A Reversibility Problem for Fleming-Viot Processes

Fleming-Viot processes incorporating mutation and selection are considered. It is well-known that if the mutation factor is of uniform type, the process has a reversible stationary distribution, and it has been an open problem to characterize the class of the processes that have reversible stationary distributions. This paper proves that if a Fleming-Viot process has a reversible stationary distribution, then the associated mutation operator is of uniform type.


Problem and Result
Fleming-Viot processes form a class of probability measure-valued diffusion processes, which are derived as a continuum limit from Markov chain models in population genetics. The processes have attracted not only probabilists but also mathematical population genetists since they are very reasonable models to analyze the geneological structure. (cf. [1], [3]). In particular, if the mutation factor is of uniform type, namely the distribution of mutants is independent of their parent's genotype, then the Fleming-Viot processes have reversible stationary distributions, that makes the geneological analysis extremely tractable. However it has been an open problem whether there exists another class of mutation operators such that the associated Fleming-Viot processes have reversible stationary distributions. In the present paper we shall solve this problem. Our result is that if a Fleming-Viot process has a reversible stationary distribution, then the mutation operator is of uniform type.
Let us begin with description of the Fleming-Viot processes. Let E be a locally compact separable space. We denote by B(E) the set of all bounded Borel measurable functions on E, and by C ∞ (E) the Banach space of bounded continuous functions vanishing at infinity if E is non-compact, which is equipped with the supremum norm · ∞ . We denote by C 0 (E) the set of continuous functions with compact support, and by C + 0 (E) the set of nonnegative functions in C 0 (E). Let M 1 (E) be the space of Borel probability measures on E endowed with the topology of weak convergence. For µ ∈ M 1 (E), we denote by µ ⊗n ∈ M 1 (E n ) the n-fold product of µ. We also use the notation µ(f ) := E f dµ for f ∈ B(E) and µ ∈ M 1 (E).
Let (A, D(A)) be the generator of a conservative Markovian Feller semigroup T t acting on C ∞ (E), which governs a mutational evolution, and let σ = (σ(x, y)) be a symmetric bounded Borel measurable function on E × E, which is interpreted as a selective density. For given (A, D(A)) and σ let us consider the following operator (L, D(L)) : where δφ(µ)/δµ(x) = lim r→0+ r −1 {φ(µ + rδ x ) − φ(µ)}, and we take D(L) to be the set of all φ ∈ C(M 1 (E)) of the form Let Ω be the space of continuous paths from [0, ∞) to M 1 (E) with the coordinate process denoted by {X t : t ≥ 0}. We furnish Ω with the compact uniform topology. Let (F , F t ) t≥0 be the natural σ-algebras on Ω generated by {X t : t ≥ 0}. It is known that for every µ ∈ M 1 (E) there is a unique probability measure Q µ on Ω such that for every φ ∈ D(L) Lφ(X s )ds is a Q µ -martingale starting at 0. Then Ω, (F t ) t≥0 , Q µ , X t defines a diffusion process in M 1 (E), which is called a Fleming-Viot process incorporating mutation and selection. Hereafter we simply write (X t , Q µ ) for the Fleming-Viot process Ω, It is convenient to describe the process (X t ) by the following stochastic equation associated with a martingale measure introduced in [8]: where σ · f ⊗ 1(x, y) = σ(x, y)f (x), and M (dsdx) is a martingale measure such that is a continuous martingale with quadratic variation process Then it holds that for every f ∈ C ∞ (E) Now we give two examples.
Then the associated Fleming-Viot process is a limit process of the step-wise mutation model of Ohta-Kimura [7]. In this case there exists no stationary distribution, instead the process exhibits a wandering phenomenon (c.f. [2]). If we start the step-wise mutation model with periodic boundary condition, then the limit process is a Fleming-Viot process associated with A is the Laplacian on T d and σ = 0. In this case there is a unique stationary distribution, but it has not been known whether the stationary distribution is reversible or not.

Example 1.2 Let A be the following jump-type generator;
Af

is a constant and P (x, dy) is a stochastic kernel on E × E. This class of Fleming-Viot processes are investigated by Ethier and Kurtz ([3]) for sample path properties and ergodic behaviors. In particular, if
the mutation operator is called of uniform type, which is the case having been well-studied from geneological view point. An advantage of the uniform mutation is that the associated Fleming-Viot process has a reversible stationary distribution which is identified with Poisson-Dirichlet distribution (c.f. [3]).
It is known that in the non-selective case; σ = 0, the Fleming-Viot process is ergodic if and only if the mutation semigroup is ergodic. In the selective case, i.e. σ = 0, it might be expected that the same conclusion holds, but it has not yet been proved completely.
In the present paper we consider a reversibility problem for Fleming-Viot processes incorporating mutation and selection, that is to characterize the mutation operator (A, D(A)) with which the associated Fleming-Viot process has a reversible stationary distribution. Now we state our main theorem.
If the Fleming-Viot process (X t , Q µ ) with the mutation operator (A, D(A)) and the selective density σ has a reversible stationary distribution Q, then (A, D(A)) is of the form (1.4) with We remark that the above irreducibility assumption seems to be natural. Because, if there exists a unique stationary distribution, it can be reduced to this case by restricting the basic space E to a smaller one. On the other hand, if there are more than two stationary distributions, the basic space E splits into several disjoint subsets, and on each subset the mutation operator will be of uniform type.
Our method of the proof is based on moment calculations. Assuming that σ = 0, we first show by second moment calculations that the barycenter of the reversible stationary distribution Q is a reversible distribution for the mutation semigroup T t , with which a regular Dirichlet space is associated. Then combining the Beurling-Deny formula for the Dirichlet form with third moment calculations we obtain the theorem in the non-selective case, which is discussed in the next section. Section 3 is devoted to the proof in the selective case, which is carried out by reducing it to the non-selective case by making use of a transformation of the probability law Q µ .

The non-selective case
In this section we assume σ = 0. Let Q be a stationary distribution of the Fleming-Viot process (X t , Q µ ). For Q we define the moment measures m n on the product space Using the independence of X 0 and M (dsdx) and (1.2) to compute m 2 (f ⊗T r g) = E(X t (f )X t (T r g)) we have By ( Finally if Q is a reversible distribution of (X t , Q µ ), denoting by Q the associated stationary Markovian probability measure on Ω with initial distribution Q, it holds Hence m is T t -reversible.
In the sequel of this section, we assume Q is a reversible stationary distribution of (X t , Q µ ), hence by Lemma 2.1 m is T t -reversible. Let L 2 (E; m) be the Hilbert space of real-valued m-square-integrable functions on E with the inner product (f, g) m := m(f g). Then T t can be extended as a symmetric contraction semigroup acting on L 2 (E; m). We denote its generator by (Ā, D(Ā)), which is a self-adjoint and non-positive definite operator on L 2 (E; m). Let

Lemma 2.2 (D[E], E) is a regular Dirichlet space, that is,
Proof. Denote by G λ (λ > 0) the resolvent operators of (A, D(A)). The Feller property of For a regular Dirichlet space, it is known that the Dirichlet form has the following expression (cf. [5]).

6)
where E c is the diffusion part which satisfies the local property; and J is the jumping measure, which is a symmetric Radon measure on the product space E × E off the diagonal ∆.

Lemma 2.4
Proof. From (2.4) with r = 0 it follows that Letting t → ∞ and using T t -symmetry of m we get which yields the desired conclusion.

Lemma 2.5 For f ∈ D(A) and g, h ∈ C ∞ (E)
Proof. We first assume f, g, h ∈ D(A). Let (X t , Q) be the reversible stationary Markov process with initial distribution Q. Note that by Itô's formula Now use the reversibility of (X t , Q) to get from which together with (2.9) it follows that Then dividing the equality by t > 0 and letting t → 0 we get Interchanging g and h, A combination of the last two equations gives the desired result for f, g, h ∈ D(A). The extension to g, h ∈ C ∞ (E) is trivial. and Proof. The former fact is found in [6], Lemma 1.4.2. If f ∈ D(A) and g, h ∈ C 0 (E), Lemmas 2.4 and 2.5 imply Moreover, if g ∈ D(A), inserting (I − A)g in place of g in the above equation we get Then the desired equality follows from the symmetry between f and g. It is trivial to extend

Lemma 2.7
The jumping measure J(dx, dy) of (2.6) is everywhere dense in E × E \ ∆, that is, Here we have used the symmetry of J for the last equality. Similarly E(g, f G 1/2 h) = 0 holds. Accordingly by Lemma 2.6 we have (2.11) Note that the irreducibility of T t implies that m is everywhere dense in E and G λ f (x) > 0 (x ∈ E) for f ∈ C + 0 (E) with f = 0, so that from (2.11) and the Feller property of G λ it follows that which yields a contradiction, because G 1/2 f (x) > 0, G 1/2 g(x) > 0 for every x ∈ E, and f and g have disjoint supports. Thus the proof is completed.
Proof of Theorem 1.1 in the non-selective case. We claim that for f, g, h ∈ C 0 (E) with mutually disjoint supports, it holds Noting that by Lemma 2.6 we have which yields (2.13) due to the symmetry of J. Next, noting that J is everywhere dense in E × E \ ∆ by Lemma 2.7, (2.13) and the Feller property of G λ imply that G 1/2 h(x) is constant outside the support of h, so that for every compact subset K, G 1/2 (x, K) is constant in x / ∈ K. Accordingly for every compact set K there exists a constant c(K) such that 14) It is easy to see that c(K) can be extended to a Borel measure on E such that (2.14) holds for every Borel set. This observation implies that there exists a constant a ≥ 0 such that for every x ∈ E G 1/2 (x, ·) = aδ x (·) + c(·).
Here note that 0 < a < 2 and a + c(E) = 2, which follow from the continuity, the irreducibility and the conservativity of T t . Inserting this to the resolvent equation we obtain so that Thus (A, D(A)) is of the type of (1.4). Moreover, it is easy to see that c agrees with m up to a constant multiplication. Therefore the proof of Theorem 1.1 in the non-selective case is completed.

The selective case
The Fleming-Viot process incorporating selection is obtained from the non-selective process by making use of Girsanov transformation. And if the selective density is of the form σ = f ⊗ g + g ⊗ f with f, g ∈ D(A), it is straightforward to show that the reversible stationary distribution of the selective model inherits the one of the non-selective model, so that the non-selective result is applicable. Furthermore, for a bounded measurable selective density σ it is possible to prove Theorem 1.1 by taking a suitable approximation.
In this section to emphasize the selective density σ, we denote by (X t , Q σ µ ) the associated Fleming-Viot process. Let M (dsdx) be the martingale measure defined by (1.1) and (1.2). We use the notation B sym (E 2 ) for the set of symmetric and bounded measurable functions on E 2 For ϕ ∈ B sym (E 2 ) we define the martingales which is a positive martingale with mean one and Let Q 0 µ be the probability measure on Ω such that Q 0 µ (F ) = Q σ µ {N (2) t (σ)1 F } for F ∈ F t and t ≥ 0.

Lemma 3.1 For f ∈ D(A)
that converges to ϕ with the L 2 (E 2 ; m 2 )-norm. Therefore by Lemma 3.2, (3.7) holds for every ϕ ∈ B sym (E 2 ), Now we can complete the proof of Theorem 1.1 in the selective case. Suppose that there exists a reversible stationary distribution Q σ of the Fleming-Viot process associated with (A, D(A)) and σ. Then by Lemma 3.3 Q 0 defined by (3.5) is a reversible stationary distribution of the non-selective Fleming-Viot process (X t , Q 0 ). Therefore, by the result of the non-selective case in the previous section, we conclude that (A, D(A)) is of the form (4), which complete the proof of Theorem 1.1 in the selective case.