Related papers: Gaussian heat kernel bounds through elliptic Moser…

We consider metric graphs with Kirchhoff boundary conditions. We study the intrinsic metric, volume doubling and a Poincar\'e inequality. This enables us to prove a parabolic Harnack inequality. The proof involves various techniques from…

We prove an $L^{p}$ estimate $$ \|e^{-itL} \varphi(L)f\|_{p}\lesssim (1+|t|)^s\|f\|_p, \qquad t\in \mathbb{R}, \qquad s=n\left|\frac{1}{2}-\frac{1}{p}\right| $$ for the Schr\"odinger group generated by a semibounded, selfadjoint operator…

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if $\phi$ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition…

We prove the global existence and the uniqueness of the $L^p\cap H_0^1-$valued ($2\leq p < \infty$) strong solutions of a nonlinear heat equation with constraints over bounded domains in any dimension $d\geq 1$. Along with the…

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms which…

The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the unit ball and simplex in $\mathbb{R}^n$, and in particular on the interval, generated by classical differential operators whose eigenfunctions are…

The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the…

Two-sided Gaussian bounds are established for the weighted heat kernels on the unit ball and simplex in $\mathbb{R}^d$ generated by classical differential operators whose eigenfunctions are algebraic polynomials.

By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of…

In this article, we describe a geometric method to study cusp forms, which relies on heat kernel and Bergman kernel analysis. This new approach of applying techniques coming from analytic geometry is based on the micro-local analysis of the…

Let $\mathcal{H}=\sum_{j=1}^{m}X_{j}^{2}-\partial_{t}$ be a heat-type operator in $\mathbb{R}^{n+1}$, where $X=\{X_{1},\ldots,X_{m}\}$ is a system of smooth H\"{o}rmander's vector fields in $\mathbb{R}^{n}$, and every $X_{j}$ is homogeneous…

We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the…

We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon>0$ there exists such a field, whose covariance is…

Let $d\geq 1$ and $\alpha \in (0, 2)$. Consider the following non-local and non-symmetric L\'evy-type operator on $\mR^d$: $$ \sL^\kappa_{\alpha}f(x):=\mbox{p.v.}\int_{\mR^d}(f(x+z)-f(x))\frac{\kappa(x,z)}{|z|^{d+\alpha}} \dif z, $$ where…

We construct a conservative and strongly local regular symmetric Dirichlet form on the pointed Gromov--Hausdorff limit space and demonstrate the stability of heat kernel estimates under this convergence. Furthermore, we establish the Mosco…

We consider the linear heat equation on a bounded domain. We study estimates of the derivatives, up to the second order, of the solution locally in time in the Lebesgue spaces. We give a self-contained proof of the estimates in the…

In this note, we derive a new logarithmic Sobolev inequality for the heat kernel on the Heisenberg group. The proof is inspired from the historical method of Leonard Gross with the Central Limit Theorem for a random walk. Here the non…

We consider a self-adjoint non-negative operator $H$ in a Hilbert space $\mathsf{L}^2(X,{\rm d}\mu)$. We assume that the semigroup $(\mathrm{e}^{-t H})_{t>0}$ is defined by an integral kernel, $p$, which allows an estimate of the form…

We prove Poisson upper bounds for the kernel $K$ of the semigroup generated by the Dirichlet-to-Neumann operator if the underlying domain is bounded and has a $C^\infty$-boundary. We also prove Poisson bounds for $K_z$ for all $z$ in the…

We derive estimates of the derivatives of the heat kernel on noncompact symmetric spaces and on locally symmetric spaces. Applying these estimates we study the $L^{p}$-boundedness of Littlewood-Paley-Stein operators and the Laplacian of the…