Feeling boundary by Brownian motion in a ball

We establish short-time asymptotics with rates of convergence for the Laplace Dirichlet heat kernel in a ball. So far, such results were only known in simple cases where explicit formulae are available, i.e., for sets as half-line, interval and their products. Presented asymptotics may be considered as a complement or a generalization of the famous “principle of not feeling the boundary” in case of a ball. Following the metaphor, the principle reveals when the process does not feel the boundary, while we describe what happens when it starts feeling the boundary.

Intrinsic Ultracontractivity for Domains in Negatively Curved Manifolds

Let M be a complete, non-compact, connected Riemannian manifold with Ricci curvature bounded from below by a negative constant. A sufficient condition is obtained for open and connected sets D in M for which the corresponding Dirichlet heat semigroup is intrinsically ultracontractive. That condition is formulated in terms of capacitary width. It is shown that both the reciprocal of the bottom of the spectrum of the Dirichlet Laplacian acting in $$L^2(D)$$ L 2 ( D ) , and the supremum of the torsion function for D are comparable with the square of the capacitary width for D if the latter is sufficiently small. The technical key ingredients are the volume doubling property, the Poincaré inequality and the Li-Yau Gaussian estimate for the Dirichlet heat kernel at finite scale.

Estimates the upper bounds of Dirichlet eigenvalues for fractional Laplacian

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^n \; (n\geq 2) $\end{document}</tex-math></inline-formula> be a bounded domain with continuous boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>. In this paper, we study the Dirichlet eigenvalue problem of the fractional Laplacian which is restricted to <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ 0&lt;s&lt;1 $\end{document}</tex-math></inline-formula>. Denoting by <inline-formula><tex-math id="M5">\begin{document}$ \lambda_{k} $\end{document}</tex-math></inline-formula> the <inline-formula><tex-math id="M6">\begin{document}$ k^{th} $\end{document}</tex-math></inline-formula> Dirichlet eigenvalue of <inline-formula><tex-math id="M7">\begin{document}$ (-\triangle)^{s}|_{\Omega} $\end{document}</tex-math></inline-formula>, we establish the explicit upper bounds of the ratio <inline-formula><tex-math id="M8">\begin{document}$ \frac{\lambda_{k+1}}{\lambda_{1}} $\end{document}</tex-math></inline-formula>, which have polynomially growth in <inline-formula><tex-math id="M9">\begin{document}$ k $\end{document}</tex-math></inline-formula> with optimal increase orders. Furthermore, we give the explicit lower bounds for the Riesz mean function <inline-formula><tex-math id="M10">\begin{document}$ R_{\sigma}(z) = \sum_{k}(z-\lambda_{k})_{+}^{\sigma} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M11">\begin{document}$ \sigma\geq 1 $\end{document}</tex-math></inline-formula> and the trace of the Dirichlet heat kernel of fractional Laplacian.</p>

Some local maximum principles along Ricci flows

In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ for some constant ${\alpha }>0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ .

Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels, in