Abstract
Consider a branching diffusion in which each individual
moves as a Markov diffusion with corresponding operator
L and branches at a (spatial) rate b into precisely two
particles at each fission point. Suppose the process
begins from an individual particle. Given any ball and
starting position, does a criteria exist which will
guarantee the ball is visited (or charged) infinitely
often by the branching process with positive/zero
probability. The answer is yes and the criteria concerns
the sign (+/-) of the minimum real number a such that
there exist a positive harmonic function with respect to
the operator (L+b-a). The number a is called the generalized
principal eigenvalue. The proofs are purely probabilistic
and conceptual, appealing to martingale techniques.
This is joint work with Janos Englander.
Contact person:Goran Peskir.