Logarithmic Sobolev and Poincaré Inequalities for the Circular Cauchy Distribution * 0.1 Circular Cauchy Distribution

In this paper, we consider the circular Cauchy distribution µx on the unit circle S with index 0 ≤ |x| < 1 and we study the spectral gap and the optimal logarithmic Sobolev constant for µx, denoted respectively by λ1(µx) and CLS(µx). We prove that 1 1+|x| ≤ λ1(µx) ≤ 1 while CLS(µx) behaves like log(1 + 1 1−|x|) as |x| → 1. Let S be the unit circle in R 2 with the Riemannian structure induced by R 2 and write ∇ S for the spherical gradient. For any x ∈ R 2 with |x| < 1, we consider the probability measure µ x on S which has density h(x, y) = 1 2π 1 − |x| 2 |y − x| 2 , y ∈ S with respect to the arc length µ on the unit circle S. The form of the density h makes µ x known as circular Cauchy distribution or wrapped Cauchy distribution (see [10, 11]). On the one hand, it enjoys the following property: if f is an integrable function on S, theñ f (x) = S f (y)dµ x (y) solves the following Cauchy problem: u = 0, in B(0, 1) u| S = f, where B(0, 1) = {y||y| < 1} is the unit ball in R 2. For this reason, µ x is also called the harmonic probability associated with x on S. Obviously µ 0 = µ. On the other hand, due to the connection with Brownian motion as first identified by Kakutani [9], harmonic probabilities play an important role in probability theory. x denotes the probability distribution of a standard two-dimensional Brow-nian motion B t starting from x, and τ the first time for B t to hit S, µ x is nothing but the distribution of B τ under P x (see [7]).


Circular Cauchy distribution
Let S be the unit circle in R 2 with the Riemannian structure induced by R 2 and write ∇ S for the spherical gradient.For any x ∈ R 2 with |x| < 1, we consider the probability measure µ x on S which has density h(x, y) = 1 2π 1 − |x| 2 |y − x| 2 , y ∈ S with respect to the arc length µ on the unit circle S. The form of the density h makes µ x known as circular Cauchy distribution or wrapped Cauchy distribution (see [10,11]).
On the one hand, it enjoys the following property: if f is an integrable function on S, then f (x) = S f (y)dµ x (y) solves the following Cauchy problem: u = 0, in B(0, 1) where B(0, 1) = {y||y| < 1} is the unit ball in R 2 .For this reason, µ x is also called the harmonic probability associated with x on S. Obviously µ 0 = µ.
On the other hand, due to the connection with Brownian motion as first identified by Kakutani [9], harmonic probabilities play an important role in probability theory.Indeed, if P x denotes the probability distribution of a standard two-dimensional Brow- nian motion B t starting from x, and τ the first time for B t to hit S, µ x is nothing but the distribution of B τ under P x (see [7]).
where β = (x 1 , x 2 ) ∈ B(0, 1) and β = (x 2 , x 1 ).Suppose that W 0 is a constant or a random variable which takes values in S and (ε n ) n≥1 are independent identically distributed random variables taking values in S with common distribution µ x0 for some x 0 ∈ B(0, 1) Kato [10] proved that µ x is the unique invariant probability of the Möbius Markov process (W n ) n≥1 .
The aim of this paper is to estimate the spectral gap and logarithmic Sobolev constants of µ x .
Let λ 1 (µ x ) be the spectral gap of the circular Cauchy distribution µ x associated with the Dirichlet form where Var µx (f ) = S f 2 dµ x − ( S f dµ x ) 2 is the variance of f with respect to µ x .The constant λ 1 (µ x ) is thus the best constant in the following Poincaré inequality CVar µx (f ) ≤ E µx (f, f ).
We say µ x satisfies a logarithmic Sobolev inequality if there exists a non-negative constant C such that for any smooth function f : S → R, is the entropy of f 2 under µ x .We will denote by C LS (µ x ) the optimal logarithmic Sobolev constant of µ x .
An effective method to prove Poincaré or logarithmic Sobolev inequalities is the Bakry-Émery curvature-dimension criterion [1].It gives, in particular, that λ 1 (µ) = C LS (µ) = 1.It is classical for the Poincaré inequality and for logarithmic Sobolev inequality as in [8].Nevertheless, this criterion cannot be applied for all x as the generalized curvature is not bounded from below when x tends to the unit circle.Another natural approach would be to use the Brownian motion interpretation of µ x together with stochastic calculus, as in [12], for which the stopping time τ was involved.In detail, in [12] with this method, G. Schechtman and M. Schmuckenschläger proved that harmonic measures µ n x on S n−1 with n ≥ 3 and |x| < 1 had a uniform Gaussian concentration.
In [3], with F. Barthe, we used another method to work on harmonic measures µ n x on the unit spheres S n−1 .Precisely, we took advantage of the fact that the density of ECP 19 (2014), paper 10.
the harmonic measures only depends on one coordinate, based on which, we proved respectively that and Here ν |x|,n is the image probability of µ n x by the map y → d(y, e 1 ) with e 1 the first component of the canonical basis in R n .From this comparison, we proved that for harmonic measures µ n x on S n−1 with n ≥ 3, with C a positive universal constant.However when n = 2, for the circular Cauchy distribution µ x , n − 2 = 0, the inequalities (0.2), (0.3) do not apply.So in this paper, we follow the main idea of [3] while adjust the estimates.
Remark 0.3.Since the diameter of the unit circle S is π, the result in [15] ensures that for any f : that is to say µ x satisfies the so called L 2 -transportation inequalities W 2 H introduced by Talagrand [13].Here W 2 d (ν, µ) is the L 2 -Wasserstein distance between ν and µ, which is defined as with π the coupling of ν and µ.However by Theorem 0.1, when x approaches S, the optimal logarithmic Sobolev constant explodes with speed log(1 + 1 1 − |x| ).That is, the circular Cauchy distribution µ x is a natural counter-example to declare the real gap between logarithmic Sobolev and W 2 H inequalities as in [3,4,14].
Lemma 1.1.Let M be a probability measure on S with where ϕ is non-negative and measurable.Let ν be the image probability of M by the map y → d(y, e 1 ), which is a probability on the interval [0, π].
Here λ 1 (ν) is the spectral gap of ν and λ DD (ν) is the first eigenvalue of ν with Dirichlet boundary conditions at 0 and π, which has a classical variational formula as Proof.Let F be any every smooth function F : [0, π] → R, and apply the Poincaré inequality for M to the function f It holds by the classical variational formula (0.1) that λ 1 (M ) ≤ λ 1 (ν) since the family of non constant functions f : S → R is larger than that of non constant functions F : [0, π] → R.
Replacing the Variaance by Entropy, we get C LS (ν) ≤ C LS (M ).
For the lower bound of λ 1 (M ), we use the notations presented at the beginning of this section.
For any f measurable on S, we have It is clear that g satisfies g(0) = g(π) = 0. Observe that Therefore .
where g is given in (1.1) and the first inequality is true since the optimal logarithmic Sobolev constant for the Bernoulli distribution with parameter 1/2 is 1.
The proof is complete now.
2 Proof of Theorem 0.1 By rotation invariance of the unit circle, without loss of generality, take x = ae 1 .Let ν a be the image probability of µ x by the map y → d(y, e 1 ).Precisely, (2.1) When a = 0, ν 0 is the uniform probability on [0, π], whose spectral gap and optimal logarithmic Sobolev constant are known to be 1.
Consider the associated Dirichlet form of ν a where the generator L a is given as for any f ∈ C 2 ([0, π]), Proof of the item (a) of Theorem 0.1.Take f (θ) = cos θ, we have which implies ECP 19 (2014), paper 10.
Therefore it follows from Theorem 1.1 in [6] that where the second inequality follows from the proportional property and the last equality holds since (1 ECP 19 (2014), paper 10.
Page 7/9 ecp.ejpecp.org The circular Cauchy distribution Finally, we have for any x with 0 ≤ |x| = a < 1, The proof of the item (a) of Theorem 0.1 is complete.Proof of the item (b) of Theorem 0.1.Recall that for the function f := cos, in the third section, it was proved that ν a .