In this paper we study some applications of the Lévy logarithmic Sobolev inequality to the study of the regularity of the solution of the fractal heat equation, i. e. the heat equation where the Laplacian is replaced with the fractional Laplacian. It is also used to the study of the asymptotic behaviour of the Lévy-Ornstein-Uhlenbeck process.
We study the degenerated Itô SDE on $\mathbb R^d$ whose drift coefficient only fulfills a mixed Osgood and Sobolev regularity. Under suitable assumptions on the gradient of the diffusion coefficient and on the divergence of the drift coefficient, we prove the existence and uniqueness of generalized stochastic flows associated to such equations. We also prove the uniqueness of solutions to the corresponding Fokker--Planck equation by using the probabilistic method.
We prove two new results connected with elliptic Fokker-Planck-Kolmogorov equations with drifts integrable with respect to solutions. The first result answers negatively a long-standing question and shows that a density of a probability measure satisfying the Fokker-Planck-Kolmogorov equation with a drift integrable with respect to this density can fail to belong to the Sobolev class~$W^{1,1}(\mathbb{R}^d)$. There is also a version of this result for densities with respect to Gaussian measures. The second new result gives some positive information about properties of such solutions: the soluti…
We prove that every probability measure $μ$ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a $μ$-integrable perturbation $v$ of the drift term $-x$ of the Ornstein-Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure $γ$ and for the density $f=dμ/dγ$ the integral of $f |\log (f+1)|^α$ against $γ$ is estimated via $\|v\|_{L^1(μ)}$ for all $α<1/4$, which is a weakened $L^1$-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolute…
We study one-dimensional functional inequalities of the type of Poincaré, logarithmic Sobolev and Wirtinger, with weight, for probability densities with polynomial tails. As main examples, we obtain sharp inequalities satisfied by inverse Gamma densities, taking values on $R_+$, and Cauchy-type densities, taking values on $R$. In this last case, we improve the result obtained by Bobkov and Ledoux in 2009 by introducing a better weight function in the logarithmic Sobolev inequality. The results are obtained by resorting to Fokker-Planck type equations which possess these densities as steady sta…
An operatorial based approach is used here to prove the existence and uniqueness of a strong solution $u$ to the time-varying nonlinear Fokker--Planck equation $u_t(t,x)-Δ(a(t,x,u(t,x))u(t,x))+{\rm div}(b(t,x,u(t,x))u(t,x))=0$ in $(0,\infty)\times \mathbb{R}$ $u(0,x)=u_0(x),\ x\in\mathbb{R}^d$ in the Sobolev space $H^{-1}(\mathbb{R}^d)$, under appropriate conditions on the $a:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}$ and $b:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}^d.$ It is proved also that, if $u_0$ is a density of a probability measure, so is $u(t,\cdot)$ for all $t\g…
We study the long time behaviour of the kinetic Fokker-Planck equation with mean field interaction, whose limit is often called Vlasov-Fkker-Planck equation. We prove a uniform (in the number of particles) exponential convergence to equilibrium for the solutions in the weighted Sobolev space H 1 ($μ$) with a rate of convergence which is explicitly computable and independent of the number of particles. The originality of the proof relies on functional inequalities and hypocoercivity with Lyapunov type conditions, usually not suitable to provide adimensional results.
In this article we prove the existence and uniqueness for degenerate stochastic differential equations with Sobolev (possibly singular) drift and diffusion coefficients in a generalized sense. In particular, our result covers the classical DiPerna-Lions flows and, we also obtain the well-posedness for degenerate Fokker-Planck equations with irregular coefficients. Moreover, a large deviation principle of Freidlin-Wenzell type for this type of SDEs is established.
Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincaré inequality. Their proof is based on the Bakry-Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra, and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli-Laplace mod…
We obtain and study new $Φ$-entropy inequalities for diffusion semigroups, with Poincaré or logarithmic Sobolev inequalities as particular cases. From this study we derive the asymptotic behaviour of a large class of linear Fokker-Plank type equations under simple conditions, widely extending previous results. Nonlinear diffusion equations are also studied by means of these inequalities. The $Γ_2$ criterion of D. Bakry and M. Emery appears as a main tool in the analysis, in local or integral forms.
We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a defi…
In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, whi…
This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield Hölder regularity of the non-local order parameter, which is improved by an iteration argu…
