Remarks on spectral gaps on the Riemannian path space

In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces.


Introduction
Let M be a complete smooth Riemannian manifold of dimension d, and Z a C 1 -vector field on M . We will be concerned with the diffusion operator where ∆ M is the Beltrami-Laplace operator on M . Let ∇ be the Levi-Civita connection and Ric the Ricci curvature tensor on M . We will denote Ric Z = Ric + ∇Z.
It is well-known that the lower bound K 2 of the symmetrized Ric s Z , that is, where Ric * Z denotes the transposed matrix of Ric Z , gives the lower bound of constants in the logarithmic Sobolev inequality with respect to the heat measure ρ t (x, dy), associated to L; more precisely, M u 2 (y) log u 2 (y) ||u|| 2 ρt ρ t (x, dy) ≤ 2 1 − e −K2t K 2 M |∇u(y)| 2 ρ t (x, dy), t > 0, (1.2) where ||u|| 2 ρt = M u 2 (y) ρ t (x, dy). Given now a finite number of times 0 < t 1 < . . . < t N , consider the probability measure ν t1,...,t N on M N defined by M N f dν t1,...,t N = M N f (y 1 , . . . , y N ) p t1 (x, dy 1 )p t2−t1 (y 1 , dy 2 ) · · · p t N −t N −1 (y N −1 , dy N ) (1.3) where f is a bounded measurable function on M N . Then with respect to the correlated metric | · | C on T M N (see definition (1.10) below), the logarithmic Sobolev inequality still holds for ν t1,...,t N , that is, there is a constant C N > 0 such that . (1.4) It was proved in [20,6] that under the hypothesis sup x∈M |||Ric Z (x)||| < +∞, (1.5) where ||| · ||| denotes the norm of matrices, the constant C N in (1.4) can be bounded, that is sup N ≥1 C N < +∞. (1.6) A natural question is whether (1.6) still holds only under Condition (1.1)? In a recent work [21], A. Naber proved that if the uniform bound (1.6) holds, then the Ricci curvature of the base manifold has an upper bound. It is well-known that Inequality (1.2) implies the lower bound (1.1), therefore Condition (1.6) implies (1.5). The main purpose in [21] is to get informations on Ric Z from the analysis of the Riemannian path space. Let's explain briefly the context.
where H Z denotes the horizontal lift of Z to O(M ), that is, π (u) · H Z (u) = Z(π(u)). It is well-known that under Condition (1.1), the life-time τ x of the SDE (1.7) is infinite. Let γ t (w) = π(u t (w)). (1.8) Then {γ t (w); t ≥ 0} is a diffusion process on M , having L as generator. The probability measure ν t1,...,t N considered in (1.3) is the law of w → (γ t1 (w), . . . , γ t N (w)) on M N . Now consider the following path space The law µ x,T on W T x (M ) of w → γ · (w) is called the Wiener measure on W T x (M ). The integration by parts formula for µ x,T was first estalished in the seminal book [5], then developed in [16,10]; the Cameron-Martin type quasi-invariance of µ x,T was first proved by B. Driver [9], completed and simplified in [18,19,13]. We consider the Cameron-Martin space where the dot denotes the derivative with respect to the time t. Let F : W T x (M ) → R be a cylindrical function in the form: F (γ) = f (γ(t 1 ), · · · , γ(t N )) for some N ≥ 1, 0 ≤ t 1 < t 2 < · · · < t N ≤ 1, and f ∈ C 1 b (M N ). The usual gradient of F in Malliavin calculus is defined by where ∂ j is the gradient with respect to the j-th component. The correlated norm of ∇f where t j ∧ t k denotes the minimum between t j and t k . Notice that the norm |∇f | C is random. The generator L x T associated to the Dirichlet form is called the Ornstein-Uhlenbeck operator. The powerful tool of Γ 2 of Bakry and Emery [3] is not applicable to L x T , the reason for this is the geometry of W T x (M ) inherted from H is quite complicated, the associated "Ricci tensor" being a divergent object (see [7,8,12]). When the base manifold M is compact, the existence of the spectral gap for L x T has been proved in [14]. The logarithmic Sobolev inequality for D τ F defined in (1.9) has been established in [2], as well as in [20] or [6] where the constant was estimated using the bound of Ricci curvature tensor of the base manifold M . The method used in [14] is the martingale representation, which takes advantage the Itô filtration; this method has been developed in [12] to deal with the problem of vanishing of harmonic forms on W T x (M ). The purpose in [21] is to proceed in the opposite direction, to get the bound for Ricci curvature tensor of the base manifold M from the analysis of the path space W T x (M ).
The organization of the paper is as follows. In section 2, we will recall briefly basic objets in Analysis of W T x (M ). On the path space W T x (M ), there exist two type of gradients: the usual one is more related to the geometry of the base manifold, while the damped one is easy to be handled. In section 3, we will make estimation of the spectral gap of L x T as explicitly as possible in function of lower bound K 2 and upper bound K 1 of Ric. In section 4, we will study the behaviour of the spectral gap SG(L x T ) as T → 0.
Roughly speaking, we will get the following result: under the following condition (4.1).

Framework of the Riemannian path space
We shall keep the notations of Section 1, and throughout this section, u t (w) denotes always the solution of (1.7) and γ t (w) the path defined in (1.8). For any h ∈ H, we introduce first the usual gradient on the path space W T x (M ), which gives Formula (1.9) when the functional F is a cylindrical function. To this end, let where Ω u is the equivariant representation of the curvature tensor on M . Let ric Z be the equivariant representation of Ric Z , that is, . Then according to [16], we define where D τ F was given in (1.9). Consider the following resolvent equation where Q * τ,s is the transpose matrix of Q τ,s . The damped gradientD τ F on the path space W T x (M ) plays a basic role in analysis of W T x (M ). Let (v t ) t≥0 be a R d -valued process, adapted to the Itô filtration F t generated by {w(s); s ≤ t} such that E( The good feature of the damped gradient is that it admits a nice martingale representation where E Ft denotes the conditional expectation with respect to F t . The following logarithmic Sobolev inequality holds ( [11,17]):

Precise lower bound on the spectral gap
The inconvenient of Inequality (2.9) is that the geometric information of the base manifold M is completely hidden. Now we use the usual gradient D t F to make involving the geometry of M . By (2.9), the matter is now to estimate Then we have the relation: Proof. From (2.5) and (2.8), we havẽ Thus, In the following we will estimate the term of I 2 and I 3 . Under the lower bound in (3.1),

ds.
Then Combining all the above inequalities, we get dt.
Next, then we compute the term J 1 (s) and J 2 (s). By direct computation, we have Adding J 1 (s) to J 1 (s) implying that The proof is completed.
Notice that as K 2 → 0, by expression (3.2), Now we study the variation of the function t → Λ(t, T ). It is quite interesting to remark that its monotonicity is dependent of the sign of K 2 . Proof. Taking the derivative of t → Λ(t, T ) gives In addition, we have From the second equality in the above, we observe that Λ(T, T ) ≥ Λ(0, T ). Moreover, and We see that Therefore there exists at most one t such that Λ (t, T ) = 0. For the case where K 2 < 0, if there exists t 0 ∈ (0, T ) such that Λ(t 0 , T ) < 0. Then by (3.5) and (3.7), the equation Λ (t, T ) = 0 has at least two solutions, it is impossible. Therefore for K 2 < 0, Λ (t, T ) ≥ 0. For K 2 > 0, we suppose t 0 such that Λ (t 0 , T ) = 0. Let β = K1 K2 , then by (3.8) The proof is completed.
for any cylindrical function F on W T x (M ).
It is well-konwn that the above logarithmic Sobolev inequality implies that the spectral gap of L x T , denoted by SG(L x T ), has the following lower bound . (3.14) Proof. Using the elementary inequality: A + B ≥ 2 √ AB to the last two terms in (3.10) yields (3.13). Inequality (3.14) is obvious.
It is quite interesting to remark that Proposition 3.6. Let ψ(T, K 1 , K 2 ) be the right hand side of (3.13) when K 2 > 0 and the right hand side of (3.14) for K 2 < 0, then ψ(T, K 1 , K 2 ) → 1 + Proof. It is easy to see that the right hand side of (3.14) tends to 1 + K1T 2 + K 2 1 T 2 8 as K 2 → 0. For the right hand side of (3.13), we first remark that ECP 22 (2017), paper 19.
Then (3.15) follows from the right hand side of (3.13).
Corollary 3.7. Assume (3.1) holds. In this section, we consider the case where Z = 0. Then Condition (3.1) can be readed as and SDE (1.7) is reduced to The path γ t (w) = π(u t (w)) is called Brownian motion path on M . Let ρ(x, y) be the Riemannian distance. By [22, p. 199], there is ε > 0 such that Assume that the curvature tensor satisfies the following growth condition where L Hi denotes the Lie derivative with respect to H i .
Let h ∈ H; then by (2.3), we have (see also [15])   Then by Fubini theorem, the term q(t, h) has the expression According to (4.6), the gradient D τ F T has the following expression: (4.7) We have where u 0 is the initial frame of (4.2).