Accelerating planar Ornstein-Uhlenbeck diffusion with suitable drift

The principal aim of this paper is to construct an explicit sequence of weighted divergence free vector fields which accelerates the rate of convergence of planar Ornstein-Uhlenbeck diffusion to its equilibrium state. The rate of convergence is expressed in terms of the spectral gap of the diffusion generator. We construct an explicit sequence of vector fields which pushes the spectral gap to infinity. The acceleration of the diffusion results from the strong oscillation of the flow lines generated by the vector field.

1. Introduction. The main objective of this paper is to study the influence of some oscillatory measure preserving drift on the rate of convergence of some diffusion process to its equilibrium state. Let us informally introduce our topic in some general setting. We assume V to be a smooth function on R d such that ϕ(x) = 1 Z e −V (x) is a probability density with respect to the Lebesgue measure on R d , where Z is a suitable normalizing constant. The resulting probability measure on R d will be denoted by γ ϕ . We are interested in the diffusions X u which are solution of the stochastic differential equation where B is a Brownian motion and u is a vector field satisfying div(ue −V ) = 0. Together with some suitable growth condition on the function V this last property ensures that the process X u has γ ϕ as its unique equilibrium measure. Note that X u is reversible if and only if u = 0.
A classical result on the asymptotic behavior of the diffusion X u is given by the following L 2 -inequality which holds for all t > 0 (see [12]): In this inequality K u is a suitable positive constant and is the spectral gap of the infinitesimal generator of the diffusion X u which acts on smooth functions with compact support as It follows from this that the spectral gap is an important indicator for understanding the speed at which the diffusion generated by A u converges to its equilibrium distribution. We want to understand which specific feature of the vector field forces this spectral gap to become large. This is interesting to know, since the spectral gap is often used to measure the performance of Markov chain Monte Carlo methods. Those algorithms use processes of the type X to approximate integrals with respect to γ ϕ through iterated simulations (see [9]). It was shown by Hwang, Hwang-Ma and Sheu in [11] and [12] that the optimal algorithms are to be found in the class of non-reversible diffusions, where the vector field u is non-zero and the generator A u is not self-adjoint. In other words, for u = 0, the process X u (t) converges to the equilibrium faster than X 0 (t) as t goes to infinity. In fact they showed that ρ u ≥ ρ 0 for all u satisfying div(ue −V ) = 0. Moreover, they proved that the inequality is strict whenever the flow generated from the vector field u does not preserve a subspace of the eigenspace associated to the first non-zero eigenvalue of the operator A 0 . An explicit asymptotic expression for the spectral gap resulting from the multiplication of the vector field u with a large constant is found in Franke, Hwang, Pai and Sheu [6]. In the same context Constantin, Kiselev, Ryzhik and Zlatos proved in [4] that the operator norm of the resulting semi group converges toward zero if and only if the operator u · ∇ dosn't have eigen-functions in the Sobolev space H 1 . This implies the convergence of the spectral gap to infinity. For example vector fields u generating weak mixing flows satisfy the above condition. However, the existence of such flows is not always guaranteed and in some cases, like for example on two dimensional spaces, those flows do not exist. Moreover, the existence of mixing flows is usually proved by non-constructive methods and thus those flows are not explicit, which makes their application in numerical simulations difficult. This leads to the question, whether one can find a sequence of explicit vector fields u n such that ρ un diverges to infinity. The problem was studied on d dimensional torus by Hwang and Pai in [14] and by Franke and Yaakoubi for various compact two-dimensional Riemannian manifolds in [7] and [8]. In those papers the authors prove that the spectral gap denoted by ρ u can be arbitrarily large with a suitable choice of u. In this paper we will study the case of the two dimensional Ornstein-Uhlenbeck diffusion (i.e.: V (x) = x 2 ), which is of some importance in probability theory, where it is used to modelize mean reverting processes. It grew from the PhD thesis of N. Yaakoubi which was directed by B. Franke and M. Damak (see [15]) and completes some of the arguments given in this thesis.
The proof is based on the construction of strongly oscillating periodic flow-lines in the plane. Those flow-lines are used to define a weighted divergence free vector field u in R 2 . This vector field u can be modified in such a way that the differential operator u · ∇ has no H 1 -eigenfunctions associated to non-zero eigenvalues. Only the the zero eigenvalue has H 1 -eigenfunctions, which then are essentially constant along the flow-lines. In this context the result from [6] yields an explicit asymptotic expression for the spectral gap ρ cu as c → ∞ in terms of some Rayleigh quotient over functions from the H 1 -eigenspaces of the operator u · ∇. Note that by the special construction of u, only functions which are invariant along the flow-lines have to be considered in this quotient. Increasing the oscillation of the flow-lines implies longer level sets for those functions which results in large gradients and thus a large Rayleigh quotient. To prove this we use the isoperimetric structure of the flow-invariant sets in R 2 to introduce some suitable comparison measure and prove some Faber-Krahn type rearrangement inequality for flow-invariant H 1 -functions.
This general scheme was used in [7] and [8] for compact two dimensional spaces. The Ornstein Uhlenbeck process being a process on an unbounded domain, a considerable amount of effort has to be spent to understand the asymptotic behavior of the isoperimetric structure of the flow-invariant sets as the oscillation increases.
It is also worthy to note that the explicit construction of the flow lines has the advantage to bypass the difficult task of integrating the vector field which is necessary to run the Monte Carlo simulations.
An interesting question is whether the method presented in this manuscript can be generalized to higher dimensional Ornstein Uhlenbeck diffusion. While most arguments can be carried over to higher dimensions, the behavior of the isoperimetric problem as oscillation grows is quite difficult to estimate in higher dimensions. be the density of the normalized 2-dimensional Gaussian measure γ ϕ = ϕγ where γ denotes the Lebesgue measure on R 2 . The weighted arc length measure with respect to the gaussian measure γ ϕ will be denoted ϕ . This means that ϕ = ϕ , where is the euclidean arc length measure on R 2 . The spectra of the operators we consider are complex valued and the eigenfunctions will be complex valued functions as well. We therefore introduce the following Hilbert space of mean-zero complex-valued functions: with scalar product and norm f gdγ ϕ and ||f || 2 = f, f .
We define the following Sobolev space of mean-zero functions: with scalar product and norm ∇f ∇gdγ ϕ and ||f || 2 1 = f, f 1 .

Note that
is the generator of the ordinary two dimensional Ornstein-Uhlenbeck process, which has γ ϕ as its invariant probability measure. This process is reversible and converges in law toward γ ϕ .
For f ∈ C ∞ (R 2 ), we define the operator where u is a vector field satisfying div(uϕ) = 0.
1.2. Main result. The notion of the spectral gap is often used to quantify the acceleration of the diffusion introduced above. The following is our main result: There exists a sequence of vector fields (u n ) n∈N with the property div(u n ϕ) = 0 such that one has lim n→∞ ρ un = ∞.

2.
Construction of the flow lines. In this section we will construct some strongly oscillating flow in the plane. Each flow line is obtained from a rotation around the origine with some simultaneous oscillation in radial direction. Later in Proposition 1 the isoperimetric behavior of those flow lines will be studied as the number of oscillations grows to infinity. Consider the C 1 -function η: For all r > 0, we note the graph of the function θ → h n (r, θ) by: for all θ ∈ [0, 2π[ and r > 0, it follows that the graphs Γ Lemma 2.1. As r → 0, one has that ϕ (Γ Proof. At first, note that, for r < 1 3 , the curve Γ (n) r oscillates n times between two concentric circle of radius (r − r 2 2 ) and of radius (r + r 2 2 ). Also, note that the gaussian density ϕ decays with r. Thus, Thanks to those considerations, there exist two constants β 1 < 0 and β 2 < 0 such that: After dividing the 3 sides of those inequalities by the arc length of ∂B r (0), we obtain .
We obtain the first assertion when taking the limit r → 0. The second assertion follows from the first by integration over r.
determines the weight of an infinitesimal arc length element of the curve Γ . We note that the arc length element at θ of Γ (n) r is increasing in ] 1 2 , r max ] and decreasing in [r max , ∞[. However, r max depends on θ and n. The biggest possible value of r max is equal to 6π 5π−1 and the smallest equals 6π 7π+1 . As a conclusion, r → ϕ (Γ Moreover, since we have that Lemma 2.2. There exist four positive constants C 1 , C 2 , C 3 and C 4 such that for all r > 0, one has that Proof. Since for all r > 0, the infinitesimal arc length element of the curve Γ (n) r in polar coordinates equals Proof. The proof will address the following three cases separately. The first case is 1 3 ≤ r c ≤ 6π 5π−1 , the second is 6π 5π−1 < r c and the last is r c < 1 3 . 1) For the first case, i.e.: 1 3 ≤ r c ≤ 6π 5π−1 , we use Lemma 2.2 to see that inf 0<c< 1 In exactly the same way, we can show that: 2) When r c > 6π 5π−1 , we use inequality (1) to see that Using the fact that r → ϕ (Γ (n) r ) is decreasing for r > 6π 5π−1 , we obtain that It follows then that From Lemma 2.2, it follows that Replacing R c = 6π 5π−1 −2 ln(1 − c) in the previous expression yields that However, the function is bounded away from zero on the interval ]0, 1 2 ]. From this follows that

MONDHER DAMAK, BRICE FRANKE AND NEJIB YAAKOUBI
Note that for a n : .
We will show that both expressions in the minimum go to infinity. First, note that for r > 1 2 we have It thus follows that uniformly in n ∈ N Since a n → ∞ this implies that For 6π 5π−1 < r ≤ a n ≤ we have that It then follows from Lemma 2.2 and the choice of a n that inf 6π 5π−1 <r≤an Thus we have proved that 3) For the case when r c < 1 3 , Lemma 2.2 yields By the above consideration, the second expression in the minimum is bigger than This shows that inf 0<c< 1 On the other hand we have from Lemma 2.2 and Remark 6 that As a conclusion, we see that rc ) c goes to infinity as n → ∞. Furthermore, we also have that goes to infinity as n → ∞. This proves our proposition.
3. Construction of the vector field. In this section we construct a sequence of vector fields u n which will have the property to generate the sequence of flow lines Γ (n) r which was introduced in the previous section. In Proposition 4 we will see that u n can be chosen in such a way that the operator u n · ∇ has no H 1 -eigenfunctions except the ones associated to the zero eigenvalue. In order to achieve this, we first use the implicit value theorem to construct a C 1 -function α * n which is constant on the orbits Γ (n) r . From this function, we obtain a vector field u 0 n by applying some symplectic gradient ∇ ⊥ to the function α * n and multiplying with ϕ −1 . This vector field is tangential to the orbit Γ (n) r and generates a flow which follows the curves Γ (n) r and also satisfies our requirement div(ϕu 0 n ) = 0. However, we have to modify this vector field by a multiplication with a suitable function S to ensure that the resulting vector field u n leads to a well defined anti-symetric operator u n · ∇ with domain containing H 1 and without H 1 eigenfunctions to non-zero eigenvalues.
For a C 1 -function F we introduce its gradient and its symplectic gradient Obviously we have the two following properties: -(∇F ) t (∇ ⊥ F ) = 0; -div(∇ ⊥ F) = 0. In the following we will use polar coordiantes (r, θ) ∈ R + × [0, 2π[ to parametrize R 2 . For n ∈ N we use the implicit function theorem to define a C 1 -function α * n : It follows from this construction, that the function α n is constant on the sets Γ (n) r for all r > 0. We denote by α n the representation of α * n with respect to cartesian coordinates in the plane; i.e.: for all (r, θ) ∈ R + × [0, 2π[ we have α * n (r, ϕ) = α n (r cos ϕ, r sin ϕ). Note that we put α n to be zero in the origine to obtain a well defined differentiable function on the plane. From this family of functions, we define a sequence of vector fields u 0 n on R 2 by: By construction u 0 n is a vector field which satisfies div(ϕu 0 n ) = 0. Moreover, it generates a flow following the curves Γ (n) r which are also the level sets of the function α n . Since the vector field u 0 n is not bounded it might not lead to a well defined operator u 0 n · ∇ on H 1 . This point will be adressed in Proposition 2. Remark 7. In order to obain some bounds on the return times of the flow generated by the vector field u 0 n we introduce polar coordinates in the plane; i.e.: (x 1 , x 2 ) = (r cos θ, r sin θ) for r > 0 and θ ∈ [0, 2π[. For a given C 1 -function f : R 2 \{0} → R we can represent its gradient ∇f with respect to the local basis ∂ r = cos θ sin θ and ∂ θ = − sin θ cos θ of the tangent space of R 2 \{0} in the point (r, θ). The gradient then can be expressed as It then also follows that Lemma 3.1. For every n ∈ N there exist constants M 1 , M 2 , M 3 , M 4 , M 5 , M > 0, 0 < L 1 < L 2 < 1, 0 < K 1 < K 2 < 1 and 0 < K < 1 such that the radial and angular components of the vector field u 0 n satisfy the following bounds : -for r > 1 2 one has M 1 e K1r 2 ≤ u 0 n , ∂ θ ≤ M 2 e K2r 2 ; -for r < 1 3 holds M 3 e L1r 2 ≤ u 0 n , ∂ θ ≤ M 4 e L2r 2 ; -for r > 1 2 one has u 0 n , ∂ r ≤ M 5 ne Kr 2 ; -for r < 1 3 one has that u 0 n , ∂ r < M .
Proof. We can partly express α * n explicitly as follows : for r > 1 2 ; no expression for 1 3 ≤ r ≤ 1 2 ; √ 1+2r sin(nθ)−1 sin(nθ) for r < 1 3 . From this one can compute the partial derivatives This implies that the angular component of ∇ ⊥ α * n satisfies the following bounds when r > 1 2 : From this follows the following bounds for the angular component of the vector field u 0 n along the level set Γ This implies : This is the bound given in the lemma for r > 1 2 . We now turn to the case r < 1 3 . The above computation of ∂ r α * n yields the following bounds for the angular component of the vector field ∇ ⊥ α * n when r < 1 3 : This implies the following bounds for the angular component for u 0 n along the orbit Γ (n) r when r < 1 3 : Using the fact r < 1 3 one obtains : This leads to the bounds given in the statement of the lemma. When computing the derivative of α * n with respect to θ one obtains for r < 1 3 .
Note, that in the case r < 1 3 an asymptotic expansion up to second order of the square root in the second term of the above expression yields ∂ θ α * n (r, θ) = n 2 r 2 when sin(nθ) → 0. From those computations we obtain the following bounds for the radial part of the vector field ∇ ⊥ α * n when r > 1 2 : This yields the following bounds for the radial part of the vector field u 0 n along the orbit Γ From this follows : This implies the bounds given in the statement.
We finally have to analyze u 0 n , ∂ r for r < 1 3 . In this case we have on Γ Remark 8. Note that for a function S ∈ ker u 0 n · ∇ ∩ C 1 (R 2 ) we have that S is constant along the level sets of the function α n . Further, we have div ϕSu 0 n = u 0 n · ∇S + Sdiv(u 0 n ) = 0.
Moreover, the vector field u n = Su 0 n has the same trajectories as u 0 n . Those trajectories are the level sets of the function α n .
Proposition 2. For all n ∈ N, there exists a function S ∈ ker u 0 n · ∇ ∩ C 1 (R 2 ) such that the two following properties hold : -the flow (φ t ) t∈R generated by the vector field u n := Su 0 n satisfies for all r > 0 that φ e r 2 (x) = x for all x ∈ Γ (n) r ; -the vector field u n satisfies div(ϕu n ) = 0 and is bounded; -there exists a constant D > 0 such that |u n (x)| ≤ D|x| 2 for all x from a neighbourhood of the origine.
Proof. For a function S ∈ ker u 0 n · ∇ ∩ C 1 (R 2 ) we introduce the flows (φ t ) t∈R and (φ 0 t ) t∈R generated by u n = Su 0 n resp. u 0 n through the equationsφ 0 t = u 0 n (φ 0 t ) resp. φ t = u n (φ t ). This implies for all t ∈ R and x ∈ R 2 the relation Proposition 4. The sequence of divergence free vector fields u n constructed in Lemma 2 satisfies that H 1 λ = {0} for all λ = 0. Proof. Let τ (x) be the return time to the point x of the flow (φ t ) t∈R generated by the vector field u n . It then follows from Proposition 3 that For the choice t = 1 we obtain from this

If we put
then it follows from the coarea formula applied to the function C 1 -function α n , which was used to introduce the vector field u 0 n , that This means that for almost all r > 0 the equation must be true for ϕ -almost all x ∈ Γ 4. Comparison arguments. The proof of our result is based on some lower bound for the gradients of functions which are invariant with respect to the flow we constructed in the previous sections. We will use the information gained in Proposition 1 to construct a suitable comparison measure having some large Cheeger constant. Further, we will prove some Faber-Krahn type inequality relating flow invariant functions on the plane with some symmetrized functions on this comparison space.
4.1. Some suitable symmetric comparison measure. In this part, we will construct some comparison probability density ϕ n on R 2 based on the function c → ϕ (Γ (n) rc ).
Proof. The following proof is motivated from the analogous construction of comparison manifolds in [2]. See also [5] for an alternative construction. Let ψ n (r) be the value of the density ϕ n on the circle ∂B r (0). In order to satisfy the stated properties, the function ψ n has to satisfy 2π Rn(c) 0 rψ n (r)dr = c.
Differentiating both sides of this equality with respect to c yields d dc R n (c) 2πR n (c)ψ n (R n (c)) = 1.
Due to the requirements on the function ϕ stated in the proposition the function ψ has to satisfy 2πR n (c)ψ n (R n (c)) = ϕ (Γ . Thus we obtain the following explicit formula for the function R(c) R n (c) = dσ.
As a conclusion, the suitable density is given by .
This ends the proof.

The rearrangement method and Faber-Krahn type inequalities. Let
A be a measurable set in R 2 . Its symmetric rearrangement A * n is the open ball centered around zero in R 2 satisfying γ ϕn (A * n ) = γ ϕ (A). For all non negative measurable function f , the following representation holds almost every where with respect to γ f (x) = ∞ 0 χ y|f (y)>t (x)dt.
We define its rearrangement function by: Here we used the notation χ A for the indicator function over the set A which is one for x ∈ A and zero for x / ∈ A. Then f * n is lower semi-continuous (since its level sets are open), and is uniquely determined by the distribution function γ ϕ {x; f (x) ≥ s} of f . 5.1. Proof of Theorem 1.1. The main argument follows the arguments given in [7]. Let γ ϕ n be the comparison probability density associated to the vector-field u n . For a given K > 0, Proposition 8 and Proposition 7 tell us that there exists N ∈ N sufficiently large such that By Proposition 6 an Remark 10 one has that: We now can use Remark 12 and Proposition 4 to see that lim a↑∞ ρ au ≥ 2K.
This implies ρ aun > K for some suitable choice of a > 0.