Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

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1 Electron. Commun. Probab. 9 4), no., 9. DOI:.4/ECP.v9-37 ISSN: X ELECTRONIC COMMUNICATIONS 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 distribution µ x on the unit circle S with index x < and we study the spectral gap and the optimal logarithmic Sobolev constant for µ x, denoted respectively by λ µ x) and C LSµ x). We prove that + x λµx) while CLSµx) behaves like log + x ) as x. Keywords: circular Cauchy distribution; spectral gap; logarithmic Sobolev inequality. AMS MSC : 6E5; 39B6; 6Dxx. Submitted to ECP on October, 3, final version accepted on February 8, 4.. Circular Cauchy distribution Let S be the unit circle in R with the Riemannian structure induced by R and write S for the spherical gradient. For any x R with x <, we consider the probability measure µ x on S which has density hx, y) = x y x, 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 [, ]). On the one hand, it enjoys the following property: if f is an integrable function on S, then fx) = S fy)dµ xy) solves the following Cauchy problem: { u =, in B, ) u S = f, where B, ) = {y y < } is the unit ball in R. For this reason, µ x is also called the harmonic probability associated with x on S. Obviously µ = 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 Brownian 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]). Support: NSFC 3783, 4, 33, 4, YETP64, 985 Projects and the Fundamental Research Funds for the Central Universities. School of Math. Sci. & Lab. Math. Com. Sys., Beijing Normal University, China. mayt@bnu.edu.cn Department of Mathematics and Statistics, Wuhan University, China. zhlzhang.math@whu.edu.cn

2 Furthermore, consider the following Möbius Markov process see []): W n = W n + β βw n + ε n, n =,,, where β = x, x ) B, ) and β = x, x ). Suppose that W is a constant or a random variable which takes values in S and ε n ) n are independent identically distributed random variables taking values in S with common distribution µ x for some x B, ) fixed. Define x + x ) + 4 x β x = β β, if < β < ;, β =. Kato [] proved that µ x is the unique invariant probability of the Möbius Markov process W n ) n. The aim of this paper is to estimate the spectral gap and logarithmic Sobolev constants of µ x. Let λ µ x ) be the spectral gap of the circular Cauchy distribution µ x associated with the Dirichlet form E µx f, f) = S f dµ x, f : S R smooth function, S which has a classical variational formula λ µ x ) = inf{ E µ x f, f) : f non constant },.) Var µx f) where Var µx f) = S f dµ x S fdµ x) is the variance of f with respect to µ x. The constant λ 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, Ent µx f ) C S f dµ x, S where Ent µx f ) := µ x f log f ) µ x f ) logµ x f )) is the entropy of f 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 []. It gives, in particular, that λ µ) = C LS µ) =. 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 [], for which the stopping time τ was involved. In detail, in [] with this method, G. Schechtman and M. Schmuckenschläger proved that harmonic measures µ n x on Sn with n 3 and x < 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. Precisely, we took advantage of the fact that the density of ECP 9 4), paper. Page /9

3 the harmonic measures only depends on one coordinate, based on which, we proved respectively that min{λ ν x,n ), n } λ µ n x) λ ν x,n ).) and C LS ν x,n ) C LS µ n x) max{c LS ν x,n ), }..3) n Here ν x,n is the image probability of µ n x by the map y dy, e ) with e the first component of the canonical basis in R n. From this comparison, we proved that for harmonic measures µ n x on Sn with n 3, λ µ n x) satisfied n λ µ n x) n and the optimal logarithmic Sobolev constant C LS µ n x) satisfied n ) log + n x ) ) C LSµ n x) C n log + x ) with C a positive universal constant. However when n =, for the circular Cauchy distribution µ x, n =, the inequalities.),.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.. For any x R with x <, the following statements hold: a) The spectral gap λ µ x ) satisfies + x λ µ x ) = λ µ). b) The optimal constant C LS µ x ) satisfies max{, log + x )} C e LSµ x ) 8 log + x ) ) +. Remark.. The estimate for λ µ x ) is sharp since when x =, the lower and upper bounds coincide with λ µ) =. Remark.3. Since the diameter of the unit circle S is, the result in [5] ensures that for any f : S R with µ x f ) =, one has W d f µ x, µ x ) 48 log + )Ent µx f ), that is to say µ x satisfies the so called L -transportation inequalities W H introduced by Talagrand [3]. Here Wd ν, µ) is the L -Wasserstein distance between ν and µ, which is defined as Wd ν, µ) = inf d x, y)dx, y), S with the coupling of ν and µ. However by Theorem., when x approaches S, the optimal logarithmic Sobolev constant explodes with speed log + ). That is, the x circular Cauchy distribution µ x is a natural counter-example to declare the real gap between logarithmic Sobolev and W H inequalities as in [3, 4, 4]. ECP 9 4), paper. Page 3/9

4 Prelimilaries Given any x S, it can be written as x = cos θ, ω sin θ), where θ [, ] is the geodestic distance dx, e ) between x and the first component of the canonical basis in R, and ω {, }. We then consider the path γ defined as γ t) = cosθ + t), ω sinθ + t)), t R, which is a path on S satisfying γ ) = x and γ ) =, then S fx) = f γ ) ). For θ, ), define := {x S; dx, e ) = θ} = {cos θ, ω sin θ), ω {, }}. The conditional probability µ θ on is a Bernoulli distribution with parameter /. Lemma.. Let M be a probability measure on S with Mdy) = ϕdy, e ))µdy), y S, where ϕ is non-negative and measurable. Let ν be the image probability of M by the map y dy, e ), which is a probability on the interval [, ]. We have respectively ). The corresponding spectral gaps satisfy min{λ ν), λ DD ν)} λ M) λ ν). ). Similarly, the optimal logarithmic Sobolev constants satisfy C LS ν) C LS M) C LS ν) + λ DD ν). Here λ ν) is the spectral gap of ν and λ DD ν) is the first eigenvalue of ν with Dirichlet boundary conditions at and, which has a classical variational formula as { λ DD ν) := inf f } ) dν νf : f) = f) =, f non constant. ) Proof. Let F be any every smooth function F : [, ] R, and apply the Poincaré inequality for M to the function fx) = F dx, e )) = F arccos x). By definition Var M f) = Var ν F ). If x ±e, f is differentiable M a.e., moreover, S f x) = f γ ) ). Clearly, fγ t)) = fcosθ + t), sinθ + t)ω) = F θ + t) and f γ ) ) = F θ). So, S f x) = F θ)) = F dx, e ))), which implies S Sf dm = F ) dν. It holds by the classical variational formula.) that λ M) λ ν) since the family of non constant functions f : S R is larger than that of non constant functions F : [, ] R. Replacing the Variaance by Entropy, we get C LS ν) C LS M). For the lower bound of λ M), we use the notations presented at the beginning of this section. For any f measurable on S, we have F θ) := fcos θ, ω sin θ)dµ θ = fcos θ, sin θ) + fcos θ, sin θ) ECP 9 4), paper. Page 4/9

5 and gθ) := fcos θ, ω sin θ)ωdµ θ = fcos θ, sin θ) fcos θ, sin θ)..) It is clear that g satisfies g) = g) =. Observe that Var M f) = Var ν F ) + Var µθ f )dνθ) = Var ν F ) + νg ). Therefore Var M f) λ ν) max F ) dν + λ DD g dν ν) } { { λ ν), λ DD ν), ) + f γ ) )ωdµ θ }dν = = min{λ ν), λ DD ν)} min{λ ν), λ DD ν)} S S f dm, ) f γ ) )dµ θ f γ ) )) dµ θ dνθ). which immediately offers λ M) min{λ ν), λ DD ν)}. Given smooth function f : S R, define G θ) := f cos θ, ω sin θ)dµθ). Notice then that ) Ent M f ) = Ent ν f dµ θ + Ent ν G ) + C LS ν) Ent µθ f )dνθ) fcos θ, sin θ) fcos θ, sin θ)) dνθ) G θ)) dνθ) + λ DD ν) g θ)) dνθ),.) where g is given in.) and the first inequality is true since the optimal logarithmic Sobolev constant for the Bernoulli distribution with parameter / is. By definition, Gθ)G θ) = fcos θ, ω sin θ)f γ ) )dµ θ, which implies fcos θ, ω sin θ)f γ ) )dµ θ ) G θ)) = G θ) f cos θ, ω sin θ)dµ θ G θ) = f γ ) ) ) dµθ. f γ ) )) dµ θ And similarly we have g θ) f γ ) ) ) dµθ. ECP 9 4), paper. Page 5/9

6 Thus from.), Ent M f ) C LS ν) + λ DD ν) ) S f dm,.3) S where implies immediately that The proof is complete now. C LS µ x ) C LS ν) + λ DD ν). Proof of Theorem. By rotation invariance of the unit circle, without loss of generality, take x = ae. Let ν a be the image probability of µ x by the map y dy, e ). Precisely, dν a θ) = a + a a cos θ dθ =: h aθ)dθ, θ [, ]..) When a =, ν is the uniform probability on [, ], whose spectral gap and optimal logarithmic Sobolev constant are known to be. Consider the associated Dirichlet form of ν a E a f, f) = f ) dν a = f L a f)dν a, where the generator L a is given as for any f C [, ]), L a fθ) = f θ) a sin θ + a a cos θ f θ). Proof of the item a) of Theorem.. Take fθ) = cos θ, we have ν a f) = a which implies = a a ν a f ) = a cos θ a + a dθ = a cos θ a + + a = a, a = a = a = a4 a / / / cos θ + a a cos θ dθ + + a + a a cos θ )dθ cos θ + a a cos θ + cos θ) + a a cos θ) )dθ + a ) cos θ + a ) 4a cos θ + = a4 4a + + a ) 4a + a ) + a ) 4a cos θ )dθ = + a,.).3) E a f, f) = sin θdν a = ν a f ) = a = ν a f ) ν a f)). ECP 9 4), paper. Page 6/9

7 Thereby by classical variational formula.), λ ν a ) E af, f) =..4) Var a f) For the upper bound of /λ ν a ), we turn to Chen s original variational formula of λ ν) see [5]). Precisely, it is λ ν a ) = inf + a a cos x ρy) ν a ρ) ρ F x [,] ρ x) x + a dy,.5) a cos y where F is the set of strictly increasing functions on [, ]. Choose then ρθ) = cos θ + a a strictly increasing function on [, ] with ν a ρ) = by.). By the expression.5), we have λ ν a ) + a a cos θ θ,) sin θ + a a cos θ = θ,) a sin θ + a a cos θ = θ,) a sin θ cos ξ + a) θ + a a cos ξ dξ θ arctan )) a + a cotθ ) arctan cot θ ) ) arctan + a a cos θ a +a ) cot θ ) θ,) a sin θ + a +a cot θ )) = + a, a + a cotθ ) )) where the first equality is due to θ + a a cos θ = a arctan and the last but second inequality holds since ) a + a cotθ ).6) arctan x arctan y x y)arctan y), y < x /. To estimate λ DD ν a ), we take ρθ) = sin θ on [, ], which satisfies ρ) = ρ) =, ρ θ) θ,/) > and ρ θ) θ /,) <. Therefore it follows from Theorem. in [6] that λ DD ν a ) x,/) x /,) sin x sin x x x + a a cos y)dy + a a cos y)dy + a a cos x x,/) /,) cos x + a a cos x = x,/) /,) a cos x / x / x x / sin u + a a cos u du sin u + a a cos u du sin u + a a cos u du + a log + a a cos x ) = + a) + a) ) log t) = log t <a/+a ) t a + a,.7) where the second inequality follows from the proportional property and the last equality holds since t ) log t) is decreasing on t [, ]. ECP 9 4), paper. Page 7/9

8 Finally, we have for any x with x = a <, + a = min{ a, + a) log +a) + a } λ µ x ). +a The proof of the item a) of Theorem. is complete. Proof of the item b) of Theorem.. Recall that for the function f := cos, in the third section, it was proved that ν a f) = a, ν a f ) = + a )/ and E a f, f) = a )/. Define g = f)/ a), then ν a g) =, ν a g ) = 3 a a), E ag, g) = E af, f) a) = + a a). Therefore with the help of an elementary inequality Ent νa g ) ν a g ) logν a g )) see [3]), we have C LS ν a ) Ent ag ) E a g, g) 3 a + a log + + a a) ) log + )..8) a Next we work on the upper bound. It is clear that θ a := arctan a +a is the median of ν a since by.6), and Define B a) := a α α,θ a) θ a + a a cos θ = arctan a + a cotθ a )) =. θa α dθ + a a cos θ log + + a a cos θ)dθ, a ) α dθ B + a) := α θ a,) α + a a cos θ log + α + a a cos θ)dθ. θ a x By the equality.6) and arctan x x, we have + x sin α + a a cos α α α a ) α e +a a cos θ dθ e +a a cos θ dθ + a a cos θ dθ + a) sin α ) ).9) + a a cos θ dθ a sin α + a a cos α a..) On the one hand, by the monotonicity of x log + b x ) in x > for any b > and.9), we obtain B + a) α θ a,) 4 + a) log + a) sin α ) log + e + a) a) + e + a) a) ) = + a ) + a) log + e + a) a) ) log + e a) ). α θ a/,/) ) + a )α a sin α ) + a )α sin α ECP 9 4), paper. Page 8/9

9 On the other hand, combining the inequality.), the monotonicity of x log + b x ) for b > fixed and the fact that we have By Theorem 3 in [], θ a sin θ a = tanθ a/) + tan θ a /) = a + a, B a) a log + e ) + a ) )θ a a sin θ a θ a log + e ) sin θ a log + e ). C LS ν a ) 4 max{b + a), B a)} 8 log + The proof is complete due to.8),.) and the classical result References C LS µ x ) λ µ x ). e )..) a) [] Bakry D. and Émery M.: Diffusions hypercontractivies, In Sém. Proba. XIX, LNM, 3, Springer 985), MR [] Barthe, F. and Roberto, C.: Sobolev inequalities for probabilty measures on the real line, Studia Mathematica, 59 3), 3), MR-535 [3] Barthe, F., Ma, Y-T. and Zhang, Z.: Logarithmic Sobolev inequalities for harmonic measures on spheres. J. Math. Pures Appl., 3) DOI:.6/j.matpur.3..8 [4] Cattiaux, P. and Guillin, A.: On quadratic transportation cost inequalities. J. Math. Pures Appl., 86, 6), MR [5] Chen, M. F.: Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. Sin. A), 48), 999), MR [6] Chen, M.F., Zhang, Y.H. and Zhao, X.L.: Dual variational formulas for the first Dirichlet eigenvalue on half-line, Sci. China, 466), 3), MR-996 [7] Durrett, R.: Brownian Motion and Martingales in analysis, Wordsworth, 984. [8] Émery, M. and Yukich, J.: A simple proof of logarithmic Sobolev inequality on the circle. Séminaire de probabilités, 97, 975), [9] Kakutani, S.: On Brownian motion in n-space. Proc. Imp. Acad. Tokyo, 9), 944), [] Kato, S.: A Markov process for circular data, J. R. Statist. Soc. 75), ), [] McCullagh, P.: Möbius transformation and Cauchy parameter estimation. Ann. Statist., 4, 996), [] Schechtman, G. and Schmuckenschläger, M.: A concentration inequality for harmonic measures on the sphere, Geometric aspects of funct. analysis Israel, ): 55-73, Oper. Theory Adv. Appl., 77, 995), Birkhäuser, Basel, B99). MR [3] Talagrand, M.: Transportation cost for Gaussian and other product measures, Geom. Funct. Anal., 6, 996), MR-3933 [4] Zhang, Z.L., Ma, Y-T. and Lei, L.: Logarithmic Sobolev inequalities for Moebius measures on spheres. Submitted, 3. [5] Zhang, Z.L., and Miao, Y.: An equivalent condition between Poincaré inequality and T - transportation cost inequality. Acta Appl. Math., ), ), MR-664 ECP 9 4), paper. Page 9/9

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