Resolvent estimates for Fleming-Viot operators and uniqueness of solutions to related martingale problems
Fleming-Viot processes arise as high-density diffusion approximations to certain interacting particle models in population genetics. They take values in a space of probability measures on a compact space \(X\). The state of the F-V process describes the empirical distribution of a large population over a set of distinct genetic types. The random changes are due to mutation and random pairwise mating. An F-V process is characterized by the generator \(A\) of a conservative Feller process, which represents the mutation and causes a drift, and a real function \(\gamma: X\times X\to \mathbb{R}\), continuous, symmetric, and positive, which represents the pairwise mating and causes a fluctuation. \(A\) and \(\gamma\) are called the drift and diffusion coefficients of the process. The coefficients \(A\) and \(\gamma\) do not depend on the measure variable \(\mu\) (the state of the process). By means of the martingale problem characterization of the F-V process, the authors consider processes which look like F-V processes with coefficients \(A\) and \(\gamma\) depending on \(\mu\). Under certain regularity assumptions on \(A(\mu)\) and \(\gamma (\mu)\), the authors prove existence and uniqueness results for a solution of the martingale problem. For this purpose, they define and study a two-parameter scale of Banach spaces contained in the space of continuous functions on the space of probability measures on \(X\), and they show that the resolvent of the F-V process is a bounded operator on the scale. Uniqueness is proved under the assumption that \(A(\mu)\) and \(\gamma (\mu)\) are sufficiently close to constant in a sense which is made precise by the norms defining the scale. The existence result does not require this restriction on the coefficients.
scientific article; zbMATH DE number 850595