Weighted Poincaré inequality and heat kernel estimates for finite range jump processes

Abstract

It is well-known that there is a deep interplay between analysis and probability theory. For example, for a Markovian infinitesimal generator ${\mathcal{L}}$ , the transition density function p(t, x, y) of the Markov process associated with ${\mathcal{L}}$ (if it exists) is the fundamental solution (or heat kernel) of ${\mathcal{L}}$ . A fundamental problem in analysis and in probability theory is to obtain sharp estimates of p(t, x, y). In this paper, we consider a class of non-local (integro-differential) operators ${\mathcal{L}}$ on ${\mathbb{R}^d}$ of the form

$$\mathcal{L}u(x) = \lim\limits_{{\varepsilon \downarrow 0}} \int\limits_{\{y\in \mathbb {R}^d: \, |y-x| > \varepsilon\}} (u(y)-u(x)) J(x, y) dy,$$

where ${\displaystyle J(x, y)= \frac{c (x, y)}{|x-y|^{d+\alpha}} {\bf 1}_{\{|x-y| \leq \kappa\}}}$ for some constant ${\kappa > 0}$ and a measurable symmetric function c(x, y) that is bounded between two positive constants. Associated with such a non-local operator ${\mathcal{L}}$ is an ${\mathbb{R}^d}$ -valued symmetric jump process of finite range with jumping kernel J(x, y). We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for them. Along the way, some new heat kernel estimates are obtained for more general finite range jump processes that were studied in (Barlow et al. in Trans Am Math Soc, 2008). One of our key tools is a new form of weighted Poincaré inequality of fractional order, which corresponds to the one established by Jerison in (Duke Math J 53(2):503–523, 1986) for differential operators. Using Meyer’s construction of adding new jumps, we also obtain various a priori estimates such as Hölder continuity estimates for parabolic functions of jump processes (not necessarily of finite range) where only a very mild integrability condition is assumed for large jumps. To establish these results, we employ methods from both probability theory and analysis extensively.