A Two-parameter Family of Infinite-dimensional Diffusions in the Kingman Simplex

In the topology of coordinatewise convergence ∇∞ is a compact, metrizable and separable space. Denote by C(∇∞) the algebra of real continuous functions on ∇∞ with pointwise operations and the supremum norm. In C(∇∞) there is a distinguished dense subspace F := R [q1, q2, . . . ] generated (as a commutative unital algebra) by algebraically independent continuous functions qk(x) := ∑∞ i=1 x k+1 i , k = 1, 2, . . . , x ∈ ∇∞. For each 0 6 α < 1 and θ > −α we define an operator A : F → F which can be written as a formal differential operator of second order with respect to the generators of the algebra F :