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Collisions of random walks

Martin T. Barlow, Yuval Peres, Perla Sousi (2012)

Annales de l'I.H.P. Probabilités et statistiques

A recurrent graph G has the infinite collision property if two independent random walks on G , started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton–Watson tree with finite variance conditioned to survive, the incipient infinite cluster in ℤ d with d ≥ 19 and the uniform spanning tree in ℤ 2 all have the infinite collision property. For power-law combs and spherically symmetric...

Compactness properties of Feller semigroups

G. Metafune, D. Pallara, M. Wacker (2002)

Studia Mathematica

We study the compactness of Feller semigroups generated by second order elliptic partial differential operators with unbounded coefficients in spaces of continuous functions in ℝ N .

Degenerate stochastic differential equations for catalytic branching networks

Sandra Kliem (2009)

Annales de l'I.H.P. Probabilités et statistiques

Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of the paper by Dawson and Perkins [Illinois J. Math.50 (2006) 323–383] to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Hölder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting.

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