Mathematics Colloquia and Seminars

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Heuristically, two stochastic processes are dual if one can study one using the other. More formally, let $(X_t)$ and $(Y_t)$ be two real valued processes and let $H$ be a measurable, $R^2 \mapsto R$ function. We say that $(X_t)$ and $(Y_t)$ are $H$-dual if $E[H(X_t,y)|X_0=x]=E[H(x,Y_t)|Y_0=y]$.

Sampling Duality is stochastic duality using a duality function $S(n,x)$ of the form ¨what is the probability that all the members of a sample of size n are of certain type, given that the number (or frequency) of that type of individuals is x¨. Implicitly, this technique can be traced back to the work of Pascal. Explicitly, it was studied in a paper of Martin Möhle in 1999 in the context of population genetics. We will discuss several examples in which this technique is useful, including Haldane's formula for the fixation probability of a beneficial mutation and the long standing open question in theoretical evolution of the rate of the Muller Ratchet.