Related papers: Semi-classical Heat Kernel Asymptotics and Morse I…

For Riemannian symmetric spaces $X=G/K$ of noncompact type, we show that for all left $K$-invariant $f\in L^1(X)$, the functions $\|h_t\|_{L^p(X)}^{-1}(f\ast h_t-M_p(f)h_t)$ (with $h_t$ being the heat kernel of $X$) converges to zero in…

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic…

Following the classical result of long-time asymptotic convergence towards the Gaussian kernel that holds true for integrable solutions of the Heat Equation posed in the Euclidean Space $\mathbb{R}^n$, we examine the question of long-time…

It is shown that the heat kernel operator for the Laplace operator on any covariantly constant curved background, i.e. in symmetric spaces, may be presented in form of an averaging over the Lie group of isometries with some nontrivial…

Let $J$ be the L\'evy density of a symmetric L\'evy process in $\mathbb{R}^d$ with its L\'evy exponent satisfying a weak lower scaling condition at infinity. Consider the non-symmetric and non-local operator $$ {\mathcal L}^{\kappa}f(x):=…

In this paper we employ the Renormalization Group (RG) method to study the long-time asymptotics of a class of nonlinear integral equations with a generalized heat kernel and with time-dependent coefficients. The nonlinearities are…

Let M be a compact Riemannian manifold with smooth boundary. We obtain the exact long time asymptotic behaviour of the heat kernel on abelian coverings of M with mixed Dirichlet and Neumann boundary conditions. As an application, we study…

In this expository work we discuss the asymptotic behaviour of the solutions of the classical heat equation posed in the whole Euclidean space. After an introductory review of the main facts on the existence and properties of solutions, we…

We study the large-time asymptotic behavior of solutions to the discrete-time heat equation, i.e., caloric functions, on affine buildings, including those without transitive group actions. For each $p \in [1, \infty]$, we introduce a notion…

We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat kernels and their connection to jump diffusions.

The heat kernel expansion for a general non--minimal operator on the spaces $C^\infty (\Lambda^k)$ and $C^\infty (\Lambda^{p,q})$ is studied. The coefficients of the heat kernel asymptotics for this operator are expressed in terms of the…

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on an Euclidean space and a compact manifold. We study the "cut locus" case, namely, the case where energy-minimizing paths which join the two points…

We study inequalities related to the heat kernel for the hypoelliptic sublaplacian on an H-type Lie group. Specifically, we obtain precise pointwise upper and lower bounds on the heat kernel function itself. We then apply these bounds to…

We present a brief overview of several approaches for calculating the local asymptotic expansion of the heat kernel for Laplace-type operators. The different methods developed in the papers of both authors some time ago are described in…

This paper studies the asymptotic behavior of the integral kernel of the Dunkl transform, the so-called Dunkl kernel, when one of its arguments is fixed and the other tends to infinity either within a Weyl chamber of the associated…

We prove pointwise and $L^p$ gradient estimates for the heat kernel on the bounded and unbounded Vicsek set and applications to Sobolev inequalities are given. We also define a Hodge semigroup in that setting and prove estimates for its…

We prove some estimations of the correlation of two local observables in quantum spin systems (with Schr\"odinger equations) at large temperature. For that, we describe the heat kernel of the Hamiltonian for a finite subset of the lattice,…

We consider Dirichlet heat kernel $p_a^{(\mu)}(t,x,y)$ for the Bessel differential operator $L^{(\mu)}=\frac{d^2}{dx^2}+\frac{2\mu+1}{2x}$, $\mu\in\mathbb{R}$, in half-line $(a,\infty)$, $a>0$, and provide its asymptotic expansions for…

In this work, we establish the small-noise asymptotic behaviour (namely, the functional law of large numbers and the large deviation principle) for multi-scale McKean--Vlasov diffusions with super-linear kernels. In this setting, the…

Let $M$ be a relatively compact connected open subset with smooth connected boundary of a complex manifold $M'$. Let $(L,h^L)\rightarrow M'$ be a positive line bundle over $M'$. Suppose that $M'$ admits a holomorphic $\mathbb{R}$-action…