On some classes of Fleming-Viot type differential operators on the unit interval
Dedicated to Professor Gheorghe Coman on the occasion of his 80th anniversary
Keywords:
Degenerate second-order elliptic differential operator, Fleming-Viot operator, Markov semigroup, approximation of semigroup, Markov operator, positive approximation process, Bernstein-Schnabl operator, generalized Kantorovich operator, Bernstein-Durrmeyer operator with Jacobi weights.Abstract
Of concern are some classes of initial-boundary value differential problems associated with one-dimensional Fleming-Viot differential operators. Among other things, these operators occur in some models from population genetics to study the fluctuation of gene frequency under the influence of mutation and selection. The main aim of this survey paper is to discuss old and more recent results about the existence, uniqueness and continuous dependence from initial data of the solutions to these problems through the theory of the C0-semigroups of operators. Other additional aspects which will be highlighted, concern the approximation of the relevant semigroups in terms of positive linear operators. The given approximation formulae allow to infer several preservation properties of the semigroups together with their asymptotic behaviour. The analysis is carried out in the context of the space C([0, 1]) as well as, in some particular cases, in L p ([0, 1]) spaces, 1 ≤ p < +∞. Finally, some open problems are also discussed.
Mathematics Subject Classification (2010): 35K65, 41A36, 47B65, 47D07.
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