This paper develops precise estimates for the potential kernel, capacities of large intervals, and the probabilities of hitting large intervals for the asymmetric Cauchy processes. These are then applied to study three problems concerning the sample paths: (i) the rate of escape of $|X_t|$ as $t \rightarrow \infty$; (ii) the sizes of the large holes in the range of the process; (iii) the asymptotic behavior of the Lebesgue measure of that part of the range of the process that is in a large interval.
"The Behavior of Asymmetric Cauchy Processes for Large Time." Ann. Probab. 11 (2) 302 - 327, May, 1983. https://doi.org/10.1214/aop/1176993598