PR ] 7 A ug 2 01 9 Functional Inequalities for Weighted Gamma Distribution on the Space of Finite Measures ∗

Let M be the space of finite measures on a Locally compact Polish space, and let G be the Gamma distribution on M with intensity measure ν ∈ M. Let ∇ext be the extrinsic derivative with tangent bundle TM = ∪η∈ML(η), and let A : TM → TM be measurable such that Aη is a positive definite linear operator on L 2(η) for every η ∈ M. Moreover, for a measurable function V on M, let dG V = eV dG . We investigate the Poincaré, weak Poincaré and super Poincaré inequalities for the Dirichlet form EA ,V (F,G) := ∫ M 〈Aη∇F (η),∇G(η)〉L2(η) dG V (η), which characterize various properties of the associated Markov semigroup. The main results are extended to the space of finite signed measures. AMS subject Classification: 60J57, 60J45.


Introduction
Let M be the class of finite measures on a locally compact Polish space E, which is again a Polish space under the weak topology. Recall that a sequence of finite measures  Since M is locally compact, the Borel σ-algebra B(M) induced by the weak topology coincides with that induced by the vague topology. Let ν ∈ M with ν(E) > 0. The Gamma distribution G with intensity measure ν is the unique probability measure on M such that for any finitely many disjoint measurable subsets {A 1 , · · · , A n } of E, {η(A i )} 1≤i≤n are independent Gamma random variables with shape parameters {ν(A i )} 1≤i≤n and scale parameter 1; that is, Consider the weighted Gamma distribution G V (dη) := e V (η) G(dη), where V is a measurable function on M. We will investigate functional inequalities for the Dirichlet form induced by G V (dη) and a positive definite linear map A on the tangent space of the extrinsic derivative. See [7] and references therein for Dirichlet forms induced by both extrinsic and intrinsic derivatives, where the intensity measure ν is the Lebesgue measure on R d such that the Gamma distribution G is concentrated on the space of infinite Radon measures on R d . In this paper, we only consider finite intensity measure ν. exists for all x ∈ E, such that ∇ ext F (η) := ∇ ext F (η)(·) L 2 (η) < ∞.
If F is extrinsically differentiable at all η ∈ M, we denote F ∈ D(∇ ext ) and call it extrinsically differentiable on M.
We write A = 1 if A η is the identity map on L 2 (η) for every η ∈ M. According to Theorem 3.1 below, the assumption (H) implies that (E A,V , FC ∞ 0 ) is closable in L 2 (G V ) and the closure (E A,V , D(E A,V )) is a symmetric Dirichlet form. If moreover ) be the associated generator. We aim to investigate functional inequalities for the Dirichlet form E A,V and the spectral gap of the generator L A,V .
We first consider the Poincaré inequality When gap(L A,V ) = 0, the following weak Poincaré inequality was introduced in [13]: We also consider the super Poincaré inequality where β : (0, ∞) → (0, ∞) is a decreasing function. The existence of super Poincaré inequality is equivalent to the uniform integrability of P A,V t for t > 0, and, when P A,V t has an asymptotic density with respect to G V , it is also equivalent to the compactness of holds for some constant C > 0. It is well known (see [2,6]) that (1.12) is equivalent to the hypercontractivity of P A,V t : as well as the exponential convergence in entropy: See [21,22,23] or [24] for more results on the super Poincaré inequalities, for instance, estimates on the semigroup P A,V t and higher order eigenvalues of the generator L A,V using the function β in (1.11).
The remainder of the paper is organised as follows. In section 2, we state the main results of the paper, and illustrate these results by a typical example with specific interactions. In Section 3, we establish the integration by parts formula which implies the closability of (E A,V , FC ∞ 0 ). Then the main results are proved in Section 4, and extended in Section 5 to the space M s of finite signed measures.

Main results and an example
We first consider E 1,0 in L 2 (G) whose restriction on M 1 := {µ ∈ M : µ(E) = 1} gives rise to the Dirichlet form of the Fleming-Viot process. Corresponding to results of [16,17] for the Fleming-Viot process, we have the following result. See also [12,26] for functional inequalities of different type measure-valued processes.  (1) gap(L 1,0 ) = 1, i.e. λ = 1 is the largest constant such that (1.9) holds for V = 0 and A = 1.
(2) If supp ν contains infinitely many points, then E 1,0 does not satisfy the super Poincaré inequality.
(3) There exists a constant c 0 > 0 such that when supp ν is a finite set, the log-Sobolev To extend this result to E A,V , we will adopt a split argument by making perturbations to E 1,0 on bounded sets and estimating the principal eigenvalue of L A,V outside. To this end, we take According to (3.1) below, we set A η φ, φ L 2 (η) , r > 0.  Obviously, σ k is non-increasing in k and might be infinite. We will see in Theorem 2.2(1) that under certain conditions σ k < ∞ implies the validity of Poincaré inequality.
We have the following extension of Theorem 2.1 to E A,V . When supp ν is finite the model reduces to finite-dimensional diffusions, for which one may derive super Poincaré inequalities by making perturbations to (2.1). As the present study mainly focusses on the infinite-dimensional model, we exclude this case in the following result.  (1) If lim k→∞ σ k < ∞ (equivalently, σ k < ∞ for all k > 0), then (2) If supp ν contains infinitely many points, then E A,V does not satisfy the super Poincaré inequality.
(3) The weak Poincaré inequality (1.10) holds for The following result shows that the condition in Theorem 2.2(1) is sharp when A η and V (η) depend only on ρ(η). Corollary 2.3. Assume (H) and (1.8). Let V (η) = v(ρ(η)) and A η = a(ρ(η))1 for large ρ(η) and some a, v ∈ C 1 ([0, ∞)) with a(r) > 0 for r ≥ 0. Then  . We will see in the proof that the rate function α is derived by comparing E A,V with E 1,0 on bounded sets B N , N > 0. However, when these two Dirichlet forms are far away, this α is less sharp. As a principle, to derive a sharper weak Poincaré inequality, one should compare E A,V with a closer Dirichlet form which satisfies the Poincaré inequality. In this spirit, we present below an alternative result on the weak Poincaré inequality. To state the result, we introduce the class H as follows.
To conclude this section, we present below a simple example to illustrate the main results. For simplicity, we only consider A η = 1. But by a simple comparison argument, the assertions apply also to A η with A η φ, φ L 2 (η) ≥ c φ 2 L 2 (η) for some constant c > 0 and all η ∈ M, φ ∈ L 2 (η). Example 2.6. Consider the following potential V 0 with interactions given by ψ i ∈ B b (E × E), i = 1, 2, 3: Assume that one of the following conditions hold: a probability measure on M, and the following assertions hold: Then there exists a constant c > 0 such that the weak Poincaré inequality (1.10) holds for Proof. Obviously, the assumptions in Theorem 2.2 hold for V and A η = 1. By definition it is easy to see that .
It is easy to see that Combining this with (2.11), we may find constants c 3 , Taking this into account and applying Theorem 2.5 for Therefore, by taking ε = 1 ∧ r 1 2(p−ν(E)) , we prove (1.10) for the desired α(r). To prove this result, we introduce the divergence operator corresponding to ∇ ext . To this end, we formulate the Gamma distribution G by using the Poisson measure πν with intensityν(dx, ds) := s −1 e −s ν(dx)ds onÊ := E × (0, ∞). Recall that πν is the unique probability measure on the configuration space Below we explain that the same argument works to the present setting. (3.5) where B + (E) is the class of nonnegative measurable functions on E. This was given by [18, (7)] when ν is atomless. In general, we decompose

The Dirichlet form
are independent under G, the distribution of η(h · 1 E0 ) under G coincides with that under G 0 (the Gamma distribution with intensity measure ν 0 ), and the distribution of η({x i }) under G coincides with the one-dimensional Gamma distribution γ ci with shape parameter c i . So, applying (3.5) for ν 0 replacing ν due to [18, (7)], and using the Laplace transform for Gamma distributions on R + , we derive e −ci log(1+h(xi)) = e −ν(log(1+h)) .
On the other hand, the Laplace transform for πν (see for instance [1]) is To establish the integration by parts formula for ∇ ext φ F , we introduce the divergence operator div ext as follows.
where η(·) stands for the integral with respect to η as in (1.1), then we write φ ∈ D(div ext ) and denote When φ(η, x) = φ(x) does not depend on η, the following integration by parts formula follows from [9,Theorem 14]. We include below a complete proof for the η-dependent φ.
Proof of Theorem 3.1. We first prove (3.2), which implies the closability of (E A,V , FC ∞ 0 ) and that the closure is a symmetric Dirichlet form in L 2 (G V ), see [4]. By the definition of E A,V and Lemma 3.3, for any F, G ∈ FC ∞ 0 we have Next, assume that (1.8) holds. It remains to find a sequence To this end, we consider ρ n := n −1 + ρ 2 , n ≥ 1. By (2.3), we have ρ n ∈ D(∇ ext ) with It is easy to see that G V (|F n − 1| 2 ) → 0 as n → ∞ and due to (1.8),

Proofs of the main results
In this section, we prove Theorems 2.1, 2.2, 2.5 and Corollary 2.3.

Proof of Theorem 2.1 and a local Poincaré inequality
Proof of Theorem 2.1. The invalidity of the super Poincaré inequality will be included in the proof of Theorem 2.2(3) for a more general case. So, we only prove (1) and (3).
(a) We first prove gap(L 1,0 ) = 1, i.e. λ = 1 is the optimal constant for the Poincaré This Poincaré inequality reduces to where according to (1.2), By the additive property of the Poincaré inequality, it suffices to prove that for every This follows from the fact that the generator of the Dirichlet form which has spectral gap 1 with the first eigenfunction u i (r) = r − ν(A i ).
(b) Let supp ν = {x 1 , · · · , x n }, we have δ = min{ν({x i }) : 1 ≤ i ≤ n} > 0. It suffices to find a universal constant c 0 > 0 such that (2.1) holds for Letting µ n and µ i be as in (4.2) for A i = {x i }, (2.1) for this F becomes By the additive property of the log-Sobolev inequality, this follows from the following Lemma 4.1.
So, the desired inequality (4.3) with c 0 = 4 follows since Therefore, for µ a,b2 (f 2 ) = 1 with µ a,b2 (f ) = 0 we have In general, for a non-zero Combining this with (4.6) and using the Poincaré inequality (4.15) below, we arrive at

Proofs of Theorem 2.2 and Corollary 2.3
Proof of Theorem 2.2. We will make a standard split argument by using the local Poincaré inequality (4.13) and the principal eigenvalue of L A,V outside B N . To estimate the principal eigenvalue, we recall Hardy's criterion for the first mixed eigenvalue. Consider the following differential operator on [0, ∞): For any k > 0 and n ≥ 1, let λ k,n be the first mixed eigenvalue of L on [k, k + n] with Dirichlet boundary condition at k and Neumann boundary condition at k + n. Define  Below we prove assertions (1)-(3) respectively.
(1) By (4.13) and a standard perturbation argument, we have If σ k < ∞ for some k > 0, it suffices to prove the Poincaré inequality where according to (4.17), (4.20) Let F ∈ FC 2 0 such that supp F ⊂ B N1 for some constant N 1 > k. For any N ≥ k, let EJP 25 (2020), paper 19. Then F N = 0 for ρ ≤ N and F N = F for ψ(ρ) ≥ ψ(N ) + 1. For n > N 1 , let u n ≥ 0 be the first mixed eigenfunction of L on [k, k + n] with Dirichlet boundary condition at k and Neumann boundary condition at k + n, such that u n (k) = u n (k + n) = 0, u n (r) > 0 for r ∈ (k, k + n), Lu n = −λ k,n u n ≤ 0.
Combining this with the definition of L we obtain So, To apply the integration by parts formula, we approximate u n as follows. Since u n (k) = u n (k + n) = 0, we may construct a sequence {u n, u n,m (r) = 0 for r ≤ k, u n,m (r) = 0 for r ≥ k + n, sup m≥1 sup r≥k |u n,m (r)| + |u n,m (r)| < ∞.
Since F N = 0 for ρ ≤ N , (4.21) implies that for any k < N , On the other hand, since A η is positive definite due to (H), for any u ∈ C 2 ([0, ∞)) with u(r) > 0 for r ≥ N , we have Combining this with (4.22) and the definition of F N , we obtain Multiplying by λ −1 k,n and letting n → ∞ leads to EJP 25 (2020), paper 19. Letting s = sup{k ∈ Z : k ≤ s} be the integer part of a real number s, we have Combining this with (4.18) and noting that G Then the proof is finished by (4.20).
(2) Assume that supp ν is an infinite set. To disprove the super Poincaré inequality, it suffices to construct a sequence {F n } ⊂ D(E A,V ) such that G V (F 2 n ) > 0 and , n ≥ 1, r > 0.
Combining this with (4.27) and letting n → ∞, we obtain 1 ≤ rC for all r > 0 which is impossible.
On the other hand, let σ k = ∞ for all k > 0. We have λ k := lim n→∞ λ k,n = 0, k > 0, where λ k,n is given in the proof of Theorem 2.2. Let u k,n be the corresponding first mixed eigenfunction of L on [k, k + n] with u k,n (r) > 0 in (k, k + n], and let such that L is symmetric in L 2 ([k, k + n], Θ v ) under the mixed boundary conditions. Then k+n k u k,n (r) 2 Θ v (dr) = 1 λ k,n k+n k ra(r)|u k,n (r)| 2 Θ v (dr).
So, for any r > 0 and N > 0 such that  η n → η if η n (f ) := E f dη n → η(f ) holds for any f ∈ C b (E), since the latter on M s might be not metrizable, see [19].
To extend the Dirichlet form ( s with a potential V on M s , we introduce below the measure G V s , the extrinsic derivative and the operator A respectively. In [18], an analogue to the Lebesgue measure was introduced on M s by using the convolution of two weighted Gamma distributions. In the same spirit, we extend the measure G to G s on M s as follows: To ensure that τ (η + ) and τ (η − ) are disjoint such that η = η + − η − is the Hahn decomposition of η, we will assume that ν is atomless. In this case, τ (η + ) ∩ τ (η − ) = ∅ for G × G-a.e. (η + , η − ). Next, we define the extrinsic derivative operator (∇ ext , D(∇ ext )) as in Definition 1.1 for M s replacing M: Let F s C ∞ 0 be the class of cylindrical functions of type F (η) := f (η + (A 1 ), · · · , η + (A n ), η − (A 1 ), · · · , η − (A n )), n ≥ 1, f ∈ C ∞ 0 (R 2n ), (5.3) where {A i } 1≤i≤n is a measurable partition of E, and η = η + − η − is the Hahn decomposition. Let It is easy to see that such a function F is extrinsically differentiable with Since for any η ∈ M s , A η+εδx = A η holds for small ε > 0 and all x ∈ E, ∇ ext F (η)(x) is again extrinsically differentiable in η with Finally, for any η ∈ M s , let A η be a positive definite bounded linear operator on L 2 (|η|), where |η| := η + + η − is the total variation of η. Consider the pre-Dirichlet form To ensure the closability of this bilinear form, we assume s is a probability measure. Moreover, for any Obviously, this assumption is satisfied if A η = F (η)1 for some positive bounded extrinsically differentiable function F such that G V s is a probability measure with

Integration by parts formula
) is a symmetric Dirichlet form with 1 ∈ D(E s A,V ) and E s A,V (1, 1) = 0. To prove this result, we introduce the divergence operator associated with ∇ ext . Definition 5.1. A measurable function φ on M s × E is said in the domain D(div ext s ), if for any x ∈ E we have φ(·, x) ∈ D(∇ ext ) and In this case, the divergence operator is given by We have the following integration by parts formula for the directional derivative Lemma 5.2. Let φ ∈ D(div ext s ). Then EJP 25 (2020), paper 19.
Next, to prove that 1 ∈ D(E s A,V ) with E s A,V (1, 1) = 0, we take {f n } n≥1 ⊂ C ∞ 0 (R) such that f n (s) = 1 for |s| ≤ n, 0 ≤ f n ≤ 1 and f n ∞ ≤ 1. Let F n (η) := f n (η(E)), n ≥ 1. Then F n ∈ FC ∞ 0 . By (H') we have G V s (|F n − 1| 2 ) → 0 as n → ∞, and lim sup Therefore, 1 ∈ D(E 2 A,V ) and E s A,V (1, 1) = 0. (5.14) Proof. By taking F (η) depending only on η + , it is easy to see that a Poincaré inequality for E s 1,0 implies the same inequality for E 1,0 . So, the optimality of (5.12), and the invalidity of the super Poincaré inequality when supp ν is infinite, follow from Theorem 2.1. It remains to prove the inequalities (5.12), (5.13) and (5.14). According to the additivity property of the Poincaré and log-Sobolev inequalities, these inequalities follow from the corresponding ones of E 1,0 . For simplicity, below we only prove the first inequality.
Moreover, one may also extend Corollaries 2.3-2.4 and Theorem 2.5. We omit the details to save space.
Theorem 5.5. In addition to (H'), assume that A η s(η, ·) is extrinsically differentiable in η such that (5.16) holds. Moreover, assume that a s andā s in (5.18) are such that a −1 s (r) is locally bounded in r ≥ 0 and (5.19) holds.
(2) If supp ν contains infinitely many points, then E s A,V does not satisfy the super Poincaré inequality.