An example of application of discrete Hardy s inequalities

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1 An example of application of discrete Hardy s inequalities Laurent Miclo Laboratoire de Statistique et Probabilités, UMR C5583 Université PaulSabatieretCNRS 8, route de Narbonne 3062 Toulouse cedex, France miclo@cict.fr Summary : After having given a short proof of weighted Hardy s inequalities on IN, adapted from the continuous case, we will show how to use them to evaluate, up to an universal factor, spectral gaps and logarithmic Sobolev constants of birth and death processes on Z. We will illustrate this method by calculations on an example. Abbreviated title : Discrete Hardy s inequalities. American Mathematical Society 99 subject classifications : Primary 60J80 and 39A2 ; secondary 26D5 and 5A60. Key words and phrases : Birth and death Markov processes, spectral gaps,weightedhardy sinequalities on discrete intervals.

2 Introduction Weighted discrete Hardy s inequalities are a simple and very efficient way to evaluate spectral gaps on trees, since theoretically this method enables one to get an estimation up to a factor 32 (or 8 if, as below, the tree is the usual one on Z). The purpose of this note is to present this approach and to give an example of such a calculation. The birth and death Markov kernels under study were considered by C. Meise, who wanted to know the order of their spectral gap, to evaluate the speed of convergence to equilibrium of an other Markov chain on the n-uples, n IN \{}, ofelementsoftheadditivegroup Z 3,whoseunionsetisgenerating(formore details about this problem, see [8], where Meise gets its needed estimation by using a method due to Zeifman [3, 5]). In fact, there is already a lot of methods to evaluate the spectral gap of a birth and death process, for instance: nonnegative matrix approach ([2]), Zeifman s method([3, 5]), optimal path considerations ([6]), Chen s coupling and distance arguments ([7]), variants of Cheeger s inequalities ([2])... But in each of these approachs, there is a good choice to be made: one has to find a function in [2] or [3, 5], a length function in [6], a distance in [7], a renormalisation in [2], and there is no universal way to make such a suitable guess, one has to try and see on the specific examples at hand. The main advantage of the Hardy s inequalities is that they do not depend on yours prediction skills, they gives you a (relatively) explicit estimation of the spectral gap in terms of yours data (birth and death rates and the associated reversible probability). Sure, this evaluation is only good up to an universal factor 8, so in particular examplesthepreviousmethodscanbemore efficient, nevertheless in practice this factor is not a crucial point (see for instance the consequences of the estimation needed by Meise), and the robustess aspect of the Hardy s inequalities seems more important. Furthermore, this approach is also valid for the estimation of the Sobolev-logarithmic constant (see the third section below), and for the latter, there is no other general method. Indeed, the Hardy s inequalities give criteria for the existence of positive spectral gap and Sobolev-logarithmic constant for birth and death processes, in the spirit of the work of Bobkov and Götze for diffusions on IR (see []). But the serious drawback of the discrete Hardy s inequalities is that they can only be applied to Markov processes whose underlying graph is a tree (nevertheless subtrees can be useful for general finite Markov processes, see for instance [9]). Let us first recall the weighted discrete Hardy s inequalities on IN, with a short proof of these bounds, directly adapted from the one given by Muckenhoupt [0] for the similar continuous result on IR +.ForthegeneralcaseofweighteddiscreteHardy sinequalitiesonrootedtrees,see the appendice of [9], where these inequalities are deduced from those given by Evans, Harris and Pick in [4] for continuous trees (whose proof was itself inspired by the approach considered by Sawyer in [] for IR + ). So let µ and ν be two positive functions on IN,whichwillbeseenasmeasures,weare interested in the smallest constant A IR + {+ } such that the following inequalities are satisfied for all functions f : IN IR, µ(x) x IN 0<yx 2 f(y) A x IN ν(x)f 2 (x) 2

3 () To estimate A, letusintroducetheconstantb defined by B = sup x>0 x µ(y) ν(y) yx Then the weighted (by µ and ν) discretehardy sinequalitiesonin just say y= Proposition The constant B is a good approximation of A, inthesensethatwearealways assured of B A 4B Proof: We begin by proving that A 4B: letf be any function of IL 2 (IN, ν), we have to see that For x IN,denote x IN 0<yx 2 f(y) µ(x) 4B x IN f 2 (x) ν(x) N(x) = x y= ν(y) We start from the left hand side of (), for which we use a Cauchy-Schwarz inequality, to obtain the following upper bound µ(x) x IN 0<yx f 2 (y)ν(y)n 2 (y) 0<zx ν(z) N 2 (z) To deal with the last sum, we use the concavity inequality saying that for all a, b > 0, b 2 a 2 2 (b a) so we are assured that for all z>0(withtheconventionthatn(0) = 0), b 2 ν(z) N 2 (z) 2(N 2 (z) N 2 (z )) Thus we get 0<zx ν(z) N 2 (z) 2 2N 2 (x) ( B µ([x, + [) ) 2 We now have to consider 2 B f 2 (y)ν(y)n 2 (y) µ(x) y IN xy µ 2 ([x, + [) 3

4 and to do so we use again the previous convexity inequality, to obtain µ(x) xy µ 2 ([x, + [) 2 µ 2 ([x, + [) µ 2 ([x +, + [) xy = 2µ 2 ([y, + [) ( ) B 2 2 N(y) Putting together all these calculations, we end up with (). To prove the simpler lower bound A B, itisenoughtoapplytheinequality 2 f(y) µ(x) A f 2 (x) ν(x) x IN 0<yx x IN with well chosen functions f. More precisely, for x 0 IN fixed, let f be defined by { y IN /ν(y), if 0 <y x0, f(y) = 0, otherwise we get therefore A 0<yx 0 ν(y) f(y) x IN 0<yx xx 0 0<yx ν(y) from where we deduce that A µ([x 0, + [) 0<yx 0 /ν(y). To get the result we want, it remains to take the supremum in x 0 IN. Now let us see how this result can be applied to get an evaluation of the spectral gap for birth and death processes on Z: solet(a(i)) i Z and (b(i)) i Z be two sequences of positive numbers, which will respectively stand for the birth and death rates. Up to a multiplicative constant, there is an unique associated reversible measure µ, i.e.satisfyingforalli Z, µ(i)a(i) =µ(i+)b(i+). Let us assume that µ is in fact of finite weight, so from now on we can restrict ourselves to the case where it is a probability. The spectral gap of this birth and death process is the ergodic constant defined by E(f,f) λ = inf f IL 2 (µ)\vect(i) µ((f µ(f)) 2 ) where the Dirichlet form E is given for all functions f IL 2 (µ) by µ(x) µ(x) E(f,f) = i Z(f(i +) f(i)) 2 µ(i)a(i) For i Z, letusdefine B + (i) = sup x>i B (i) = sup x<i x y=i+ ( i y=x µ(y) µ(y)b(y) yx ) µ(y) µ(y)a(y) yx 4

5 Let m denote a median of µ (i.e. an integer m Z such that µ(],m[) /2 and µ(]m, + [) /2), and let B = B + (m) B (m) Its interest appears in the following estimation: Proposition 2 With the previous notations, we have 4B λ 2 B Proof: We first prove the lower bound on the spectral gap: let f IL 2 (µ) \ Vect(I) be given, we consider the two functions defined by and x Z, We have for them f (x) = (f(x) f(m))i ],m] (x) f + (x) = (f(x) f(m))i [m,+ [ (x) E(f,f) = E(f f(m)i,f f(m)i) = E(f,f )+E(f +,f + ) µ((f µ(f)) 2 ) µ((f f(m)i) 2 ) = µ(f 2 )+µ(f 2 +) so E(f,f) µ((f µ(f)) 2 ) min ( E(f,f ) µ(f 2 ) (A + A ) ; E(f ) +,f + ) µ(f+) 2 where A + (respectively A )isthebestconstantsuchthatthefollowing inequalities are satisfied for all functions g : IN IR, µ(m + x) x IN 0<yx 2 g(y) A + µ(m +x)a(m +x)g 2 (x) x IN = A + µ(m + x)b(m + x)g 2 (x) x IN (resp. x IN µ(m x) ( 0<yx g(y) ) 2 A x IN µ(m x)a(m x)g2 (x)). This fact shows that the bound λ /(4B) followsfromthehardy sinequalities. For the reversed inequality λ 2/B, bysymmetryoftheproblemwecanassumeforinstance that B = B + (m), and we begin by treating the case where B<+. Let 0 < ɛ <B + (m) be 5

6 given and consider a function g : IN IR (without any restriction, we could assume that g is taking values in IR + )suchthat (2) x IN µ(m + x) ( 0<yx g(y) ) 2 x IN µ(m + x)b(m + x)g2 (x) where the numerator can be supposed to be finite and non-zero. We define a non-constant function f : Z IR by setting x Z, f(x) = A + ɛ { 0, if x m g() + + g(x m), if x>m As µ({f =0}) µ(],m]) /2, we get by a Cauchy-Schwarz inequality that µ(f) 2 µ({f >0})µ(f 2 ) µ(f 2 )/2, so µ((f µ(f)) 2 ) = µ(f 2 ) µ(f) 2 µ(f 2 )/2 (A + ɛ)e(f,f)/2 (B + ɛ)e(f,f)/2 (B + ɛ)λµ((f µ(f)) 2 )/2 from which we get λ 2(B + ɛ).lettingɛ go to 0 +,weobtaintheexpectedresult. IfB =+, we proceed in same manner, but we have to consider functions g such that the left hand side of (2) is finite but very large. In practice, it can be difficult to find exactly a median, and sometimes it is better to consider the constant B = inf i Z (B +(i) B (i)) because we are still assured of the same bounds (which also show that B and B are of the same order). Proposition 3 As before, we can approximate λ by /B : λ 2 4B B Proof: The first part of the proof of the proposition 2 shows in fact that for all i Z (cut the function f f(i)i ini instead of cutting f f(m)i inm), we are assured of λ (4(B + (i) B (i))). On the other hand, the bound λ 2/B follows from the trivial inequality B B. This result shows that if one wants to get a good lower bound of the spectral gap (which is often the critical point), one only need to guess an adequate choice of x and to apply the estimate λ 4(B + (x) B (x)) 6

7 Of course, the previous considerations are still valid if Z is replaced by a infinite or finite discrete interval (this situation has to be considered for instance if the a(i) or the b(i), i Z, are only assumed to be non-negative), as it will be the case below. In fact, when the interval is not the whole Z, but the rates of birth and death are positive on this interval(except on the frontier for one of them), we have furthermore unicity, up to a multiplicative constant, for the invariant measure µ, which is then reversible. This was not necessarily true for Z. 2 The example It is now time to introduce the example of Meise: it is a familly of birth and death processes indexed by n IN \{}, takingvaluesin{,,n}. More precisely, for all n IN \{}, the birth and death rates are respectively given by i<n, a n (i) = <i n, b n (i) = i(n i) n(n ) i(i ) pn(n ) where p>0isafixedparameter. Meise was interested in showing that the spectral gaps λ n of these chains are of order /n: there exists two constants 0 <c <c 2 < + such that for all n 2, we have the bounds (3) c n λ n c 2 n (in fact Meise has got a more precise result, as he can show that lim n nλ n =,notethat such equivalences are out of reach by using only Hardy s bounds, but the goal of the following calculations is only to illustrate this method because they have some interesting features, maybe should we have chosen a more terrible example!). In order to prove them, let us denote by µ n the associated reversible probability on {,,n}. It is quite immediate to calculate that it is given by i n, µ n (i) = ((+p) n ) p i ( n i ) = p i n! (( + p) n )i!(n i)! For n>+ /p, letusdefinem n as z n +,where z n =(pn )/( + p) (and denote the integer part), it is an interesting point in {,,n} because µ n is non-decreasing (respectively decreasing) on {,,m n } (resp. {m n,,n}). More precisely, as we have for all i<n, µ n (i +) µ n (i) = a n (i) b n (i +) = p(n i) i + this ratio is decreasing on {,,n} and take the value only for the real number i = z n. Using this remark, making the convention that =0forx {,,n}, anddefining l n =2 n +3,weseethatforalln>+ /p, µ n (m n + l n ) µ n (m n ) = ρ ( p(n zn n) z n + n ( ) n p/ n +/ n ) n 7

8 where ρ =exp( p /2) < (wehavemadeuseoftheconcavityinequalitiesx/(x +) ln( + x) x, validforallx> ). With a similar calculation, we see that n>+ /p, µ n (m n l n ) µ n (m n ) ρ 2 with ρ 2 =exp( p/( + p)) < (infact,itcouldbepossibletobenumericallymoreprecise here, and via Stirling formula, one can explicit the limit values, as n tends to infinity, of the previous ratios). We deduce from these bounds that we have for n>+ /p, m n x n, x m n, µ n (x + l n ) µ n (x l n ) ρ ρ 2 But these estimations are only good near m n (this is related to the fact that if by an affine transformation, we repace [,n]by[ m n / n, (n m n )/ n], then the image of µ n looks like a centered Gaussian distribution with variance p/( + p) 2 ), and as we are away from this point with a distance of order n, itiseasytogetbetterestimates: forinstance,as µ n (k n ) µ n (k n ) 2 with k n = p(n +)/(2 + p), wehave x k n, µ n (x ) 2 In fact these informations are mainly the only ones needed to get a lower bound on λ n by using the Hardy s inequalities from the point m n. The heuristic reason is the following one: on {m n,,n} and on {k n,,m n } (in reverse order), in a space scale n, µ n decreases somewhat faster than power laws (whose parameters ρ and ρ 2 do not depend on n)andtheratesoftransition in the direction of m n are bounded below by a constant which does not depend on n, andthis contributes to an estimate of the spectral gap of order /n. On the other hand, on the interval {,,k n } (also looked at in reverse order), in a space scale, µ n decreases in some sense faster than a power law (of parameter /2), but the rates of transition toward k n are at least /n (which reduce the speed of escape from by the same factor), so this also gives an estimation of the spectral gap of order /n. Herearethemoredetailedcalculations:webeginbyevaluating B n,+ = max x>m n x y=m n+ µ n ([x, n]) µ n (y)b n (y) It is easy to explicit a n 0 + /p large enough such that for all n n 0 and all m n <y n, b(y) p/(2(p +) 2 ). Let x be fixed in {m n,,n}, thenwehave p 2(p +) 2 x y=m n+ µ n (y)b n (y) x y=m n+ = µ n (y) i IN m n (x (i+)l n)<ym n (x il n) µ n (y) 8

9 while so we end up with Then we look at µ n ([x, n]) B n,+ B n, = max x<m n l n i IN : x il nm n µ n (x il n ) l n ρ i i IN : x il nm n l n ρ i IN n (x+il n)y<n (x+(i+)l n) i IN : x+il nn i IN : x il nn l n ρ 2(p +)2 p ( mn y=x l n µ n (x + il n ) l n ρ i µ n(x) ( ln ρ ) 2 µ n (y) ) µ n ([,x]) µ n (y)a n (y) and we first consider the case of x {k n,,m n }, becauseitisthenpossibletofindexplicitly a n n 0 large enough such that for all n n,weareassuredofinf kny<m n a n (y) p/(2(2+p) 2 ), and thus we have as before, for n n, p m n 2(p +2) 2 y=x µ n (y)a n (y) l n ρ 2 so max k nx<m n ( mn y=x µ n ([,x]) ) µ n ([,x]) µ n (y)a n (y) l n ρ 2 2(p +2)2 p ( ln ρ 2 ) 2 For x {,,k n },weusethefollowingbounds,whichareprovedinsameway k n y=x µ n (y) µ n ([,x]) /2 /2 µ n(x) to obtain, via the inequality a n (y) /n, satisfiedforall y<n, k n max x<k n y=x µ n (y)a n (y) µ n([,x]) 4n 9

10 but we are also assured of m n max x<k n y=k n µ n (y)a n (y) µ n([,x]) m n y=k n 2(p +2)2 p µ n (y)a n (y) µ n([,k n ]) ( ln ρ 2 ) 2 so in the end B n, 4n + 2(p +2)2 p ( ln ρ 2 from where we easily get the existence of a constant C > 0suchthat B n, B n,+ C n i.e. the above mentionned lower bound for λ n is proved with c =/(4C ), at least for n n. To prove a related upper bound, just use the definition of λ n with the particular function I {}, it gives λ n (n[ pn/(( + p) n )]) /n, forn large. ) 2 3 Logarithmic-Sobolev inequalities The weighted Hardy s inequalities are also an efficient method to evaluate logarithmic Sobolev constants on trees. By using some ideas and results of Bobkov and Götze [], who have seen logarithmic Sobolev inequalities as Poincaré s inequalities in the Orlicz space associated to the Young function Ψ : IR + x x ln( + x), we have presented this approach for the discrete setting in [9]. Let us give these estimations in the special case of Z endowed with its natural tree structure. So we come back to the general birth and death process of positive rates (a(i)) i Z and (b(i)) i Z,reversiblewithrespecttoprobabilityµ. The associated logarithmic Sobolev constant is defined by E(f,f) α = inf f IL 2 (µ)\vect(i) Ent(f 2,µ) where the entropy of the function f 2 with respect to µ is the quantity Ent(f 2,µ)= f 2 ln(f 2 / f 2 IL 2 (µ) ) dµ. To give an approximation of this ergodic constant α, we introduce for all i Z, the numbers and we consider C + (i) = sup x>i C (i) = sup x<i Then it can be shown that x y=i+ ( i y=x µ([x, + [) ln(µ([x, + [)) µ(y)b(y) ) µ(],x]) ln(µ(],x])) µ(y)a(y) C = inf i Z (C (i) C + (i)) 0

11 Proposition 4 The constant C is of the same order as α, sincethefollowingboundsarealways satisfied 20 C α 4 ( ) C If we restrict ourselves to the case of a finite interval, one deduces from the lower bound that α λ 40 ln(µ ) where µ is the minimum value taken by µ on the finite interval. But there is a better result in this direction, due to Diaconis and Saloff-Coste [3], saying that for all finite and irreducible Markov kernels (without assumptions on the underlying graph), α 2µ ln(µ ) λ where λ, α and µ are respectively the spectral gap, the logarithmic Sobolev constant and the minimum value of the invariant probability. Of course, there is not a related upper bound (up toanuniversalconstant,butweareassured of α λ/2), nevertheless, in the example of Meise, this gives the right order of the logarithmic Sobolev constants α n :therearetwoconstants0<c 3 <c 4 < +, suchthatforalln IN \{}, c 3 n 2 α n c 4 n 2 To get the lower bound, just note that µ n, = µ n () µ(n) =(pn p n )/(( + p) n ), and the upper bound can be obtained by considering the function I {} in the definition of α n. References [] S.G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. Preprint, July 997. [2] M.-F. Chen and F.-Y Wang. Cheeger s inequalities for general symmetric forms and existence criteria for spectral gap. Preprint from the Beijing Normal University, 998. [3] P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains. The Annals of Applied Probability, 6(3): ,996. [4] W.D. Evans, D.J. Harris, and L. Pick. Weighted Hardy and Poincaré inequalities on trees. Journal of the London Mathematical Society, 52(2):2 36,995. [5] B.L. Granovsky and A.I. Zeifman. The N-limit of spectral gap of random walks associated with complete graphs. Preprint, 998. [6] N. Kahale. A semidefinite bound for mixing rates of Markov chains. Preprint, September 995.

12 [7] C. Meise. On spectral gap estimates of a Markov chain via hitting times and coupling. Preprint, 998. [8] C. Meise. Spectral gap estimates via distance arguments. Preprint, 998. [9] L. Miclo. Relations entre isopérimétrie et trou spectral pour les chaînes de Markov finies. Preprint, 998. [0] B. Muckenhoupt. Hardy s inequality with weights. Studia Mathematica, XLIV:3 38, 972. [] E. Sawyer. Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Transactions of the American Mathematical Society, 28(): ,984. [2] E. Seneta. Non-negative Matrices and Markov Chains. SpringerSeriesinStatistics. Springer- Verlag, 98. [3] A.I. Zeifman. Some estimates of the rate of convergence for birth and death processes. J. Appl. Prob., 28: ,99. 2