Positive Markov processes in Laplace duality
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435
ENS
The purpose of this talk, based on joint work with Matija Vidmar, is to investigate the class of positive Markov processes that admit a Laplace duality relationship: the Laplace transforms of the processes are related through an exchange of the argument and the initial state. This type of duality naturally emerges for instance in systems with branching phenomena. Beyond the classical branching framework, we show that a wide variety of processes and generators fall within the scope of Laplace duality.
First, from a theoretical perspective, we establish that a process admits a Laplace dual if and only if its semigroup leaves invariant the space of completely monotone functions (subject to conventions for 0 × ∞ and ∞ × 0). In a more constructive direction, we then identify seven fundamental building blocks from which such duality can be constructed. The associated processes can be viewed as generalisations of continuous-state branching processes and include several models — sometimes discovered independently of the duality perspective — used to represent random environments, immigration, collisions, and other dynamics. A key analytical tool, developed here in a general and unifying setting, is the notion of a Laplace symbol associated with a generator.