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The asymptotic behavior of fragmentation processes

Jean Bertoin (2003)

Journal of the European Mathematical Society

The fragmentation processes considered in this work are self-similar Markov processes which are meant to describe the evolution of a mass that falls apart randomly as time passes. We investigate their pathwise asymptotic behavior as t → ∞ . In the so-called homogeneous case, we first point at a law of large numbers and a central limit theorem for (a modified version of) the empirical distribution of the fragments at time t . These results are reminiscent of those of Asmussen and Kaplan [3] and Biggins...

The behavior of a Markov network with respect to an absorbing class: the target algorithm

Giacomo Aletti (2009)

RAIRO - Operations Research

In this paper, we face a generalization of the problem of finding the distribution of how long it takes to reach a “target” set T of states in Markov chain. The graph problems of finding the number of paths that go from a state to a target set and of finding the n-length path connections are shown to belong to this generalization. This paper explores how the state space of the Markov chain can be reduced by collapsing together those states that behave in the same way for the purposes of calculating...

The boundary Harnack principle for the fractional Laplacian

Krzysztof Bogdan (1997)

Studia Mathematica

We study nonnegative functions which are harmonic on a Lipschitz domain with respect to symmetric stable processes. We prove that if two such functions vanish continuously outside the domain near a part of its boundary, then their ratio is bounded near this part of the boundary.

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