AI 中文总结
本文将满足次高斯热核上界的无杀伤扩散过程的首出时估计推广到一般Borel右过程的首中时，给出上下界并刻画常数关系，还研究了小时间渐近行为。
AI 中文摘要
本文研究了度量测度空间$(E,d,μ)$上热核满足次高斯界的Borel右过程的首中时。众所周知，若$X=(X_t)_{t\neq 0}$是无杀伤的扩散过程且其热核满足次高斯上界，那么在体积增长条件$μ(B(x,r))\times r^α$下，它满足：\\[ \IP^x[τ_{B(x,r)}\le t]\le C_1\exp\left\{-C_2(r^β/t)^{1/(β-1)}\right\}, \\] 其中$B(x,r):=\{y\in E: d(y,x)<r\}$，$τ_{B(x,r)}:=\inf\{t>0:X_t\notin B(x,r)\}$，$β$是次高斯热核估计中出现的游走维数。我们将这一结果推广到一般Borel右过程，证明在体积增长的上界条件下，有：\\[ \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\}, \\] 其中$B$是近Borel集，$σ_B$表示$B$的首次首中时，$\widetilde d(x,B)$表示去除$B$的极子集影响后$x$到$B$的距离。此外，我们证明对于具有次高斯热核下界的Borel右过程，其首中时分布满足对应的下界：\\[ \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\}. \\] 我们还刻画了常数$C_i$（$3\leq i\leq 6$）与对应热核界指数中出现的常数之间的关系。作为这些首中时估计的应用，我们进一步研究了$t\downarrow 0$时$\IP^x[σ_B\leq t]$的小时间渐近行为。
英文摘要
In this paper, we study the hitting times of Borel right processes on a metric measure space $(E,d,μ)$ whose heat kernels satisfy sub-Gaussian bounds. It is well known that if $X=(X_t)_{t\geq 0}$ is diffusion process without killing whose heat kernel satisfies a sub-Gaussian upper bound, then, under the volume growth condition $μ(B(x,r))\asymp r^α$, 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 $B(x,r):=\{y\in E: d(y,x)<r\}$, $τ_{B(x,r)}:=\inf\{t>0:X_t\notin B(x,r)\}$, and $β$ 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, $σ_B$ denotes the first hitting time of $B$, and $\widetilde d(x,B)$ 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 $C_i$, $3\leq i\leq 6$, 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 $t\downarrow 0$.
