Related papers: Nash's $G$ bound for the Kolmogorov equation

We prove a Gaussian upper bound for the fundamental solutions of a class of ultra-parabolic equations in divergence form. The bound is independent on the smoothness of the coefficients and generalizes some classical results by Nash, Aronson…

We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven…

Two-sided Gaussian estimates for the fundamental solution of a second order linear parabolic differential equation are upper and lower bounds in terms of the fundamental solution of the classical heat conduction equation. In his seminal…

The theory of De Giorgi (1958) and Nash (1959) solves Hilbert's 19th problem and constitutes a major advance in the analysis of PDEs in the 20th century. This theory concerns the H\"older regularity of solutions to elliptic and parabolic…

We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hormander condition. The bound is independent of the smoothness of the coefficients and generalizes…

We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H{\"o}lder regularity and establish a Harnack inequality for…

We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows H{\"o}lder estimates and the Harnack…

In this article, we give a trajectorial proof of a kinetic Poincar\'e inequality which plays an important role in the De Giorgi-Nash-Moser theory for kinetic equations. The present work improves a result due to J. Guerand and C. Mouhot [10]…

The aim of this work is to prove a Harnack inequality and the H\"older continuity for weak solutions to the Kolmogorov equation $\mathscr{L} u = f$ with measurable coefficients, integrable lower order terms and nonzero source term. We…

We prove that the Harnack inequality fails for nonlocal kinetic equations. Such equations arise as linearized models for the Boltzmann equation without cutoff and are of hypoelliptic type. We provide a counterexample for the simplest…

We present a survey on the regularity theory for classic solutions to subelliptic degenerate Kolmogorov equations. In the last part of this note we present a detailed proof of a Harnack inequality and a strong maximum principle.

In this paper we continue the study on intrinsic Harnack inequality for non- homogeneous parabolic equations in non-divergence form initiated by the first author in [1]. We establish a forward-in-time intrinsic Harnack inequality, which in…

In this paper, we prove a kinetic Nash type inequality and adapt it to a new functional inequality for functions in a kinetic Sobolev space with absorbing boundary conditions on the half-space. As an application, we address the boundary…

We study the De Giorgi-Moser-Nash estimates of higher-order parabolic equations in divergence form with complex-valued, measurable, bounded, uniformly elliptic (in the sense of G$\mathring{a}$rding inequality) and time-independent…

We construct critical trajectories in kinetic geometry, i.e. curves in $\mathbb{R}^{1+2n}$ that are: tangential to the vector fields $\partial_t+v\cdot \nabla_x$ and $\nabla_v$, connecting any two given points, respecting the underlying…

The aim of this work is to prove the existence of a fundamental solution associated to the Kolmogorov equation L u = f with measurable coefficients in the dilation invariant case. Moreover, we prove Gaussian upper and lower bounds for it,…

Under suitable conditions, we obtain some characterization of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator $G(t,s)$ associated to a class of nonautonomous second order parabolic equations with…

We extend the De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations arising naturally in kinetic theory, in which a first-order transport operator is coupled with an elliptic nonlocal operator involving fractional…

We prove a differential Harnack inequality for the solution of the parabolic Allen-Cahn equation $ \frac{\partial f}{\partial t}=\triangle f-(f^3-f)$ on a closed n-dimensional manifold. As a corollary we find a classical Harnack inequality.…

This article deals with kinetic Fokker-Planck equations with essentially bounded coefficients. A weak Harnack inequality for non-negative super-solutions is derived by considering their Log-transform and following S. N. Kruzhkov (1963).…