Diffusion processes, Feller semigroups and Wentzell boundary conditions.
Different approaches to the study of many diffusion processes in Genetics involve Probability, Functional Analysis and Partial Differential Equations, as in the case of changes in gene frequency due only to random sampling or under random fluctuation of selective advantages. In the one-dimensional case, a unified treatment of them was given by Feller. For particular classes of Markov processes, Taira showed that these different approaches are equivalent even in the N-dimensional case. It follows that the generator of a Feller semigroup on the space of real-valued continuous functions C(D), where D is a bounded domain of RN with smooth boundary, can be identified with a particular Markov transition function. Under suitable assumptions, Taira, Favini and the author proved that some classes of degenerate elliptic operators with Wentzell boundary condition generate Feller semigroups on C(D), in such a way that the diffusion phenomenon of viscosity occurs at each point of the boundary.
Romanelli
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Dipartimento di Matematica, Università di Bari, Via E. Orabona 4, 70125 Bari.
D005787:Gene Frequency; D006801:Humans; D008390:Markov Chains; D008957:Models, Genetic; D011336:Probability; D013269:Stochastic Processes