# Survival probability for jump processes in unbounded domains on metric measure spaces

## TL;DR

This paper investigates the long-term decay of survival probabilities for symmetric jump processes in unbounded metric measure spaces, providing explicit bounds and conditions related to the spectrum's bottom.

## Contribution

It establishes explicit asymptotic bounds for survival probabilities in unbounded domains, linking spectral properties to geometric conditions for jump processes.

## Key findings

- Explicit asymptotic bounds for survival probabilities.
- Geometric conditions for positive spectral bottom.
- Sharp decay rates in horn-shaped domains.

## Abstract

We study the large time behavior of the survival probability $\mathbb{P}_x\left(\tau_D>t\right)$ for symmetric jump processes in unbounded domains with a positive bottom of the spectrum. We prove asymptotic upper and lower bounds with explicit constants in terms of the bottom of the spectrum $\lambda(D)$. Our main result applies to symmetric jump processes in general metric measure spaces. For $\alpha$-stable processes in unbounded uniformly $C^{1,1}$ domains, our results provide a probabilistic interpretation and an equivalent geometric condition for $\lambda(D)>0$. In the case of increasing horn-shaped domains, the exponential rate of decay for the survival probability is sharp. We also present examples of unbounded domains where our results apply.