In this paper, we consider a large class of subordinate random walks X on integer lattice Z d via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions.
Let be a symmetric simple random walk on the integer lattice ℤ. For a Bernstein function we consider a random walk which is subordinated to. Under a certain assumption on the behaviour of at zero we establish global estimates for the transition probabilities of the random walk. The main tools that we apply are a parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.
In this article, we establish an almost sure invariance principle for the capacity and cardinality of the range for a class of α-stable random walks on the integer lattice Z d with d > 5α/2 and d > 3α/2, respectively. As a direct consequence, we conclude Khintchine's and Chung's laws of the iterated logarithm for both processes.2010 Mathematics Subject Classification. 60F17 60F05, 60G50, 60G52 . Key words and phrases. the range of a random walk, capacity, an almost sure invariance principle, the law of the iterated logarithm.
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