order parabolic equations on the whole space. A novelty is also our class of solutions in which uniqueness holds.     

Matrix-exponential groups and Kolmogorov–Fokker–Planck equations

Aim of this paper is to provide new examples of H?rmander operators L{\mathcal{L}} to which a Lie group structure can be attached making L{\mathcal{L}} left invariant. Our class of examples contains several subclasses of operators appearing in literature and arising both in theoretical and in applied fields: evolution Kolmogorov operators, degenerate Ornstein–Uhlenbeck operators, Mumford and Fokker–Planck operators, Ornstein–Uhlenbeck operators with time-dependent periodic coefficients. Our examples basically come from exponential of matrices, as well as from linear constant-coefficient ODE’s, in \mathbbR{\mathbb{R}} or in \mathbbC{\mathbb{C}} . Furthermore, we describe how these groups can be combined together to obtain new structures and new operators, also having an interest in the applied field.     

On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity

Decay and stability of solutions to the Fokker–Planck–Boltzmann equation in

In this paper, we use the method of constructing the compensating function introduced by Kawashima and the standard energy method to study the global existence of solutions to the Fokker–Planck–Boltzmann equation in the whole space. The time decay and uniform stability of solutions to the global Maxwellian are also obtained.     

Contraction of general transportation costs along solutions to Fokker–Planck equations with monotone drifts

We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker–Planck equations in ${\mathbb{R}}^d$, when the drift is a monotone (or λ-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solutions without requiring stronger regularity, and it extends the Wasserstein theory of Fokker–Planck equations with gradient drift terms, started by Jordan, Kinderlehrer and Otto (1998) [14], to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions.     

Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces

We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, the existence and uniqueness of a measure valued solution, for the corresponding Fokker–Planck equation. In particular, we verify the Chapman–Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.     

On the rates of decay to equilibrium in degenerate and defective Fokker–Planck equations

We establish sharp long time asymptotic behaviour for a family of entropies to defective Fokker–Planck equations and show that, much like defective finite dimensional ODEs, their decay rate is an exponential multiplied by a polynomial in time. The novelty of our study lies in the amalgamation of spectral theory and a quantitative non-symmetric hypercontractivity result, as opposed to the usual approach of the entropy method.     

On Non-Uniqueness of Probability Solutions to the Two-Dimensional Stationary Fokker–Planck–Kolmogorov Equation

The problem of uniqueness of probability solutions to the two-dimensional stationary Fokker–Planck–Kolmogorov equation is considered. Under broad conditions, it is proved that the existence of two different probability solutions implies the existence of an infinite set of linearly independent probability solutions.     

New results on Fokker–Planck equations of fractional order

It is shown that the fractional Fokker–Planck equations proposed recently in the literature (by merely substituting time fractional derivative for time derivative) give rise to some problems in the sense that they provide probability densities which may have negative values. In the same way, one shows that the Kramers–Moyal equation can be thought of as related to fractal processes, but it is well known that it yields also negative densities. It seems that the key of this trouble is the misuse of the Chapman Kolmogorov equation on the one hand, and of the fractional difference on the other hand. In fact, there is a complete identification between Kramers–Moyal equation and Fokker–Planck equation of fractional order. After a careful analysis, one arrives at the conclusion that the fractional derivative in Liouville–Riemann (L–R) sense should be replaced by a slightly finite fractional derivative which involves finite difference, whilst L–R fractional derivative refers to difference of infinite order. The new fractional Fokker–Planck equation so obtained is displayed, and its solution via separation of variables is outlined. It seems that there is no alternative but to work via non-standard analysis, that is to say infinitesimal discretization in time.     

On the fundamental solution of the Fokker–Planck–Kolmogorov equation

The Fokker–Planck–Kolmogorov parabolic second-order differential operator is considered, for which its fundamental solution is derived in explicit form. Such operators arise in numerous applications, including signal filtering, portfolio control in financial mathematics, plasma physics, and problems involving linear-quadratic regulators.     

Estimates for Solutions to Fokker–Planck–Kolmogorov Equations with Integrable Drifts

The result of this paper states that every probability measure satisfying the stationary Fokker–Planck–Kolmogorov equation obtained by a -integrable perturbation of the drift term–x of the Ornstein–Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure γ and \(f = \frac{{d\mu }}{{d\gamma }}\) for the density the integral of

$$f\left| {\log } \right|{\left( {f + 1} \right)^\alpha }$$

with respect to γ is estimated via \({\left\| v \right\|_{{L^1}\left( \mu \right)}}\) for all α < \(\frac{1}{4}\). This shows that stationary measures of infinite-dimensional diffusions whose drifts are integrable perturbations of–are absolutely continuous with respect to Gaussian measures. A generalization is obtained for equations on Riemannian manifolds.

     

Self-similar solutions of stationary Navier–Stokes equations

In this paper, we mainly study the existence of self-similar solutions of stationary Navier–Stokes equations for dimension $n=3,4$. For $n=3$, if the external force is axisymmetric, scaling invariant, $C^{1,\alpha }$ continuous away from the origin and small enough on the sphere $S^2$, we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class $C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$. Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular ($U\in C_{loc}^{3,\alpha }({\mathbb{R}}^3\backslash 0)$) axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier–Stokes equations with any scaling invariant external force in $L^{4/3,\infty }({\mathbb{R}}^4)$.     

On uniqueness of stationary solutions to the Navier–Stokes equations in exterior domains

The aim of this paper is to prove a uniqueness criterion for solutions to the stationary Navier–Stokes equation in 3-dimensional exterior domains within the class $u\in L_{3,\infty }$ with $?u\in L_{3/2,\infty }$, where $L_{3,\infty }$ and $L_{3/2,\infty }$ are the Lorentz spaces. Our criterion asserts that if $u$ and $v$ are the solutions, $u$ is small in $L_{3,\infty }$ and $u,v\in L_p$ for some $p>3$, then $u=v$. The proof is based on analysis of the dual equation with the aid of the bootstrap argument.