Laws of the iterated logarithm for symmetric jump processes

Abstract

Abstract. Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are obtained for β-stable-like processes on α-sets with β > 0.

Lamplighter Random Walks on Fractals

“…The next theorem is an analogue of [12, Proposition 4.8]. Since our proof is different from that of [12,Proposition 4.8] which uses a scaling argument, we give the proof below.…”

“…Since our random walk does not satisfy this property, we need non-trivial modifications for the proof. Quite recently, Kim, Kumagai and Wang [12,Theorem 4.14] proved the LIL of the range for jump processes without using self-similarity of the process. By easy modifications, we can apply their argument to our random walk.…”

Bottom crossing probability for symmetric jump processes

“…However, the scaling property and the rotation invariance of symmetric stable processes on R d played a crucial role in his proof. Instead of these properties, we make use of the full heat kernel estimates by following [26] and [32]. Our results are applicable to symmetric stable-like processes (see [9,11,12]) and a class of symmetric jump processes on (unbounded) fractals and fractal-like spaces (see Section 3 below for details).…”

Rate Functions for Symmetric Markov Processes via Heat Kernel

“…In this paper, we will get the zero-one laws of rate functions by developing the approach of the results as mentioned before. In particular, for the upper rate functions, we use a similar approach of Kim, Kumagai and Wang [27].…”