A property of Petrov's diffusion

Petrov constructed a diffusion process in the Kingman simplex whose unique stationary distribution is the two-parameter Poisson-Dirichlet distribution of Pitman and Yor. We show that the subset of the simplex comprising vectors whose coordinates do not sum to 1 acts like an entrance boundary for the diffusion.

1 Introduction Petrov (2009) constructed an infinite-dimensional diffusion process in the compact Kingman simplex ∇ ∞ := x = (x 1 , x 2 , . . .) : x 1 ≥ x 2 ≥ · · · ≥ 0, ∞ i=1 x i ≤ 1 depending on two parameters, α and θ with 0 ≤ α < 1 and θ > −α. Its generator is 1 acting on the subalgebra of C(∇ ∞ ) generated by the sequence of functions ϕ 1 , ϕ 2 , ϕ 3 , . . . defined by More precisely, for ϕ ∈ D(A), Aϕ is evaluated on the dense subset and extended to ∇ ∞ by continuity. For example, The unique stationary distribution of Petrov's diffusion is Pitman and Yor's (1997) two-parameter generalization of the Poisson-Dirichlet distribution, which we denote by PD α,θ and regard as a Borel probability measure on ∇ ∞ that is concentrated on ∇ ∞ . The special case α = 0 (and hence θ > 0) is the unlabeled infinitely-manyneutral-alleles diffusion model of population genetics; see Ethier and Kurtz (1981). Its unique stationary distribution is of course the (one-parameter) Poisson-Dirichlet distribution, PD 0,θ , of Kingman (1975). Petrov's diffusion was not motivated by population genetics; however, see De Blasi, Ruggiero, and Spanò (2014). Feng, Sun, Wang, and Xu (2011) derived an explicit formula for the transition density p(t, x, y) of Petrov's diffusion with respect to PD α,θ , which had earlier been done in the special case α = 0 by Ethier (1992). Recently, Zhou (2013) found an elegant simplification of this formula. However, all we will need here is the fact that the Feller semigroup {T (t)} on C(∇ ∞ ) generated by the closure of A can be expressed as for all f ∈ C(∇ ∞ ). In particular, letting {X t , t ≥ 0} denote Petrov's diffusion, it follows that P x (X t ∈ ∇ ∞ ) = 1 for every x ∈ ∇ ∞ and t > 0, where the subscript x denotes the initial state. A question left open by Petrov (2009) holds.
In the special case α = 0, (2) was proved by Ethier and Kurtz (1981). It was later realized that this result has a simple interpretation. The unlabeled infinitely-many-neutral-alleles diffusion model has a more informative labeled version, namely the Fleming-Viot process in P(S) (the set of Borel probability measures on the compact set S with the topology of weak convergence) with mutation operator where ν 0 ∈ P(S) is nonatomic. The unlabeled model is a transformation of the labeled one. The transformation takes µ ∈ P(S) to x ∈ ∇ ∞ , where x is the vector of descending order statistics of the sizes of the atoms of µ. Then (2) is equivalent to the assertion that the Fleming-Viot process, regardless of its initial state, instantly becomes purely atomic and remains so forever. If α > 0, there is no such interpretation of (2) because whether there is a Fleming-Viot process corresponding to Petrov's diffusion is unknown. In fact, this is an open problem that was posed by Feng (2010, p. 112).
Notice that another way to express (2) is to say that ∇ ∞ − ∇ ∞ acts like an entrance boundary for the diffusion. Technically, this is not quite accurate because ∇ ∞ has no interior, so its boundary is all of ∇ ∞ . But what we mean is simply that, starting at a state x ∈ ∇ ∞ − ∇ ∞ , the process instantly enters ∇ ∞ and never exits.
A weaker version of (2) was obtained by Feng and Sun (2010) using the theory of Dirichlet forms. They showed that ∇∞ P y (X t ∈ ∇ ∞ for all t > 0) PD α,θ (dy) = 1, which is to say that the stationary version of Petrov's diffusion has ∇ ∞ as its natural state space.
In the next section we will use (1) and (3) to prove (2).

Entrance boundary
Let us begin by showing why Ethier and Kurtz's (1981) proof of (2) when α = 0 fails when α > 0. First, we extend the domain of A. Let H := {h ∈ C 2 [0, 1] : h(0) = h ′ (0) = 0}, and for h ∈ H define ψ h ∈ C(∇ ∞ ) by Let A + be A acting on the subalgebra of C(∇ ∞ ) generated by {1} ∪ {ψ h : h ∈ H }. Again, for ϕ ∈ D(A + ), A + ϕ is evaluated on ∇ ∞ and extended to ∇ ∞ by continuity. It is easy to see that A + ⊂ A. We also notice that ϕ m ∈ D(A + ) for all real m ≥ 2 (not just integers), where This leads to the conclusion that, for every real m ≥ 2, is a continuous square-integrable martingale with increasing process A difficulty occurs when trying to extend this last conclusion to 1 < m < 2. Fix such an m and define h ε ∈ H for ε > 0 by h ε (u) := (u + ε) m − ε m − mε m−1 u. Then A + ψ hε converges pointwise as ε → 0+ but not boundedly because of two awkward terms: Both sums are monotonically increasing as ε decreases to 0, but their coefficients have opposite signs, so the monotone convergence theorem does not apply if α > 0, and we cannot show that t 0 ϕ m−1 (X s ) ds ∈ L 2 for all t > 0. We are now ready for the proof of (2).