We apply Peter–Weyl theory to obtain necessary and sufficient conditions for a probability measure on a compact group to have a square-integrable density. Applications are given to measures on the d-dimensional torus

Applebaum, D.

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promoting access to White Rose research papers    White Rose Research Online eprints@whiterose.ac.uk    Universities of Leeds, Sheffield and York http://eprints.whiterose.ac.uk/    This is an author produced version of a paper published in Bulletin of the London Mathematical Society.    White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/9803     Published paper Applebaum, D. (2008) Probability measures on compact groups which have square-integrable densities, Bulletin of the London Mathematical Society, 40 (6), pp. 1038-1044  http://dx.doi.org/10.1112/blms/bdn088   Probability Measures on Compact Groupswhich have Square-Integrable DensitiesDavid ApplebaumAbstractWe apply Peter-Weyl theory to obtain necessary and sufficientconditions for a probability measure on a compact group to have asquare-integrable density. Applications are given to measures on thed-dimensional torus.MSC 2000: 60B15, 60E07, 43A05, 43A301 IntroductionGiven a probability measure on a Riemannian manifold it is highly beneficialif we can establish that it has a density with respect to the Riemannianvolume measure. This is vital for statistical inference but a density mayalso contain important information about the topology and geometry of themanifold, e.g. the heat kernel is the density of Brownian motion on a manifold(see e.g. [6], [18].) Much work by stochastic analysts has been focussedon finding densities for solutions of stochastic differential equations drivenby Brownian motion. A key result here is Hörmander’s theorem [10] whichgives a sufficient condition (called “hypoellipticity”) for existence of a smoothdensity involving the Lie algebra generated by the driving vector fields. Thesearch for equivalent probabilistic conditions for hypoellipticity led to theMalliavin calculus [17]. Ongoing research is now developing this into a toolto investigate densities of more sophisticated systems such as solutions ofstochastic differential equations driven by non-Gaussian noise (see e.g. [11])and solutions of stochastic partial differential equations (see e.g. [19] andreferences therein.)In this short note we focus on the case of a compact group G. Re-cently Liao (see Chapter 4 in [13] or [14]) has established existence of square-integrable densities for G-valued Lévy processes (i.e. processes with inde-pendent and stationary increments) in the case where G is a Lie group and1the laws are either conjugate invariant or invariant under the inverse map.His approach makes extensive use of Fourier analysis - especially the Pe-ter Weyl theorem - and requires the hypoellipticity assumption to hold forthe diffusion part. More recently, working together with L.Wang [15], hehas obtained densities of convolution semigroups of probability measures onsymmetric spaces. The main purpose of this note is to adapt Liao’s ap-proach to arbitrary probability measures on a compact group. We give fourequivalent necessary and sufficient conditions for such a measure to have asquare-integrable density with respect to Haar measure. These effectivelyrequire summability of squares of matrix elements taken over all irreduciblerepresentations of the associated non-commutative Fourier transform (forbackground on which see [12], section 3.2. in [7], [8], [21] or [2].) This canto some extent be seen as analogous to the well-known result that a prob-ability measure on Euclidean space has a density if its Fourier transform(characteristic function) is absolutely integrable (see e.g. Theorem 3.2.2 in[16].) As an application of our result we are able to show that a large class ofinfinitely divisible probability measures on the d-dimensional torus Πd havean L2-density if they have a non-degenerate Gaussian component. In factthis condition is equivalent to hypoellipticity and this result, together withthose of Liao which were discussed above leads us to conjecture that hypoel-lipticity is a sufficient condition for the existence of a density for an arbitraryinfinitely divisible probability measure on a compact Lie group. In the lastpart of the paper we give some examples of infinitely divisible measures onΠd which have no Gaussian component but still have an L2-density.Notation. If T is a bounded linear operator acting in a complex Hilbertspace H then T ∗ denotes its adjoint. When H is finite-dimensional then Tmay be identified with a complex matrix and T ∗ is the complex conjugate ofthe transpose of T . If z is a complex number then <(z) and =(z) are its realand imaginary parts (respectively).2 PreliminariesLet G be a locally compact group with neutral element e and let Rep(G)be the set of all equivalence classes of unitary representations of G in acomplex Hilbert space. Irr(G) is the subset obtained from all irreduciblerepresentations and those that are non-trivial are denoted by Irr+(G). Wewill, as usual, identify each equivalence class with a representative element.If π ∈ Rep(G), then for each g ∈ G, π(g) is a unitary operator in the Hilbertspace Vπ. The σ-algebra of all Borel sets in G is denoted by B(G). Let M(G)2denote the set of all Borel probability measures on G. It is a monoid withrespect to the binary operation of convolution where the identity element isthe Dirac mass at e. We recall that the convolution µ1 ∗µ2 of µ1, µ2 ∈M(G)is the unique Borel probability measure for which∫Gf(σ)(µ1 ∗ µ2)(dσ) =∫G∫Gf(στ)µ1(dσ)µ2(dτ),for all real-valued bounded Borel functions f defined on G. The reversedmeasure µ̃ that is associated to each µ ∈M(G) is defined by µ̃(A) = µ(A−1)for each A ∈ B(G). Note that µ̃ ∈M(G) and that µ̃1 ∗ µ2 = µ̃2 ∗ µ̃1 for eachµ1, µ2 ∈ M(G). µ ∈ M(G) is said to be symmetric if µ̃ = µ. In particulareach of µ̃ ∗ µ and µ ∗ µ̃ are symmetric for arbitrary µ ∈ M(G), but thesemeasures may be distinct if G is not abelian.Fix µ ∈ M(G). The non-commutative Fourier transform of µ is definedas a Bochner integral in the Banach space of bounded linear operators on Vπby the prescriptionµ̂(π) =∫Gπ(g)µ(dg),for each π ∈ Rep(G). It is easy to see that(F1) µ̂(π) is a contraction on Vπ,(F2) ̂̃µ(π) = µ̂(π)∗,(F3) µ̂1 ∗ µ2(π) = µ̂1(π)µ̂2(π), for each µ1, µ2 ∈M(G).For explicit proofs and further investigations into these ideas see [8], [21],[2].3 Densities on Compact GroupsFrom now on we will take G to be compact. In this case for each π ∈Irr(G), dπ := dim(Vπ) < ∞. Furthermore any Haar measure defined onG is finite and bi-invariant. We choose normalised Haar measure and usethe standard notation dg to denote its “differential”. The corresponding L2space is denoted by L2(G,C). The celebrated Peter-Weyl theorem tells usthat {d12π πij; 1 ≤ i, j ≤ dπ, π ∈ Irr(G)} is a complete orthonormal basis for3L2(G,C) where the co-ordinate functions πij are defined by the prescriptionπij(g) := π(g)ij, for each 1 ≤ i, j ≤ dπ. Hence if f ∈ L2(G,C),f = 〈f〉+∑π∈Irr+(G)dπdπ∑i,j=1〈f, πij〉πij, (3.1)where 〈f〉 := ∫Gf(σ)dσ. Straightforward algebraic manipulation yields thefollowing alternative representation (see e.g. [4]):f = 〈f〉+∑π∈Irr+(G)dπtr(f̂(π)π), (3.2)where f̂(π) :=∫Gf(g)π(g−1)dg.We say that µ ∈ M(G) has a square-integrable density if µ is absolutelycontinuous with respect to Haar measure and the Radon-Nikodým derivativedµdglies in the space L2(G,R). If this is the case, we observe that if we writef := dµdg, thenf̂(π) =∫Gπ(g−1)µ(dg) =∫Gπ(g)µ̃(dg) = ̂̃µ(π).An easy consequence of (3.1) and Parseval’s identity is that||f ||2 = 1 +∑π∈Irr+(G)dπdπ∑i,j=1|µ̂(π)ij|2. (3.3)The main result of this paper is the following:Theorem 3.1 µ ∈ M(G) has a square-integrable density if and only if anyone of the following equivalent conditions is satisfied.(i)∑π∈Irr+(G) dπ∑dπi,j=1 |µ̂(π)ij|2 < ∞,(ii)∑π∈Irr+(G) dπtr(µ̂(π)∗µ̂(π)) < ∞,(iii)∑π∈Irr+(G) dπtr(̂̃µ ∗ µ(π)) < ∞.(iv)∑π∈Irr+(G) dπtr(µ̂ ∗ µ̃(π)) < ∞.If this is the case we havedµdg= 1 +∑π∈Irr+(G)dπtr(µ̂(π)∗π). (3.4)4Proof. The equivalence of (i) and (ii) is easy linear algebra while that of(ii) and (iii) is straightforward from (F2) and (F3). That of (iii) and (iv)follows similarly since for each π ∈ Irr+(G),tr(µ̂ ∗ µ̃(π)) = tr(µ̂(π)µ̂(π)∗) = tr(µ̂(π)∗µ̂(π)) = tr(̂̃µ ∗ µ(π)).Necessity follows from (3.3). For sufficiency we define f := 1+∑π∈Irr+(G) dπtr(µ̂(π)∗π),so that f ∈ L2(G,C). By uniqueness of the Peter-Weyl expansion we havef̂(π) = µ̂(π)∗.By the Peter-Weyl theorem, if g ∈ C(G,C) then its Fourier series (asgiven by (3.1) or (3.2)) converges uniformly (see e.g. [4]). Hence by Plancherel’stheorem for each g ∈ C(G,C),∫Gg(σ)f(σ)dσ =∑π∈Irr(G)dπdπ∑i,j=1〈g, πij〉〈πij, f〉=∑π∈Irr(G)dπtr(ĝ(π)f̂(π)∗)=∑π∈Irr(G)dπtr(ĝ(π)µ̂(π)).However by (3.2) and Fubini’s theorem∫Gg(σ)µ(dσ) =∫G∑π∈Irr(G)dπtr(ĝ(π)π(σ))µ(dσ)=∑π∈Irr(G)dπtr(ĝ(π)µ̂(π)).It follows that ∫Gg(σ)f(σ)dσ =∫Gg(σ)µ(dσ). (3.5)Consequently we have∫Gg(σ)=(f)(σ)dσ = 0 for all g ∈ C(G,C). SinceC(G,C) is dense in L2(G,C), it follows that =(f) = 0 (a.e.). Now assumethat B := f−1((−∞, 0)) has positive Haar measure and choose g ∈ C(G,R)to be strictly negative with supp(g) ⊆ B. Note that such a g can always befound since Haar measure is regular. We then obtain0 <∫Bg(σ)f(σ)dσ =∫Gg(σ)f(σ)dσ =∫Gg(σ)µ(dσ) =∫Bg(σ)µ(dσ) ≤ 0.We have derived a contradiction and hence can deduce that f ≥ 0 (a.e.) Wenow apply the Riesz representation theorem (see e.g. Theorem 7.2.8 in [5])to both sides of (3.5) to deduce that f(σ)dσ = µ(dσ) as was required. ¤54 Infinitely Divisible Measures on the d-dimensionalTorusWe recall that a probability measure on a locally compact group G is infinitelydivisible if it has a convolution nth root for all n ∈ N. We denote the set ofall such measures by ID(G). It is well known (see e.g. [1], [3], [20]) that thereis a one-to-one correspondence between ID(Rd) and the set of all continuous,hermitian, negative definite functions η : Rd → C for which η(0) = 0 via theprescription ∫Rdeiu·xµ(dx) = e−η(u),for each u ∈ Rd. The generic form of η is given by the Lévy-Khinchineformulaη(u) = ib · u + 12u · Au +∫Rd−{0}(1− eiu·y + iu · y1B1(y))ν(dy), (4.6)where b ∈ Rd, A is a non-negative definite symmetric d × d matrix, ν is aBorel measure on Rd − {0} for which ∫Rd−{0}(|y|2 ∧ 1)ν(dy) < ∞ and B1is the open ball of radius one centred at the origin. The triple (b, A, ν) iscalled the characteristics of µ. µ ∈ M(Rd) is Gaussian (in the usual sense)if and only if µ ∈ ID(Rd) with characteristics (b, A, 0) and more generallyan arbitrary µ ∈ ID(Rd) with characteristics (b, A, ν) has a non-degenerateGaussian component if A is non-singular. Now consider the d-dimensionaltorus Πd = Rd/(2πZ)d and let γ : Rd → Πd be the canonical surjection. Ifµ ∈ ID(Rd) then it is easily verified that µγ ∈ ID(Πd) where µγ := µ ◦ γ−1.Any ρ ∈ ID(Πd) which arises in this way will be called projective. There areprobability measures on Πd which are infinitely divisible but not projectivee.g. normalised Haar measure. If µ has a density f then it is easily verifiedthat µγ has a density fγ where for all x ∈ Πd,fγ(x) =∑n∈Zdf(x + 2πn). (4.7)The irreducible representations on Πd are precisely the characters n =(n1, . . . , nd) ∈ Zd and for each µ ∈ ID(Rd), n ∈ Zd we haveµ̂γ(n) =∫Πdein·xµγ(dx) =∫Rdein·yµ(dy) = e−η(n). (4.8)Proposition 4.1 If µ ∈ ID(Rd) has a non-degenerate Gaussian componentthen µγ has an L2 density.6Proof. Using condition (i) in Theorem 3.1 and (4.8), we see that we mustshow that ∑n∈Zd|e−η(n))|2 =∑n∈Zde−2<(η(n)) < ∞.Now by (4.6)<(η(n)) = 12n · An +∫Rd−{0}(1− cos(n · y))ν(dy)≥ 12n · An,and so ∑n∈Zd|e−η(n))|2 ≤∑n∈Zde−n·An ≤∑n∈Zde−λn·n < ∞,where λ > 0 is the minimum eigenvalue of A. ¤Remarks1. If µ satisfies the hypothesis of Proposition 4.1, then it is easily verifiedthat its characteristic function is absolutely integrable and so µ hasa density. Consequently the fact that µγ has a density is immediatefrom (4.7). The new information that we have gained from Proposition4.1 is the square integrability and (3.4) is then the usual Fourier seriesexpansion.2. If µ is a standard Gaussian measure (i.e. A = I), then it is well-knownthat µγ is a product of Jacobi theta functions - indeed this is easilyverified from (4.7) by using the Poisson summation formula (see e.g.[9], p.375).3. The condition that the Gaussian component of µ is non-degenerateis easily seen to be equivalent to hypoellipticity of the vector fields{Y1, . . . , Yd}, where Yi =∑dj=1 Aij∂j, for 1 ≤ i ≤ d. Indeed an alterna-tive proof of Proposition 4.1 could be obtained by making use of thisfact and applying Lemma 4.1 in [13].For an example of a family of infinitely divisible probability measureson Π1 each of which have an L2 density but with no Gaussian component(i.e. A = 0), we take µ ∈ ID(R) to be any α-stable distribution for which0 < α < 2. In this case there exists σ > 0 such that (see e.g. Theorem 14.15in [20])7∑n∈Z|e−η(n))|2 =∑n∈Ze−2σ|n|α= 1 + 2∞∑n=1e−2σnα. (4.9)The convergence of this series follows by comparison with∞∑n=11n2sincelimn→∞ n2e−2σnα= limx→∞ x2α e−2σx = 0.Now suppose that for 0 < α < 2, µα is a rotationally invariant (non-Gaussian) α-stable distribution on Rd. Then (4.6) can be written in the formηα(u) = c||u||α for all u ∈ Rd, where c > 0 and || · || denotes the Euclideannorm. For later reference we note that µα has characteristics (0, 0, να) whereνα(dx) =K||x||α+d for some K > 0 (see e.g. section 3.14 in [20].) We can nowshow that µγα has an L2-density on Πd, indeed by Hölder’s inequality we have∑n∈Zde−2c||n||α ≤∑n∈Zdexp{−κd∑i=1|ni|α}=(∑m∈Ze−κ|m|α)d< ∞ (4.10)by the remarks following (4.9) where κ := 2cd22−α .We close this paper with a result that allows us to obtain L2-densitiesfor a wider class of infinitely divisible distributions on Πd. First we need adefinition. Let ν1 and ν2 be Lévy measures on Rd − {0}. We say that ν1dominates ν2 if there exists k > 0 such thatν1(U) ≥ kν2(U)for all Borel sets U which are bounded away from zero in the sense that0 /∈ U . For a simple example we take ν2 ≡ να and ν1(dx) = h(x)να(dx)where h(x) ={ec||x|| if ||x|| < 11 if ||x|| ≥ 1 and c > 0. In this case we may take0 < k ≤ 1.The idea for the following result was suggested to the author by MingLiao.Theorem 4.1 Let µ be an infinitely divisible distribution on Rd with char-acteristics (b, A, ν). If ν dominates να for some 0 < α < 2 then µγ has asquare-integrable density on Πd.Proof. Let (Um,m ∈ N) be an increasing sequence of Borel sets that arebounded away from zero with Um ↑ Rd − {0} as m →∞.8Using (4.6) we have for each m ∈ N, n ∈ Zd,<(η(n)) = 12n · An +∫Um(1− cos(n · x))ν(dx) +∫Rd−{0}−Um(1− cos(n · x))ν(dx)≥∫Um(1− cos(n · x))ν(dx)≥ k∫Um(1− cos(n · x))να(dx).Taking limits as m →∞, we deduce that<(η(n)) ≥ k∫Rd−{0}(1− cos(n · x))να(dx) = kηα(n).The required result now follows from (4.10). ¤We remark that the verification of Theorem 3.1(i) for the case of measureson SU(2) may at least in principle be carried out using the fact that all matrixelements of irreducible representations can be expressed in terms of Jacobipolynomials (see e.g. [22], p.97).References[1] D.Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Uni-versity Press (2004)[2] D.Applebaum, Lévy processes in Euclidean spaces and groups in Quan-tum Independent Increment Processes I: From Classical Probability toQuantum Stochastic Calculus, Springer Lecture Notes in Mathematics ,Vol. 1865 M Schurmann, U. Franz (Eds.)(2005), 1-99[3] C. Berg, G. Forst, Potential Theory on Locally Compact Abelian Groups,Springer-Verlag (1975).[4] T. Bröcker, T. tom Dieck, Representations of Compact Lie Groups,Springer-Verlag New York (1985)[5] D. L. Cohn, Measure Theory, Birkhaüser (1980)[6] K. D. Elworthy, Geometric aspects of diffusions on manifolds. In Ecoled’Eté de Probabilités de St Flour XVII, ed. P. L. Hennequin, LectureNotes in Mathematics 1362, Springer-Verlag (1988), 276-4259[7] U.Grenander, Probabilities on Algebraic Structures, John Wiley andSons Inc (1963)[8] H.Heyer, L’analyse de Fourier non-commutative et applications àla théorie des probabilités, Ann. Inst. Henri Poincaré (Prob. Stat.)4(1968), 143-68[9] H.Heyer, Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin-Heidelberg (1977)[10] L.Hörmander, Hypoelliptic second order differential operators, ActaMath. 119(1967), 147-71[11] Y.Ishikawa, H.Kunita, Malliavin calculus on the Wiener-Poisson spaceand its application to canonical SDE with jumps, Stoch. Proc. Applns116(2006), 1743-69[12] Y.Kawada, K. Itô, On the probability distribution on a compact groupI, Proc. Phys.-Mat. Soc. Japan 22 (1940), 977-98[13] M.Liao, Lévy Processes in Lie Groups, Cambridge University Press(2004).[14] M.Liao, Lévy processes and Fourier analysis on compact Lie groups,Ann. Prob. 32 (2004), 1553-73[15] M.Liao, L.Wang, Lévy-Khinchine formula and existence of densitiesfor convolution semigroups on symmetric spaces, Potential Analysis 27(2007), 133-50[16] E.Lukacs, Characteristic Functions (second edition) C.Griffin and Co.Ltd London (1970)[17] P.Malliavin, Stochastic calculus of variations and hypoelliptic operators,in Proc. Inter. Symposium on SDE, Kyoto 1976, 195-263 J. Wiley andSons (1978)[18] S.Rosenberg, The Laplacian on a Riemannian Manifold, CambridgeUniversity Press (1997)[19] M.Sanz-Solé, Malliavin Calculus with Applications to Stochastic PartialDifferential Equations EPFL Press Lausanne, distributed by CRC Press(2005)10[20] K-I.Sato, Lévy Processes and Infinite Divisibility, Cambridge UniversityPress (1999)[21] E.Siebert, Fourier analysis and limit theorems for convolution semi-groups on a locally compact group, Advances in Math. 39 (1981), 111-54[22] D.P. Želobenko, Compact Lie Groups and their Representations, Trans-lations of Mathematical Monographs Volume 40, American Math. Soc.Providence, Rhode Island (1973)Acknowledgements I would like to thank Boguslaw Zegarlinski whoinvited me to lecture on Lévy processes on Lie groups at Imperial CollegeLondon in the summer of 2007. This revived my interest in this subject andled directly to this work. I would also like to thank Ming Liao for someextremely valuable suggestions, Wilfrid Kendall for a useful observation andthe referee for very helpful comments.Probability and Statistics Department,University of Sheffield,Hicks Building, Hounsfield Road,Sheffield, England, S3 7RHe-mail: D.Applebaum@sheffield.ac.uk11

Probability measures on compact groups which have square-integrable densities

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Probability measures on compact groups which have square-integrable densities

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