Exponential convergence in the Wasserstein metric $W_1$ for one dimensional diffusions

In this paper, we find some general and efficient sufficient conditions for the exponential convergence $W_{1,d}(P_t(x,\cdot), P_t(y,\cdot) )\le Ke^{-\delta t}d(x,y)$ for the semigroup $(P_t)$ of one-dimensional diffusion. Moreover some sharp estimates of the involved constants $K\ge 1, \delta>0$ are provided. Those general results are illustrated by a series of examples.

Let C ∞ 0 (I) be the space of infinitely differentiable real functions f on I with compact support and C ∞ 0,N (I) be the space of all functions f in C ∞ 0 (I) such that f ′ | ∂I = 0 (i.e. satisfying the Neumann boundary condition). Let (X t ) be the diffusion process on the interval I generated by L with initial value x.
We will assume : (H2) The diffusion process is non-explosive or equivalently ( [14]) : Throughout this paper we assume that (H1)-(H4) are satisfied. In this case, µ(dx) := m ′ (x) m(I) dx is the unique invariant probability measure of the diffusion (X t ). Let (P t ) be the transition semigroup of (X t ), L 2 be the generator of (P t ) on L 2 (I, µ) with domain D(L 2 ), which is an extension of (L, C ∞ 0,N (I)). For any two probability measures µ, ν on I, the Wasserstein distance between µ and ν w.r.t. a given metric d(x, y) on I is defined by d(x, y)π(dx, dy) where π runs over all couplings of µ and ν, i.e. all probability measures π on I 2 with the first and second marginal distributions µ and ν, respectively. We say that (P t ) is exponential convergent in W 1,dρ if there exist some constants K ≥ 1 and δ > 0 such that By Kantorovich duality relation, this is equivalent to In this paper, we are interested in the exponential convergence of (P t ) in W 1,dρ . When ρ ∈ L 1 (I, µ), consider the Banach space C Lip(ρ),0 := {f : I → R : f Lip(ρ) < +∞ and µ(f ) = 0} equipped with the Lipschitzian norm · Lip(ρ) . We have immediately Proposition 1.1. If (P t ) is exponential convergent in W 1,dρ , i.e. (1.1) holds and ρ ∈ L 2 (I, µ), then the Poisson operator (−L 2 ) −1 = +∞ 0 P t dt : C Lip(ρ),0 → C Lip(ρ),0 is bounded, and Recall the following result in Djellout and the third named author [9].
Theorem 1.2. Assume (H1)-(H4) and let ρ be a function on I such that ρ ∈ C 1 (I) ∩ L 2 (I, µ), ρ ′ > 0 everywhere, the Poisson operator is well defined and bounded if and only if (iff in short) In that case, its norm is given by . In other words a necessary condition for the exponential convergence of (P t ) in W 1,dρ is c W (ρ) < +∞. The objective of this paper is to show that this necessary condition becomes sufficient in a quite general situation and some sharp estimates of the involved constants δ, K could be obtained.
1.3. Some comments on the literature. For the one-dimensional diffusions, the Poincaré inequality (equivalent to the exponential convergence in L 2 (µ)) and the log-Sobolev inequality (equivalent to the exponential convergence in entropy) can be characterized by means of the generalized Hardy inequality ( [1,4]). For the characterization of Latala-Oleszkiewicz inequality ( [15]) between Poincaré and log-Sobolev, see Barthe and Robertho [1]. See [7] for the characterization of the Sobolev inequality.
For the transport and isoperimetric inequalities, see Djellout and Wu [9] for sharp estimates of constants.
On the other hand the exponential convergence in the L p -Wasserstein metric is also a very active subject of study. For the early study on this question, the reader is referred to [2,5,6,4]. Renesse and Sturm [18] showed that for L = ∆ − ∇V · ∇ on a Riemannian manifold, if the exact exponential convergence (1.1) in W 1,d (with K = 1, d being the Riemannian metric) holds, then the Bakry-Emery's curvature must be bounded from below by δ. If one allowed K > 1 in (1.1), Eberle [11] found sharp sufficient conditions for high dimensional interacting diffusions, by means of reflected coupling. The reader is referred to this last paper for an overview of literature.
The general idea of proving the exponential convergence in W 1,d for a high dimenioanl diffusion is to use the reflection coupling (X t , Y t ) and then to compare d(X t , Y t ) with some one-dimensional diffusion. But curiously a general study on the exponential convergence in W 1,d of one-dimensional diffusion is absent : that is the objective of our study.
1.4. Organization. Our paper is organized as follows. In the next section, we state the main result and present several corollaries. In Section 3, we provide several examples to illustrate our theorem. In Section 4, we prove the main result in the compact case. The proof of the general case is given in Section 5.

Main Result
For a function f on I, the sup-norm of f is defined by f ∞ = sup x∈I |f (x)|. Recall that the conditions (H1)-(H4) are always assumed.
when α → 0+, our estimate on the exponential convergence rate δ is sharp.
If moreover λ 1 > 0 is associated with an eigenfunction ρ, which can be chosen to be increasing ( [2]), then one verifies easily that u(x) = 1 and we get from Part (b) above that

Corollaries and examples.
We present now a corollary for illustrating the extra condition (C) in Part (a) of Theorem 2.1.
Proof. For x > L, by the mean value theorem of Cauchy, there exists some ξ ∈ (x, +∞) such that For x < −L, by the similar proof, we also have u(x) ≤ C 1 . Then c W (ρ) < +∞. Hence by Part (b) of Theorem 2.1, the exponential convergence (2.4) in the metric W 1,dρ holds. Furthermore, if moreover b ′ ≤ M for some non-negative constant M, applying Theorem 2.1(a) for ϕ = 1, we get the exponential convergence in W 1,dρ . In the case where (2.7) is satisfied, we have u(x) ∈ [1/C 2 , C 2 ] for all x ∈ R, for some constant C 2 ≥ C 1 by the mean value theorem of Cauchy. Then Thus we get the exponential convergence in W 1,dρ -metric : A curious point in Corollary 2.4 above is that our sufficient condition above for the exponential convergence in W 1 -metric associated with the Euclidean distance, depends very few upon the volatility coefficient a(x) (except the conditions (H1)-(H4)).
We give an example of Corollary 2.4 : It is easy to see that the hypotheses (H1)-(H4) are all satisfied, and µ( Since and when L > |µ(ρ)| large enough, by Corollary 2.4, (P t ) generated by L is exponential convergent in W 1,dρ .
We present now another example to illustrate the extra condition (C).

Main idea.
We explain now the main idea in Theorem 2.1. The crucial point is that in the actual one-dimensional case, we would have formally the following commutation relation t is the semigroup generated by L D g = (ag ′ + bg) ′ . Then by the Kantorovitch duality relation, the exponential convergence of (P t ) in W 1,dρ is equivalent to that of P D t to 0 in the Banach space b V B of all Borel-measurable functions g such that the norm g V := sup x∈I . An easy sufficient condition to this last exponential convergence is L D U ≤ −δU for some positive constant C and some function U such that ρ ′ ≤ U ≤ Cρ ′ .
To see the meaning of the necessary condition c W (ρ) < +∞, notice that u is a particular solution of L D u = −ρ ′ , u should be bounded by Cρ ′ if P D t converges exponentially to 0 in the norm · ρ ′ . Moreover our extra condition (C) says simply that there is some function ϕ ≥ ερ ′ in the domain of L D in b V B.
However the formal approach above is very difficult to be realized in the general case. It can be realized rigorously in the compact case (I = [x 0 , y 0 ]) when a, b are quite regular : see §4. The general case can be treated by approximation, as the involved constants δ, K have explicit expressions.

Several Examples
is the intrinsic metric d X of the diffusion (X t ). In this section, we present several examples and study their exponential convergence in the W 1 -metric associated with the intrinsic distance d X .
where V (x) = C 1 |x| r , C 1 and r are positive constants, and a ∈ C 1 (R) which is bounded i.e. 1 C 2 ≤ a(x) ≤ C 2 for some constant C 2 ≥ 1. Then µ(dx) = 1 Za(x) e B(x) dx (Z being the normalization constant), For this example it is well known that the spectral gap exists (i.e. λ 1 > 0) iff r ≥ 1. About the exponential convergence in W 1 associated to the Euclidean metric, we have the following result : Corollary 3.3. In the above Example 3.2, for ρ(x) = x, (P t ) is exponential convergent in W 1,dρ iff r ≥ 2.
Proof. At first we can check easily that all assumptions (H1)-(H4) hold. For the necessity, we only need to prove that c W (ρ) = +∞ when r < 2. By the L'Hospital criterion, By the definition of c W (ρ), we have c W (ρ) = +∞.
x . This process arises as diffusion limit of discrete space branching process. For ρ = √ 2x, d ρ = d X , we see that and by calculus we have by Part (b) of Theorem 2.1 the exponential convergence (2.2) w.r.t. d X holds with δ = 1 and K = 1.
Moreover from (3.1) we see that the increasing function ρ − µ(ρ) is an eigenfunction of −L, and its associated eigenvalue 1 must be the spectral gap λ 1 . This example shows again that Theorem 2.1 is sharp.

Compact case
In this section, we prove the main result Theorem 2.1 in the compact case i.e. I = [x 0 , y 0 ] is a bounded closed interval of R.
in other words v(t, x) satisfies the Dirichlet boundary condition. For all g ∈ C ∞ D [x 0 , y 0 ] := {g ∈ C ∞ [x 0 , y 0 ] : g| {x 0 ,y 0 } = 0}, we define L D as follows : where (1) (X t ) is the diffusion generated by L with the Neumann boundary condition : where (B t ) is the Brownian motion, L x 0 t (resp. L y 0 t ) is the local time of (X t ) at x 0 (resp. y 0 ) ; (2) τ ∂I = inf{t ≥ 0 : X t ∈ {x 0 , y 0 }} is the first hitting time to the boundary ; (3) for t < τ ∂I , satisfying X D t = X t for t < τ ∂I and X D t = X τ ∂I for t ≥ τ ∂I , is the killed process at ∂I. First, by the Markov property of (X t ), it is easy to see that (P D t ) is a C 0 -semigroup on C D [x 0 , y 0 ]. Then we prove (4.2). When 0 ≤ t < τ ∂I , for any g ∈ C ∞ D [x 0 , y 0 ], by Itô formula, d(D t g(X t )) = D t dg(X t ) + g(X t )dD t + d D, g(X) t = D t a(X t )g ′′ (X t )dt + b(X t )g ′ (X t )dt is a martingale. And for t ≥ τ ∂I , D t g(X D t ) = 0. Then for x ∈ (x 0 , y 0 ), Then by the uniqueness,P D t g(x) = P D t g(x) .
Recall that for an everywhere positive function V , the V -norm of f is defined by Lemma 4.2. If there exists some positive constant δ and a C 2 -function V : [x 0 , y 0 ] → R + such that C 1 ≤ V ρ ′ ≤ C 2 (C 1 , C 2 are positive constants) and (aV ′ + bV ) ′ ≤ −δV on (x 0 , y 0 ), (4.5) The only delicate point is that the test function V does not necessarily belong to the domain of definition of the generator L D , in fact V (x 0 ), V (y 0 ) may be different of 0. At first, we prove (4.6). For this purpose it is enough to show we only need to prove that (Y t ) is a supermartingale. For t < τ ∂I , by Itô formula and (4.5), where M t is a local martingale up to τ ∂I . Then (Y t ) is a supermartingale by Fatou's lemma. Thus by Lemma 4.1, Now we prove (4.7), which is equivalent to At first for f ∈ C ∞ N [x 0 , y 0 ], we have by (4.6), Now for every f ∈ C ∞ [x 0 , y 0 ] and n ∈ N + , let f n = f (x 0 ) + x x 0 ψ n (y)f ′ (y)dy where ψ n (x) = 1 for x ∈ [x 0 + 1/n, y 0 − 1/n] and ψ n is C ∞ -smooth, valued in [0, 1], with compact support contained in (x 0 , y 0 ). For each n ∈ N + , as f n satisfies the Neumann boundary condition and f n Lip(ρ) ≤ f Lip(ρ) , we have where (4.8) follows by letting n → ∞.

Now we turn to :
Proof of Theorem 2.1 in the compact and C ∞ -case. Part (a). Since By Lemma 4.2, we only need to find a C 2 -function V : [x 0 , y 0 ] → R + such that C 1 ≤ V ρ ′ ≤ C 2 for some positive constants C 1 , C 2 and (aV ′ + bV ) ′ ≤ −δV on (x 0 , y 0 ). (4.9) Consider the following equation It is explicitly solvable and the unique solution satisfying the Dirichlet boundary condition is It is easy to see u(x) > 0 for all x ∈ (x 0 , y 0 ). Since (aϕ ′ + bϕ) ′ ≤ Mρ ′ , for any constant α ∈ (0, 1 M ), we can choose V = αϕ + u.
First notice that sup x∈I then by Lemma 4.2, we get the desired result. Part (b). It is enough to show (1.2) holds with δ = 1 c W (ρ) , K = 1. Consider the unique solution u of (4.10) with Dirichlet boundary condition. Since c W (ρ) < +∞, then u.
Sinceρ ′ = u, by Lemma 4.2 with V = u, we have First reduction : (c is a fixed constant in [x 0 , y 0 ]), we see that u ε (x) → u(x) uniformly over [x 0 , y 0 ] as ε → 0, then c W (ρ, ε) defined in (1.3) associated with (a ε , b ε ) converges to c W (ρ) associated with (a, b) as ε → 0. Moreover the condition in (2.1) is satisfied for (a ε , b ε ) with some constant M ε > M and M ε → M as ε → 0. By the result of the C ∞ -case in Part (a) of Theorem 2.1, the semigroup (P ε t ) generated by L ε = a ε d 2 Mε ). Obviously, δ ε → δ and K ε → K as ε → 0. Now we only need to show for all f ∈ C ∞ [x 0 , y 0 ], But the process (X ε t ) generated by L ε with the refelection Neumann boundary condition converges in law to (X t ) (well-known in the theory of SDE, [13]). Then the convergence above holds.
with the same initial value x. Then for any f ∈ C[x 0 , y 0 ], we have by Girsanov's formula, Let ξ 0 ∈ I such that ρ(ξ 0 ) = µ(ρ). Fix some N and δ > 0 so that [ξ 0 − δ, ξ 0 + δ] ⊂ I N . Notice that for n ≥ N, if x ∈ I n and x ≥ ξ 0 + δ, u n (x) ≤ s ′ (x) (Notice that lim inf n→∞ c W (ρ, n) ≥ c W (ρ), lim inf n→∞ K n ≥ K hold always.) Now for the exponential convergence in Part (a), it remains to show that for any f ∈ C ∞ 0 (x 0 , y 0 ) and x ∈ (x 0 , y 0 ), lim Denote the first hitting time of (X t ) to the boundary of I n by τ ∂In = inf{t ≥ 0 : X t ∈ {x n , y n }}, we have X (n) t = X t , ∀t ∈ [0, τ ∂In ).
By the non-explosion assumption (H2), we have for any t ≥ 0 and x ∈ I fixed, Part (b). For any f ∈ C ∞ 0 (x 0 , y 0 ) and x < y in (x 0 , y 0 ), the support of f and {x, y} are contained in I N for some N ≥ 1 large enough. Then if N ≤ n → ∞, recalling thatρ ′ (x) = u(x), sup x∈(xn,yn) for u n → u uniformly over I N . Since lim where the desired result follows by Kantorovitch duality characterization.