Reversing time as an analytical tool: Isn't that just Radon-Nikodym?

Phil Pollett

Abstract: Let X=(X(t), t in T) and X*=(X*(t), t in T) be two stochastic processes with the same parameter set T and the same state space (E,E). I begin by explaining what it means for X* to be the time reverse of X and then discuss some implications of this definition. I will then characterize the property for Markov processes on a general state space with a given transition function p, thereby introducing the idea of a reverse transition function. I will show how the reverse transition function might be used, outside the context of mere time reversal, as an analytical tool for establishing the existence of invariant measures.

The major problem I will consider is that of establishing the existence of a reverse transition function p* in terms of a given sigma-finite subinvariant measure m on (E,E), something which, for countable-state Markov chains (in discrete or continuous time) is almost trivial. We will see that, while the Radon-Nikodym theorem does indeed throw up a candidate p*, it is far from clear that p* is a transition function. I will prove that the candidate p* is a transition function under the condition that m be totally finite, thus establishing the existence and uniqueness (m-almost everywhere) of a reverse transition function. Whilst the finititude of m is a rather ugly assumption, it does help us avoid technicalities. I will indicate how this assumption can be relaxed, but we will need to equip E will a special, but very general, topology.

Several musical examples will be used to motive the ideas of the talk.

Acknowledgement: This work is supported by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems

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Last modified: 11th May 2006