Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

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ISSN:1083-589X in PROBABILITY

Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

^∗

Yutao Ma

^†

Zhengliang Zhang

^‡

Abstract

In this paper, we consider the circular Cauchy distributionµx on the unit circle S with index 0≤ |x|< 1and we study the spectral gap and the optimal logarithmic Sobolev constant forµx, denoted respectively byλ1(µx)andCLS(µx).We prove that

1

1+|x| ≤λ1(µx)≤1whileCLS(µx)behaves likelog(1 +_1−|x|¹ )as|x| →1.

Keywords:circular Cauchy distribution; spectral gap; logarithmic Sobolev inequality.

AMS MSC 2010:60E15; 39B62; 26Dxx.

Submitted to ECP on October 11, 2013, final version accepted on February 8, 2014.

0.1 Circular Cauchy distribution

LetS be the unit circle inR² with the Riemannian structure induced by R² ^and write ∇_S for the spherical gradient. For any x ∈ R² ^with |x| < 1, we consider the probability measureµxonSwhich has density

h(x, y) = 1 2π

1− |x|²

|y−x|², y∈S

with respect to the arc lengthµon the unit circleS.The form of the densityhmakesµx

known as circular Cauchy distribution or wrapped Cauchy distribution (see [10, 11]).

On the one hand, it enjoys the following property: iff is an integrable function on S, thenf˜(x) =R

Sf(y)dµ_x(y)solves the following Cauchy problem:

(4u= 0, inB(0,1) u|S =f,

whereB(0,1) ={y||y|<1} is the unit ball inR². For this reason,µxis also called the harmonic probability associated withxonS.Obviouslyµ0=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, ifP^xdenotes the probability distribution of a standard two-dimensional Brow- nian motionB_tstarting fromx, andτ the first time forB_tto hitS,µ_xis nothing but the distribution ofB_τ underP^x^{(see [7]).}

∗Support: NSFC 11371283, 11201040, 11101313, 11101040, YETP0264, 985 Projects and the Fundamen- tal Research Funds for the Central Universities.

†School of Math. Sci.&Lab. Math. Com. Sys., Beijing Normal University, China. E-mail:[email protected]

‡Department of Mathematics and Statistics, Wuhan University, China. E-mail:[email protected]

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Furthermore, consider the following Mo¨bius Markov process (see [10]):

W_n= W_n−1+β

βW¯ _n−1+ 1ε_n, n= 1,2,· · · ,

whereβ= (x1, x2)∈B(0,1)andβ¯= (x2, x1).Suppose thatW0is a constant or a random variable which takes values in S and (ε_n)_n≥1 are independent identically distributed random variables taking values inSwith common distributionµ_x₀for somex₀∈B(0,1) fixed. Define

x=







|x0| −1 +p

(1− |x0|)²+ 4|x0||β|²

2|β|² β, if0<|β|<1;

0, |β|= 0.

Kato [10] proved thatµ_x is the unique invariant probability of the Möbius Markov process(Wn)_n≥1.

The aim of this paper is to estimate the spectral gap and logarithmic Sobolev constants ofµx.

Letλ1(µx)be the spectral gap of the circular Cauchy distributionµxassociated with the Dirichlet form

Eµx(f, f) = Z

S

|∇Sf|²dµx, ∀f :S→R smooth function, which has a classical variational formula

λ1(µx) = inf{Eµ_x(f, f)

Varµ_x(f):f non constant}, (0.1) whereVarµ_x(f) = R

Sf²dµx−(R

Sf dµx)² is the variance of f with respect to µx. The constantλ1(µx)is thus the best constant in the following Poincaré inequality

CVarµ_x(f)≤ Eµ_x(f, f).

We say µ_x satisfies a logarithmic Sobolev inequality if there exists a non-negative constantCsuch that for any smooth functionf :S→R^,

Ent_µ_x(f²)≤2C Z

S

|∇_Sf|²dµ_x, where

Ent_µ_x(f²) :=µ_x(f²logf²)−µ_x(f²) log(µ_x(f²))

is the entropy off²underµx.We will denote byCLS(µx)the optimal logarithmic Sobolev constant ofµx.

An effective method to prove Poincaré or logarithmic Sobolev inequalities is the Bakry-Émery curvature-dimension criterion [1]. It gives, in particular, that λ₁(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 allxas the gener- alized curvature is not bounded from below when xtends to the unit circle. Another natural approach would be to use the Brownian motion interpretation ofµx together with stochastic calculus, as in [12], for which the stopping timeτ was involved. In de- tail, in [12] with this method, G. Schechtman and M. Schmuckenschläger proved that harmonic measuresµⁿ_xonSⁿ⁻¹ withn≥3and|x|<1had a uniform Gaussian concentration.

In [3], with F. Barthe, we used another method to work on harmonic measures µⁿ_x on the unit spheresSⁿ⁻¹. Precisely, we took advantage of the fact that the density of

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the harmonic measures only depends on one coordinate, based on which, we proved respectively that

min{λ1(ν_|x|,n), n−2} ≤λ₁(µⁿ_x)≤λ₁(ν_|x|,n) (0.2) and

C_LS(ν_|x|,n)≤C_LS(µⁿ_x)≤max{CLS(ν_|x|,n), 1

n−2}. (0.3)

Here ν_|x|,n is the image probability of µⁿ_x by the map y → d(y, e1) with e1 the first component of the canonical basis in Rⁿ. From this comparison, we proved that for harmonic measuresµⁿ_x onSⁿ⁻¹ withn≥3, λ1(µⁿ_x)satisfied ⁿ⁻²₂ ≤λ1(µⁿ_x)≤n−1 and the optimal logarithmic Sobolev constantC_LS(µⁿ_x)satisfied

1

2(n−1)log(1 + 2

n(1− |x|))≤CLS(µⁿ_x)≤ C

n log(1 + 1 1− |x|) withCa positive universal constant.

However whenn= 2,for the circular Cauchy distributionµx, 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.

Our main results are the following:

Theorem 0.1. For anyx∈R²^with0≤ |x|<1, the following statements hold:

(a) The spectral gapλ1(µx)satisfies

1

1 +|x| ≤λ1(µx)≤1 =λ1(µ).

(b) The optimal constantC_LS(µ_x)satisfies

max{1,1

2log(1 + 1

1− |x|)} ≤CLS(µx)≤8πlog(1 + e²π

2(1− |x|)) + 2.

Remark 0.2. The estimate forλ1(µx)is sharp since whenx= 0,the lower and upper bounds coincide withλ1(µ) = 1.

Remark 0.3. Since the diameter of the unit circleSisπ,the result in [15] ensures that for anyf :S→R^withµx(f²) = 1,one has

W_d²(f²µ_x, µ_x)≤4(8 log 2 +π)Ent_µ_x(f²),

that is to sayµ_xsatisfies the so calledL²-transportation inequalitiesW₂Hintroduced by Talagrand [13]. HereW_d²(ν, µ)is theL²-Wasserstein distance betweenνandµ,which is defined as

W_d²(ν, µ) = inf

π

Z

S²

d²(x, y)dπ(x, y),

withπ the coupling of ν and µ. However by Theorem 0.1, when xapproaches S, the optimal logarithmic Sobolev constant explodes with speedlog(1 + 1

1− |x|).That is, the circular Cauchy distribution µx is a natural counter-example to declare the real gap between logarithmic Sobolev andW2H inequalities as in [3, 4, 14].

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1 Prelimilaries

Given any x ∈ S, it can be written as x = (cosθ, ωsinθ), where θ ∈ [0, π] is the geodestic distanced(x, e₁)betweenxand the first component of the canonical basis in R²^{, and}ω∈ {−1,1}. We then consider the pathγ₀defined as

γ0(t) = (cos(θ+t), ωsin(θ+t)), t∈R,

which is a path onSsatisfyingγ0(0) =xand|γ⁰₀(0)|= 1,then∇Sf(x) = (f◦γ0)⁰(0).

Forθ∈(0, π), define

S(θ) :={x∈S;d(x, e1) =θ}={(cosθ, ωsinθ), ω∈ {−1,1}}.

The conditional probabilityµ_θonS(θ)is a Bernoulli distribution with parameter1/2.

Lemma 1.1. LetM be a probability measure onSwith M(dy) = 1

2πϕ(d(y, e1))µ(dy), y∈S,

whereϕis non-negative and measurable. Letν be the image probability ofM by the mapy→d(y, e₁),which is a probability on the interval[0, π].

We have respectively

(1). The corresponding spectral gaps satisfy

min{λ₁(ν), λ^DD(ν)} ≤λ₁(M)≤λ₁(ν).

(2). Similarly, the optimal logarithmic Sobolev constants satisfy CLS(ν)≤CLS(M)≤CLS(ν) + 1

λ^DD(ν).

Here λ1(ν)is the spectral gap of ν and λ^DD(ν)is the first eigenvalue of ν with Dirichlet boundary conditions at0andπ,which has a classical variational formula as

λ^DD(ν) := inf ( Rπ

0(f⁰)²dν

ν(f²) : f(0) =f(π) = 0, f non constant )

.

Proof. Let F be any every smooth functionF : [0, π] →R, and apply the Poincaré inequality forM to the functionf(x) =F(d(x, e₁)) =F(arccosx). By definitionVar_M(f) = Var_ν(F). Ifx6=±e₁,f is differentiableM−a.e., moreover,

|∇Sf|²(x) =|(f◦γ0)⁰(0)|².

Clearly,f(γ0(t)) =f(cos(θ+t),sin(θ+t)ω) =F(θ+t)and(f◦γ0)⁰(0) =F⁰(θ). So,

|∇_Sf|²(x) = (F⁰(θ))²= (F⁰(d(x, e₁)))², which impliesR

S|∇Sf|²dM =Rπ

0(F⁰)²dν. It holds by the classical variational formula (0.1) thatλ₁(M)≤λ₁(ν)since the family of non constant functionsf :S →R^{is larger} than that of non constant functionsF : [0, π]→R.

Replacing theVariaancebyEntropy, we getCLS(ν)≤CLS(M).

For the lower bound ofλ1(M), we use the notations presented at the beginning of this section.

For anyf measurable onS,we have F(θ) :=

Z

S(θ)

f(cosθ, ωsinθ)dµθ= 1

2f(cosθ,sinθ) +1

2f(cosθ,−sinθ)

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and

g(θ) :=

Z

S(θ)

f(cosθ, ωsinθ)ωdµθ= 1

2f(cosθ,sinθ)−1

2f(cosθ,−sinθ). (1.1) It is clear thatgsatisfiesg(0) =g(π) = 0.Observe that

VarM(f) = Varν(F) + Z π

0

Varµ_θ(f|_S(θ))dν(θ) = Varν(F) +ν(g²).

Therefore

VarM(f)≤ 1 λ1(ν)

Z π 0

(F⁰)²dν+ 1 λ^DD(ν)

Z π 0

g⁰²dν

≤max 1

λ1(ν), 1 λ^DD(ν),

Z π 0

Z

S(θ)

(f◦γ₀)⁰(0)dµ_θ ²

+ Z

S(θ)

(f◦γ₀)⁰(0)ωdµ_θ ²

dν

= 1

min{λ1(ν), λ^DD(ν)}

Z π 0

Z

S(θ)

(f ◦γ0)⁰(0))²dµθdν(θ)

= 1

min{λ1(ν), λ^DD(ν)}

Z

S

|∇_Sf|²dM,

.

which immediately offersλ1(M)≥min{λ1(ν), λ^DD(ν)}.

Given smooth functionf :S→R,defineG²(θ) :=R

S(θ)f²(cosθ, ωsinθ)dµ(θ).Notice then that

EntM(f²) = Entν

Z

S(θ)

f²dµθ

! +

Z π 0

Entµ_θ(f²|S(θ))dν(θ)

≤Ent_ν(G²) +1 2

Z π 0

(f(cosθ,sinθ)−f(cosθ,−sinθ))²dν(θ)

≤2CLS(ν) Z π

0

(G⁰(θ))²dν(θ) + 2 λ^DD(ν)

Z π 0

(g⁰(θ))²dν(θ),

(1.2)

whereg is given in (1.1) and the first inequality is true since the optimal logarithmic Sobolev constant for the Bernoulli distribution with parameter1/2is 1.

By definition,

2G(θ)G⁰(θ) = 2 Z

S(θ)

f(cosθ, ωsinθ)(f◦γ0)⁰(0)dµθ,

which implies

(G⁰(θ))²=

R

S(θ)f(cosθ, ωsinθ)(f◦γ₀)⁰(0)dµ_θ ²

G²(θ)

≤ R

S(θ)f²(cosθ, ωsinθ)dµθ

G²(θ)

Z

S(θ)

(f◦γ₀)⁰(0) ²

dµ_θ

= Z

S(θ)

(f ◦γ0)⁰(0)2

dµθ.

And similarly we have

g⁰(θ)²≤ Z

S(θ)

(f◦γ0)⁰(0)² dµθ.

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Thus from (1.2),

Ent_M(f²)≤2(C_LS(ν) + 1 λ^DD(ν))

Z

S

|∇_Sf|²dM, (1.3)

where implies immediately that

CLS(µx)≤CLS(ν) + 1 λ^DD(ν). The proof is complete now.

2 Proof of Theorem 0.1

By rotation invariance of the unit circle, without loss of generality, takex=ae₁.Let νa be the image probability ofµxby the mapy→d(y, e1).Precisely,

dν_a(θ) = 1 π

1−a²

1 +a²−2acosθdθ=:h_a(θ)dθ, θ∈[0, π]. (2.1) Whena = 0, ν0 is the uniform probability on[0, π],whose spectral gap and optimal logarithmic Sobolev constant are known to be1.

Consider the associated Dirichlet form ofνa

Ea(f, f) = Z π

0

(f⁰)²dνa= Z π

0

f(−Laf)dνa,

where the generatorLa is given as for anyf ∈C²([0, π]), Laf(θ) =f⁰⁰(θ)− 2asinθ

1 +a²−2acosθf⁰(θ).

Proof of the item (a) of Theorem 0.1.Takef(θ) = cosθ,we have νa(f) = 1−a²

π Z π

0

cosθ

1 +a²−2acosθdθ= 1−a² 2aπ

Z π 0

(−1 + 1 +a²

1 +a²−2acosθ)dθ

=−1−a²

2a +1 +a² 2a =a,

(2.2)

νa(f²) = 1−a² π

Z π 0

cos²θ

1 +a²−2acosθdθ

= 1−a² π

Z π/2 0

( cos²θ

1 +a²−2acosθ + cos²(π−θ)

1 +a²−2acos(π−θ))dθ

= 1−a² π

Z π/2 0

2(1 +a²) cos²θ (1 +a²)²−4a²cos²θ

= 1−a⁴ 2a²π

Z π/2 0

(−1 + (1 +a²)²

(1 +a²)²−4a²cos²θ)dθ

=−1−a⁴

4a² +(1 +a²)²

4a² = 1 +a² 2 ,

(2.3)

which implies Ea(f, f) =

Z π 0

sin²θdνa = 1−νa(f²) = 1−a²

2 =νa(f²)−(νa(f))².

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Thereby by classical variational formula (0.1), λ1(νa)≤ Ea(f, f)

Vara(f) = 1. (2.4)

For the upper bound of1/λ1(νa),we turn to Chen’s original variational formula of λ1(ν)(see [5]). Precisely, it is

λ1(νa)⁻¹= inf

ρ∈F sup

x∈[0,π]

1 +a²−2acosx ρ⁰(x)

Z π x

ρ(y)−νa(ρ)

1 +a²−2acosydy, (2.5) whereFis the set of strictly increasing functions on[0, π].

Choose then ρ(θ) =−cosθ+aa strictly increasing function on[0, π]withνa(ρ) = 0 by (2.2). By the expression (2.5), we have

1

λ1(νa) ≤ sup

θ∈(0,π)

1 +a²−2acosθ sinθ

Z π θ

(−cosξ+a) 1 +a²−2acosξdξ

= sup

θ∈(0,π)

1 +a²−2acosθ 2asinθ

π−θ−2 arctan 1−a

1 +acot(θ 2)

= sup

θ∈(0,π)

1 +a²−2acosθ asinθ

arctan

cot(θ

2)

−arctan 1−a

1 +acot(θ 2)

≤ sup

θ∈(0,π)

1 +a²−2acosθ asinθ

(1−^1−a_1+a) cot(^θ₂) 1 + (^1−a_1+acot(^θ₂))²

= 1 +a,

where the first equality is due to Z π

θ

1

1 +a²−2acosθ = 2

1−a²arctan 1−a

1 +acot(θ 2)

(2.6) and the last but second inequality holds since

arctanx−arctany≤(x−y)(arctany)⁰, ∀0≤y < x≤π/2.

To estimateλ^DD(νa),we takeρ(θ) = sinθon[0, π],which satisfies ρ(0) =ρ(π) = 0, ρ⁰(θ)|_{θ∈(0,π/2)}>0andρ⁰(θ)|_{θ∈(π/2,π)}<0.

Therefore it follows from Theorem 1.1 in [6] that 1

λ^DD(νa) ≤ sup

x∈(0,π/2)

1 sinx

Z x 0

(1 +a²−2acosy)dy Z π/2

x

sinu

1 +a²−2acosudu

∨ sup

x∈(π/2,π)

1 sinx

Z π x

(1 +a²−2acosy)dy Z x

π/2

sinu

1 +a²−2acosudu

≤ sup

x∈(0,π/2)∪(π/2,π)

1 +a²−2acosx cosx

Z π/2 x

sinu

1 +a²−2acosudu

= sup

x∈(0,π/2)∪(π/2,π)

1 +a²−2acosx

2acosx log( 1 +a² 1 +a²−2acosx)

= sup

|t|<2a/(1+a²)

(1−1

t) log(1−t) = (1 +a)²

2a log(1 +a)² 1 +a² ,

(2.7)

where the second inequality follows from the proportional property and the last equality holds since(1−¹_t) log(1−t)is decreasing ont∈[−1,1].

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Finally, we have for anyxwith0≤ |x|=a <1, 1

1 +a = min{ 2a (1 +a)²log^(1+a)_1+a2²

, 1

1 +a} ≤λ1(µx)≤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 functionf := cos,in the third section, it was proved thatνa(f) =a, νa(f²) = (1 +a²)/2andEa(f, f) = (1−a²)/2.

Defineg= (1−f)/(1−a),then

νa(g) = 1, νa(g²) = 3−a

2(1−a), Ea(g, g) = Ea(f, f)

(1−a)² = 1 +a 2(1−a).

Therefore with the help of an elementary inequalityEnt_ν_a(g²)≥ν_a(g²) log(ν_a(g²))(see [3]), we have

2C_LS(ν_a)≥Ent_a(g²)

Ea(g, g) ≥ 3−a

1 +alog(1 + 1 +a

2(1−a))≥log(1 + 1

1−a). (2.8) Next we work on the upper bound. It is clear thatθa := 2 arctan^1−a_1+a is the median of ν_a since by (2.6),

1−a² π

Z π θ_a

1

1 +a²−2acosθ = 2

πarctan(1−a 1 +acot(θ_a

2 )) = 1 2. Define

B−(a) := sup

α∈(0,θa)

Z α 0

dθ

1 +a²−2acosθlog 1 + e²π (1−a²)Rα

0 1

1+a²−2acosθdθ

!

· Z θa

α

(1 +a²−2acosθ)dθ,

B₊(a) := sup

α∈(θa,π)

Z π α

dθ

1 +a²−2acosθlog 1 + e²π (1−a²)Rπ

α 1

1+a²−2acosθdθ

!

· Z α

θa

(1 +a²−2acosθ)dθ.

By the equality (2.6) and x

1 +x² ≤arctanx≤x, we have sinα

1 +a²−2acosα≤ Z π

α

1

1 +a²−2acosθdθ≤ 2

(1 +a)²sin^α₂ ^(2.9)

and Z α

0

1

1 +a²−2acosθdθ≤ π

1−a² − sinα

1 +a²−2acosα ≤ π

1−a². (2.10) On the one hand, by the monotonicity ofxlog(1 +^b_x)inx >0for anyb >0and (2.9), we obtain

B+(a)≤ sup

α∈(θ_a,π)

2

(1 +a)²sin(^α₂)log

1 +e²π(1 +a) 2(1−a)

(1 +a²)α−2asinα

≤ 4 (1 +a)²log

1 +e²π(1 +a) 2(1−a)

sup

α∈(θa/2,π/2)

(1 +a²)α sinα

=2π(1 +a²)

(1 +a)² log(1 +e²π(1 +a) 2(1−a) )

≤2πlog(1 + e²π 2(1−a)).

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On the other hand, combining the inequality (2.10), the monotonicity ofxlog(1 + _x^b)for b >0fixed and the fact that

2

πθa≤sinθa= 2 tan(θ_a/2)

1 + tan²(θa/2) =1−a² 1 +a², we have

B−(a)≤ π

1−a²log(1 +e²) (1 +a²)θa−asinθa

≤πlog(1 +e²) θ_a sinθa

≤ π²

2 log(1 +e²).

By Theorem 3 in [2],

C_LS(ν_a)≤4 max{B₊(a), B₋(a)} ≤8πlog(1 + e²π

2(1−a)). (2.11) The proof is complete due to (2.8),(2.11) and the classical result

CLS(µx)≥ 1

λ1(µx) ≥1.

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Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution