Resolvent Estimates for Fleming-Viot Operators and Uniqueness of Solutions to Related Martingale Problems

We define a two-parameter scale of Banach spaces contained in the contininuous functions on the space of probability measures on a compact metric space X, and show that the resolvent of the Fleming-Viot process is a bounded operator in the scale. We use this result to prove uniqueness of solutions to the martingale problem for Fleming-Viot operators whose mutation operator (Aμ, A) and sampling rate γ(μ, x, y) may depend on the measure variable μ ∈ M1(X), provided they are sufficiently close to constant in a sense made precise by the norms defining the scale. We prove an existence result with no such restriction on the coefficients.