Lamperti Transforms of Self-Similar Measure-Valued Processes and Simple Coalescents

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Abstract

In this presentation, Alejandro will share his collaborative work with Arno Siri-Jégousse in which they derived a Lamperti transform for self-similar processes that take values in normed vector spaces. This transformation involves a random time change followed by a ‘(log) polar’ decomposition of the state space. The resulting process is a Markov additive process (MAP) in which the ‘(log) norm’ coordinate is additive-homogeneous. Lamperti originally studied the case of self-similar processes taking values in the positive reals, resulting in a MAP that is essentially a Lévy process. Alili et al. in 2017 extended his work to processes taking values in R^d, where now the ‘argument/angle’ coordinate becomes non-trivial. He will demonstrate an application of their generalization to processes taking values in the space of positive measures, thereby reframing and expanding upon the results of Birkner et al. in 2005. They showed that for self-similar measure-valued branching processes, the time-changed and renormalized process (i.e., the ‘argument’ in our setting) is a Fleming-Viot process, which is in duality with the Beta Coalescent. We strengthen this result and obtain Fleming-Viot processes that are in duality with general Lambda-Coalescents.

About Alejandro

Alejandro studied genomic sciences during his undergraduate studies, mostly focusing on molecular and computational biology. Then he did a masters and PhD studies in mathematics, focusing on Population Genetics. Now, during his postdoc, he is incursionating in theoretical aspects of the learning of manifolds from data, and in stochastic control theory.