On hitting time distributions of Markov processes with sub-Gaussian heat kernel bounds

Abstract

In this paper, we study the hitting times of Borel right processes on a metric measure space $(E,d,\mu )$ whose heat kernels satisfy sub-Gaussian bounds. It is well known that if $X=(X_t{)}_{t\ge 0}$ is diffusion process without killing whose heat kernel satisfies a sub-Gaussian upper bound, then, under the volume growth condition $\mu (B(x,r))\asymp r^{\alpha }$, it satisfies \[ \IP^x[τ_{B(x,r)}\le t]\le C_1\exp\left\{-C_2(r^β/t)^{1/(β-1)}\right\}, \] where $B(x,r):=\{y\in E:d(y,x)<r\}$, ${\tau }_{B(x,r)}:=\inf \{t>0:X_t\notin B(x,r)\}$, and $\beta$ is the walk dimension appearing in the sub-Gaussian heat kernel estimate. We extend this result to general Borel right processes, showing that under an upper bound condition on the volume growth, \[ \IP^x[σ_B\le t]\le C_3\exp \left\{-C_4\left(\frac{\widetilde d(x, B)^β}{t}\right)^{1/(β-1)}\right\}, \] where $B$ is a nearly Borel set, ${\sigma }_B$ denotes the first hitting time of $B$, and $\tilde{d}(x,B)$ represents the distance from $x$ to $B$ after removing the influence of polar subsets of $B$. Furthermore, we show that for a Borel right process with a sub-Gaussian heat kernel lower bound, the hitting time distribution satisfies the corresponding lower bound \[ \IP^x[σ_B\le t]\ge C_5 \exp\left\{-C_6\cdot \left(\frac{\widetilde d(x,B)^β}{t}\right)^{1 /(β-1)}\right\}. \] We also characterize the relationship between the constants $C_i$, $3\le i\le 6$, and the constants appearing in the exponents of the corresponding heat kernel bounds. As an application of these hitting time estimates, we further study the small-time asymptotic behavior of $\IP^x[σ_B\leq t]$ as $t\downarrow 0$.