Spectral condition, hitting times and Nash inequality

TL;DR

This paper establishes a spectral condition linking the finiteness of certain expected functionals of Hunt processes to the spectral measure of their generators, deriving Nash inequalities and convergence rates for diffusions and general processes.

Contribution

It provides necessary and sufficient spectral conditions for moments of exit times and derives Nash inequalities for killed processes, extending results to multi-dimensional diffusions and general Hunt processes.

Findings

Spectral conditions characterize finiteness of expected functionals.

Nash inequalities are derived for killed processes and diffusions.

Convergence rates in $L^2$ are established based on moments of exit times.

Abstract

Let $X$ be a $\mu$-symmetric Hunt process on a LCCB space E. For an open set G $\subseteq$ E, let $\tau_G$ be the exit time of $X$ from G and $A^G$ be the generator of the process killed when it leaves G. Let $r:[0,\infty[\to[0,\infty[$ and $R (t) = \int_0^t r(s) ds$. We give necessary and sufficient conditions for $\E_{\mu} R (\tau_G)<\infty$ in terms of the behavior near the origin of the spectral measure of $-A^G.$ When $r(t)=t^l$, $l>0$, by means of this condition we derive the Nash inequality for the killed process. In the case of one-dimensional diffusions, this permits to show that the existence of moments of order $l$ for $\tau_G$ implies the Nash inequality of order $p=\frac{l+2}{l+1}$ for the whole process. The associated rate of convergence of the semi-group in $L^2(\mu)$ is bounded by $t^{-(l+1)}$. For diffusions in dimension greater than one, we obtain the Nash inequality of the same order under an additional non-degeneracy condition (local Poincar\'e inequality). Finally, we show for general Hunt processes that the Nash inequality giving rise to a convergence rate of order $t^{-(l+1)}$ of the semi-group, implies the existence of moments of order $l+1 -\epsilon$ for $\tau_G$, for all $ \epsilon>0$.