Fourier transform of a gaussian measure on the Heisenberg group

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www.elsevier.com/locate/anihpb

Fourier transform of a Gaussian measure on the Heisenberg group

Mátyás Barczy, Gyula Pap

^∗

Faculty of Informatics, University of Debrecen, Pf. 12, 4010 Debrecen, Hungary Received 5 July 2004; received in revised form 8 July 2005; accepted 12 July 2005

Available online 7 December 2005

Abstract

An explicit formula is derived for the Fourier transform of a Gaussian measure on the Heisenberg group at the Schrödinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two Gaussian measures to be a Gaussian measure.

Résumé

Une formule explicite est donnée pour la transformée de Fourier de la mesure gaussienne sur le groupe d’Heisenberg dans la représentation de Schrödinger. A l’aide de celle-ci, des conditions nécessaires et suffisantes sont données pour que la convolée de deux mesures gaussiennes soit gaussienne.

MSC:60B15

Keywords:Heisenberg group; Fourier transforms; Gaussian measures

1. Introduction

Fourier transforms of probability measures on a locally compact topological group play an important role in several problems concerning convolution and weak convergence of probability measures. Indeed, the Fourier transform of the convolution of two probability measures is the product of their Fourier transforms, and in case of many groups the continuity theorem holds, namely, weak convergence of probability measures is equivalent to the pointwise convergence of their Fourier transforms. Moreover, the Fourier transform is injective, i.e., if the Fourier transforms of two probability measures coincide at each point then the measures coincide. (See the properties of the Fourier transform, e.g., in Heyer [7, Chapter I].) In case of a locally compact Abelian group, an explicit formula is available for the Fourier transform of an arbitrary infinitely divisible probability measure (see Parthatsarathy [11]). The case of non-Abelian groups is much more complicated. For Lie groups, Tomé [16] proposed a method how to calculate Fourier transforms based on Feynman’s path integral and discussed the physical motivation, but explicit expressions have been derived only in very special cases.

* Corresponding author.

E-mail addresses:[email protected] (M. Barczy), [email protected] (G. Pap).

doi:10.1016/j.anihpb.2005.07.002

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In this paper Gaussian measures will be investigated on the 3-dimensional Heisenberg groupH which can be obtained by furnishingR³with its natural topology and with the product

(g₁, g₂, g₃)(h₁, h₂, h₃)=

g₁+h₁, g₂+h₂, g₃+h₃+1

2(g₁h₂−g₂h₁)

.

The Schrödinger representations {π_±_λ: λ >0} of H are representations in the group of unitary operators of the complex Hilbert spaceL²(R)given by

π_±_λ(g)u

(x):=e^±^i(λg³⁺

√λg2x+λg1g2/2)u x+√

λg₁

(1.1) forg=(g₁, g₂, g₃)∈H,u∈L²(R)andx∈R. The value of the Fourier transform of a probability measureμonH at the Schrödinger representationπ_±_λis the bounded linear operatorμ(πˆ _±_λ):L²(R)→L²(R)given by

ˆ

μ(π_±_λ)u:=

H

π_±_λ(g)uμ(dg), u∈L²(R), interpreted as a Bochner integral.

Let(μ_t)_t₀ be a Gaussian convolution semigroup of probability measures onH(see Section 2). By a result of Siebert [13, Proposition 3.1, Lemma 3.1],(μ_t(π_±_λ))_t₀is a strongly continuous semigroup of contractions onL²(R) with infinitesimal generator

N (π_±_λ)=α₁I+α₂x+α₃D+α₄x²+α₅(xD+Dx)+α₆D²,

whereα₁, . . . , α₆are certain complex numbers (depending on(μ_t)_t₀, see Remark 3.1),I denotes the identity operator onL²(R),xis the multiplication by the variablex, andDu(x)=u(x). One of our purposes is to determine the action of the operators

μt(π_±λ)=e^{t N (π}^±^λ⁾, t0,

onL²(R). (Here the notation(e^{t A})t0means a semigroup of operators with infinitesimal generatorA.) WhenN (π_±λ) has the special form¹₂(D²−x²), the celebrated Mehler’s formula gives us

e^{t (D}²⁻^x²^)/2u(x)= 1

√2πsinht

R

exp

−(x²+y²)cosht−2xy 2 sinht

u(y)dy

for allt >0, u∈L²(R)andx∈R, see, e.g., Taylor [15], Davies [4]. Our Theorem 3.1 in Section 3 can be regarded as a generalization of Mehler’s formula.

It turns out thatμ_t(π_±_λ)=e^{t N (π}^±^λ⁾, t0 are again integral operators onL²(R)ifα₆is a positive real number. One of the main results of the present paper is an explicit formula for the kernel function of these integral operators (see Theorem 3.1). We apply a probabilistic method using that the Fourier transformμ(πˆ _±_λ)of an absolutely continuous probability measureμonHcan be derived from the Euclidean Fourier transform ofμconsideringμas a measure onR³(see Proposition 4.1). We note that a random walk approach might provide a different proof of Theorem 3.1, but we think that it would not be simpler than ours.

The second part of the paper deals with convolutions of Gaussian measures onH. The convolution of two Gaussian measures on a locally compact Abelian group is again a Gaussian measure (it can be proved by the help of Fourier transforms; see Parthatsarathy [11]). We prove that a convolution of Gaussian measures on H is almost never a Gaussian measure. More exactly, we obtain the following result (using our explicit formula for the Fourier transforms).

Theorem 1.1.Letμandμbe Gaussian measures onH. Then the convolutionμ∗μis a Gaussian measure onH if and only if one of the following conditions holds:

(C1) there exist elementsY₀, Y₀, Y₁, Y₂in the Lie algebra ofHsuch that[Y₁, Y₂] =0, the support ofμis contained inexp{Y₀+R·Y₁+R·Y₂}and the support ofμis contained inexp{Y₀+R·Y₁+R·Y₂}.(Equivalently, there exists an Abelian subgroupGofHsuch thatsupp(μ)andsupp(μ)are contained in “Euclidean cosets”

ofG.)

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(C2) there exist a Gaussian semigroup(μ_t)_t₀andt, t0and a Gaussian measureνsuch thatsupp(ν)is con- tained in the center ofHand eitherμ=μ_t,μ=μ_t∗νorμ=μ_t∗ν,μ=μ_tholds.(Equivalently,μ andμare sitting on the same Gaussian semigroup modulo a Gaussian measure with support contained in the center ofH.)

We note that in case of (C1),μandμare Gaussian measures also in the “Euclidean sense” (i.e., considering them as measures onR³). Moreover, Theorem 6.1 contains an explicit formula for the Fourier transform of a convolution of arbitrary Gaussian measures onH.

The structure of the present work is similar to Pap [10]. Theorems 1.1 and 3.1 of the present paper are generaliza- tions of the corresponding results for symmetric Gaussian measures onHdue to Pap [10]. We summarize briefly the new ingredients needed in the present paper. Comparing Lemma 6.1 in Pap [10] and Proposition 5.1 of the present paper, one can realize that now we have to calculate a much more complicated (Euclidean) Fourier transform (see (5.5)).

For this reason we generalized a result due to Chaleyat-Maurel [3] (see Lemma 5.2). We note that using Lemma 6.2 one can easily derive Theorem 1.1 in Pap [10] from Theorem 1.1 of the present paper.

It is natural to ask whether we can prove our results for non-symmetric Gaussian measures using only the results for symmetric Gaussian measures. The answer is no. The reason for this is that in case ofHthe convolution of a symmetric Gaussian measure and a Dirac measure is in general not a Gaussian measure. For example, ifa=(1,0,0)∈Hand (μ_t)_t₀is a Gaussian semigroup with infinitesimal generatorX₁²+X²₂, then using Lemma 4.2, one can easily check thatμ₁∗ε_ais not a Gaussian measure onH,whereε_adenotes the Dirac measure concentrated on the elementa∈H. (For the definition of an infinitesimal generator andX₁, X₂, X₃,see Section 2.)

We note that if the convolution of two Gaussian measures on H is again a Gaussian measure on H, then the corresponding infinitesimal generators not necessarily commute, nor even if the infinitesimal generator corresponding to the convolution is the sum of the original infinitesimal generators. Now we give an illuminating counterexample. Let μandμbe Gaussian measures onHsuch that the corresponding Gaussian semigroups have infinitesimal generators

N=1

2(X1+X2)² and N=1

2(X1+X2)²+X1X3, respectively.

Using Theorem 6.2 and Lemma 6.2, μ∗μ is a symmetric Gaussian measure onHsuch that the corresponding Gaussian semigroup has infinitesimal generatorN+N.ButNandNdo not commute. Indeed,NN−NN=

−(X₁+X₂)X²₃=0.

At the end of our paper we formulate Theorem 1.1 in the important special case of centered Gaussian measures for which the corresponding Gaussian semigroups are stable in the sense of Hazod. This kind of Gaussian measures arises in a standard version of central limit theorems onHproved by Wehn [17]. In this special case Theorem 1.1 can be derived from the results for symmetric Gaussian measures in Pap [10].

2. Preliminaries

The Heisenberg groupHis a Lie group with Lie algebraH, which can be realized as the vector spaceR³furnished with multiplication

(p₁, p₂, p₃), (q₁, q₂, q₃)

=(0,0, p1q₂−p₂q₁).

An elementX∈Hcan be regarded as a left-invariant differential operator onH, namely, for continuously differentiable functionsf:H→Rwe put

Xf (g):=lim

t→0t⁻¹ f

gexp(t X)

−f (g)

, g∈H,

where the exponential mapping exp :H→His now the identity mapping.

A family (μ_t)_t₀ of probability measures onHis said to be a (continuous)convolution semigroupif we have μs∗μt=μs+t for alls, t0, and limt↓0μt=μ0=εe, wheree=(0,0,0)is the unit element ofH.Itsinfinitesimal generatoris defined by

(Nf )(g):=lim

t↓0t⁻¹

H

f (gh)−f (g)

μ_t(dh), g∈H,

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for suitable functionsf:H→R. (The infinitesimal generator is always defined for infinitely differentiable func- tions f:H→R with compact support.) A convolution semigroup (μ_t)_t₀ is called a Gaussian semigroup if limt↓0t⁻¹μ_t(H\U )=0 for all (Borel) neighbourhoodsUofe. Let{X₁, X₂, X₃}denote the natural basis inH(that is, expX₁=(1,0,0), expX₂=(0,1,0)and expX₃=(0,0,1)). It is known that a convolution semigroup(μ_t)_t₀is a Gaussian semigroup if and only if its infinitesimal generator has the form

N= 3 k=1

a_kX_k+1 2

3 j=1

3 k=1

b_j,kX_jX_k, (2.1)

wherea=(a₁, a₂, a₃)∈R³ andB=(b_j,k)₁_j,k₃is a real, symmetric, positive semidefinite matrix. A probability measureμ onHis called aGaussian measure if there exists a Gaussian semigroup(μ_t)_t₀such thatμ=μ1. A Gaussian measure on H can be embedded only in a uniquely determined Gaussian semigroup (see Baldi [2], Pap [9]). (Neuenschwander [8] showed that a Gaussian measure onHcan not be embedded in a non-Gaussian convolution semigroup.) Thus for a vector a=(a₁, a₂, a₃)∈R³ and a real, symmetric, positive semidefinite matrix B=(b_j,k)₁_j,k₃we can speak about the Gaussian measureμwith parameters(a, B)which is by definitionμ:=μ₁, where(μ_t)_t₀is the Gaussian semigroup with infinitesimal generatorN given by (2.1). Ifν is a Gaussian measure with parameters(a, B)and(ν_s)_s₀is the Gaussian semigroup with infinitesimal generatorNgiven by (2.1) thenν_tis a Gaussian measure with parameters(t a, t B)for allt0, sinceμ_s:=ν_st,s0 defines a Gaussian semigroup with infinitesimal generatort N. Henceν_t=μ₁, so it will be sufficient to calculate the Fourier transform ofμ₁.

Let us consider a Gaussian semigroup(μ_t)_t₀with parameters(a, B)onH. Its infinitesimal generatorN can be also written in the form

N=Y₀+1 2

d j=1

Y_j², (2.2)

where 0d3 and Y0=

3 k=1

akXk, Yj= 3 k=1

σk,jXk, 1jd,

whereΣ=(σ_k,j)is a 3×d matrix with rank(Σ )=rank(B)=d. Moreover,B=Σ·Σ . Then the measureμ_t can be described as the distribution of the random vectorZ(t )=(Z1(t ), Z2(t ), Z3(t ))with values inR³, where

Z1(t )=a1t+ d k=1

σ1,kW_k(t ), Z2(t )=a2t+ d k=1

σ2,kW_k(t ),

Z3(t )=a3t+ d k=1

σ3,kWk(t )+1 2

t 0

Z1(s)dZ2(s)−Z2(s)dZ1(s)

=a3t+ d k=1

σ3,kW_k(t )+ d k=1

(a2σ1,k−a1σ2,k)W_k^∗(t )+

1k< d

(σ1,kσ2, −σ1, σ2,k)W_k,(t ), where(W₁(t ), . . . , W_d(t ))_t₀is a standard Wiener process inR^dand

W_k^∗(t ):=1 2

t 0

W_k(s)ds− t 0

sdWk(s)

,

W_k,(t ):=1 2

t 0

W_k(s)dW (s)− t 0

W (s)dWk(s)

.

(See, e.g., Roynette [12].) The process(W_k,(t ))_t₀ is the so-called Lévy’s stochastic area swept by the process (W_k(s), W (s))_s_∈[_0,t_]onR².

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3. Fourier transform of a Gaussian measure

The Schrödinger representations are infinite dimensional, irreducible, unitary representations, and each irreducible, unitary representation is unitarily equivalent with one of the Schrödinger representations or with χ_α,β for some α, β∈R, whereχα,βis a one-dimensional representation given by

χ_α,β(g):=e^i(αg¹⁺^βg²⁾, g=(g₁, g₂, g₃)∈H.

The value of theFourier transformof a probability measureμonHat the representationχ_α,βis ˆ

μ(χ_α,β):=

H

χ_α,β(g)μ(dg)=

H

e^i(αg¹⁺^βg²⁾μ(dg)= ˜μ(α, β,0),

whereμ˜:R³→Cdenotes theEuclidean Fourier transformofμ,

˜

μ(α, β, γ ):=

H

e^i(αg¹⁺^βg²⁺^{γ g}³⁾μ(dg).

Let us consider a Gaussian semigroup(μ_t)_t₀ with parameters(a, B)onH. The Fourier transform ofμ:=μ₁ at the one-dimensional representations can be calculated easily, since the description of(μ_t)_t₀given in Section 2 implies that

ˆ

μ(χ_α,β)=^Eexp

i(αa1+βa₂)+i

α d k=1

σ_1,kW_k(1)+β d k=1

σ_2,kW_k(1)

forα, β∈R. The random variable _d

k=1

σ1,kW_k(1), d k=1

σ2,kW_k(1)

has a normal distribution with zero mean and covariance matrix σ_1,1 . . . σ_1,d

σ_2,1 . . . σ_2,d ⎡⎣

σ_1,1 σ_2,1 ... ... σ_1,d σ_2,d

⎤

⎦=

b_1,1 b_1,2 b_2,1 b_2,2

,

sinceΣ Σ =B. Consequently, ˆ

μ(χ_α,β)=exp

i(αa1+βa₂)−1 2

b_1,1α²+2b1,2αβ+b_2,2β² .

One of the main results of the present paper is an explicit formula for the Fourier transform of a Gaussian measure on the Heisenberg groupHat the Schrödinger representations.

Theorem 3.1.Letμbe a Gaussian measure onHwith parameters(a, B). Then μ(πˆ _±_λ)u

(x)=

⎧⎪

⎪⎨

⎪⎪

⎩

R

K_±_λ(x, y)u(y)dy ifb_1,1>0, L_±_λ(x)u

x+√ λa₁

ifb_1,1=0, foru∈L²(R),x∈R, where

K_±_λ(x, y):=C_±_λ(B)exp

−1

2z D_±_λ(a, B)z

, z:=(x, y,1) ,

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where, withδ:=

b1,1b2,2−b²_1,2,δ1:=b1,1b2,3−b1,2b1,3,δ2:=a1b1,2−a2b1,1,

C_±_λ(B):=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ 1

2π λb1,1

ifδ=0,

δ

2π b1,1sinh(λδ) ifδ >0,

andD_±_λ(a, B)=(d_j,k^±^λ(a, B))₁_j,k₃are symmetric matrices defined forb_1,1>0andδ=0by d_1,1^±^λ(a, B):=λ⁻¹±ib1,2

b_1,1 , d_1,2^±^λ(a, B):= − 1

λb_1,1, d_2,2^±^λ(a, B):=λ⁻¹∓ib1,2

b_1,1 , d_1,3^±^λ(a, B):=a1±iλb1,3

√λb1,1

±i

√λδ2

2b1,1

, d_2,3^±^λ(a, B):= −a1±iλb1,3

√λb1,1

±i

√λδ2

2b1,1

, d_3,3^±^λ(a, B):=(a1±iλb1,3)²

b1,1 + λ²δ₂²

12b1,1+λ²b_3,3∓2iλa3, and forδ >0by

d_1,1^±^λ(a, B):=δcoth(λδ)±ib1,2

b_1,1 , d_1,2^±^λ(a, B):= − δ

b_1,1sinh(λδ), d_2,2^±^λ(a, B):=δcoth(λδ)∓ib1,2

b_1,1 , d_1,3^±^λ(a, B):=a₁±iλb1,3

√λb1,1

+ λδ₁±iδ2

√λb1,1δcoth(λδ/2), d_2,3^±^λ(a, B):= −a₁±iλb1,3

√λb1,1

+ λδ₁±iδ2

√λb1,1δcoth(λδ/2),

d_3,3^±^λ(a, B):=(a1±iλb1,3)²

b1,1 −(λδ1±iδ2)² λb_1,1δ³

λδ−2 tanh(λδ/2)

+λ²b_3,3∓2iλa3, and

L_±λ(x):=exp

±i√ λ 2

√

λ(2a3+a1a2)+2a2x

−λ² 6

3b3,3+3a1b2,3+a²₁b2,2

−λ^3/2

2 (2b2,3+a₁b_2,2)x−λ 2b_2,2x²

. We prove this theorem in Section 5.

Remark 3.1. Consider a Gaussian convolution semigroup (μt)t0 with infinitesimal generator N given in (2.1).

Siebert [13, Proposition 3.1, Lemma 3.1] proved that(μ_t(π_±_λ))_t₀is a strongly continuous semigroup of contractions onL²(R)with infinitesimal generator

N (π_±λ)= 3 k=1

akXk(π_±λ)+1 2

3 j=1

3 k=1

bj,kXj(π_±λ)Xk(π_±λ), where

X(π_±_λ)u:=lim

t→0t⁻¹ π_±_λ

exp(t X) u−u for all differentiable vectorsu. Hence

X1(π_±λ)u (x)=√

λu(x)=√

λDu(x), X2(π_±λ)u

(x)= ±i√ λxu(x), X₃(π_±_λ)u

(x)= ±iλu(x)

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for allx∈R. Consequently,

N (π_±λ)=α1I+α2x+α3D+α4x²+α5(xD+Dx)+α6D², where

α₁= −1

2λ²b_3,3±iλa3, α₂= −λ^3/2b_2,3±iλ^1/2a₂, α₃=λ^1/2a₁±iλ^3/2b_1,3, α₄= −1

2λb_2,2, α₅= ±i

2λb_1,2, α₆=1 2λb_1,1.

4. Absolute continuity and singularity of Gaussian measures

A probability measure μ onH is said to be absolutely continuous or singular if it is absolutely continuous or singular with respect to the normalized Haar measure onHwhich coincides with the Lebesgue measure onR³. The following proposition is the same as Proposition 2.1 in Pap [10]. But the proof given here is simpler, we do not use Weyl calculus.

Proposition 4.1.Ifμis absolutely continuous with densityf then the Fourier transformμ(πˆ _±_λ)is an integral oper- ator onL²(R),

μ(πˆ _±_λ)u (x)=

R

K_±_λ(x, y)u(y)dy

with kernel functionK_±_λ:R²→Cgiven by K_±λ(x, y):= 1

√λf˜2,3

y−x

√λ ,±√ λ

y+x 2

,±λ

, where

f˜_2,3(s₁,s˜₂,s˜₃):=

R²

eⁱ⁽^s^˜²^s²^+˜^s³^s³⁾f (s₁, s₂, s₃)ds2ds3, (s₁,s˜₂,s˜₃)∈R³

denotes a partial Euclidean Fourier transform off.

Proof. Using the definition of the Schrödinger representation we obtain μ(πˆ _±λ)u

(x)=

R³

e^±^i(λs³⁺

√λs2x+λs1s2/2)u(x+√

λs1)f (s1, s2, s3)ds1ds2ds3

= 1

√λ

R³

e^±^i(λs³⁺

√λs₂x+√

λ(y−x)s₂/2)u(y)f y−x

√λ , s2, s3

dyds2ds3

=

R

K_±λ(x, y)u(y)dy, where

K_±λ(x, y)= 1

√λ

R²

e^±^i(λs³⁺

√λ(x+y)s2/2)f y−x

√λ , s2, s3

ds2ds3

= 1

√λf˜2,3

y−x

√λ ,±√ λ

y+x 2

,±λ

. Hence the assertion. 2

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The partial Euclidean Fourier transformf˜_2,3can be obtained by the inverse Euclidean Fourier transform:

f˜_2,3

s₁,s˜₂,s˜₃

= 1 2π

R

e⁻^is¹^s^˜¹f˜

˜

s₁,s˜₂,s˜₃ d˜s₁,

s₁,s˜₂,s˜₃

∈R³, (4.1)

wheref˜denotes the (full) Euclidean Fourier transform off: f˜

˜

s₁,s˜₂,s˜₃ :=

R³

eⁱ⁽^s^˜¹^s¹^+˜^s²^s²^+˜^s³^s³⁾f (s₁, s₂, s₃)ds1ds2ds3

for(s˜1,s˜2,s˜3)∈R³. Moreover,μ(πˆ _±_λ)is a compact operator. If the densityf ofμbelongs to the Schwarz space then ˆ

μ(π_±λ)is a trace class (i.e., nuclear) operator.

In order to apply Proposition 4.1 we shall need the description of the set of absolutely continuous Gaussian measures onH. Using a general result due to Siebert [14, Theorem 2] one can prove the following lemma as in Pap [10, Lemma 3.3].

Lemma 4.1.A Gaussian measureμonHwith parameters(a, B)is either absolutely continuous or singular. More precisely,μis absolutely continuous ifb_1,1b_2,2−b²_1,2>0and singular ifb_1,1b_2,2−b²_1,2=0.

The next lemma describes the support of a Gaussian measure onH.

Lemma 4.2.Let(μt)t0be a Gaussian semigroup onHwith infinitesimal generatorN given by(2.2). According to the structure ofN we can distinguish five different types of Gaussian semigroups:

(i) N=Y₀+¹₂(Y₁²+Y₂²+Y₃²)withY₁,Y₂andY₃linearly independent. Then the semigroup is absolutely continuous andsupp(μt)=Hfor allt >0. Moreover,rank(B)=3,b_1,1b_2,2−b²_1,2=0.

(ii) N=Y₀+¹₂(Y₁²+Y₂²)withY₁andY₂linearly independent and[Y₁, Y₂] =0. Then the semigroup is absolutely continuous andsupp(μt)=Hfor allt >0. Moreover,rank(B)=2,b_1,1b_2,2−b_1,2² =0.

(iii) N=Y₀+¹₂(Y₁²+Y₂²)withY₁andY₂linearly independent and[Y₁, Y₂] =0. Then the semigroup is singular, it is a Gaussian semigroup onR³ as well, and it is supported by a ‘Euclidean coset’ of the same closed normal subgroup, namely,

supp(μt)=exp(t Y0+R·Y₁+R·Y₂)

for allt >0. Moreover,rank(B)=2,b_1,1b_2,2−b²_1,2=0.

(iv) N=Y₀+¹₂Y₁². Then the semigroup is singular, it is a Gaussian semigroup onR³as well, and it is supported by a “Euclidean coset” of the same closed normal subgroup, namely,

supp(μt)=exp

t Y₀+R·Y₁+R· [Y₀, Y₁] for allt >0. Moreover,rank(B)=1,b_1,1b_2,2−b²_1,2=0.

(v) N=Y₀. Then the semigroup is singular and consists of point measures, namely,μ_t=ε_{exp(t Y}₀₎for allt0.

Proof. From general results due to Siebert [14, Theorem 2 and Theorem 4], it follows that a Gaussian measureμon His absolutely continuous if and only ifG:=L(Yi,[Yj, Yk]: 1id, 0j kd)=R³, whereL(·)denotes the linear hull of the given vectors, andYi∈H, 0idare described in (2.2). Moreover, the support ofμt is

supp(μt)= ∞ n=1

Mexp

t Y₀ n

n

for allt >0,

whereMis the analytic subgroup ofHcorresponding to the Lie subalgebra generated by{Y_i: 1ir}and the bar denotes the closure inH. Clearly[Y_i, Y_i] =0,[Y_i, Y_j] =(σ_1,iσ_2,j−σ_1,jσ_2,i) X₃for 1i < jdand[Y, Z] ∈L(X₃) for allY, Z∈H.

We prove only the cases (iii) and (iv), the other cases can be proved similarly.

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In case of (iii) we have G=L(Y₁, Y₂,[Y₀, Y₁],[Y₀, Y₂]). Clearly[Y₀, Y₁],[Y₀, Y₂] ∈L(X₃). Since [Y₁, Y₂] =0, we have σ_1,1σ_2,2−σ_1,2² =0, so Y₁ and Y₂ are linearly dependent in their first two coordinates, thus their linear independence yieldsX3∈L(Y1, Y2). SoG=L(Y1, Y2)=R³, i.e., the semigroup(μt)t0is singular.

To obtain the formula for the support ofμtit is sufficient to prove that(Mexp(_n^tY0))ⁿ=exp(t Y0+R·Y1+R·Y2) for allt >0 andn∈N, where nowM=exp(R·Y1+R·Y2). The multiplication inHcan be reconstructed by the help of the Campbell–Haussdorf formula

exp(X)exp(Y )=exp

X+Y+1 2[X, Y]

, X, Y∈H.

Applying induction byngives the assertion. Indeed, forn=1 we haveMexp(t Y0)=exp(R·Y₁+R·Y₂)exp(t Y0)= exp(t Y0+R·Y₁+R·Y₂), sinceX₃∈L(Y₁, Y₂). Suppose that(Mexp(_n₋^t₁Y₀))ⁿ⁻¹=exp(t Y0+R·Y₁+R·Y₂) holds. Using the Campbell–Haussdorf formula and the induction hypothesis we get(Mexp(_n^tY₀))ⁿ=exp(ⁿ⁻_n¹t Y₀+ R·Y₁+R·Y₂)exp(_n^tY₀+R·Y₁+R·Y₂). SinceX₃∈L(Y₁, Y₂)and[Y, Z] ∈L(X₃)for allY, Z∈H, application of the Campbell–Haussdorf formula once more gives the assertion.

The case (iv) can be obtained similarly. Indeed, we have G=L(Y₁,[Y₀, Y₁])=R³, M=exp(R·Y₁), hence supp(μt)=exp(t Y0+R·Y₁+R· [Y₁, Y₀])for allt >0. 2

5. Euclidean Fourier transform of a Gaussian measure and the proof of Theorem 3.1

Now we investigate the processes (W_k^∗(t ))_t₀ and(W_k,(t ))_t₀ (defined in Section 2). Lett >0 be fixed. We prove thatW_k^∗(t )andW_k,(t )can be constructed by the help of infinitely many independent identically distributed real random variables with standard normal distribution. Because of the self-similarity property of the Wiener process it is sufficient to prove the caset=2π.

Lemma 5.1.Let(W₁(s), . . . , W_d(s))_s_∈[_0,2π_]be a standard Wiener process inR^don a probability space(Ω,A, P ). Let us consider the orthonormal basisf_n(s)=(2π )⁻^1/2e^ins,s∈ [0,2π],n∈Zin the complex Hilbert spaceL²([0,2π]).

If(g(s))_s_∈[_0,2π_]is an adapted, measurable, complex valued process, independent of(W₁(s), . . . , W_d(s))_s_∈[_0,2π_]such thatE(_2π

0 |g(s)|²ds) <∞then 2π

0

g(s)dWj(s)=

n∈Z

g, fn 2π 0

fn(s)dWj(s) a.s., j=1, . . . , d, (5.1)

where·,·denotes the inner product inL²([0,2π])and the convergence of the series on the right-hand side of (5.1) is meant inL²(Ω,A, P ).

Proof. Let 1j d be arbitrary, but fixed. First we prove that the right-hand side of (5.1) is convergent in L²(Ω,A, P ).Using that the processes(g(s))_s_∈[_0,2π_] and(W₁(s), . . . , W_d(s))_s_∈[_0,2π_]are independent of each other, forn, m∈Z, n=m,we get

E

g, f_n 2π 0

f_n(s)dWj(s)g, f_m 2π 0

f_m(s)dWj(s)

=^E

g, fng, fm E

2π 0

fn(s)dWj(s) 2π 0

fm(s)dWj(s)

=^E

g, f_ng, f_m^2π

0

f_n(s)f_m(s)ds=0.

Using again the independence of(g(s))_s_∈[_0,2π_]and(W₁(s), . . . , W_d(s))_s_∈[_0,2π_], we have

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E g, f_n

2π 0

f_n(s)dWj(s)

²=^E g, f_n ²E ^2π

0

f_n(s)dWj(s) ²

=^E g, f_n ²^2π

0

f_n(s) ²ds=^E g, f_n ². SinceE(_2π

0 |g(s)|²ds) <∞, Parseval’s identity inL²([0,2π])gives us that

n∈Z

g, f_n ²= 2π 0

g(s) ²ds a.s.

This implies that

n∈Z

E g, f_n ²=^E 2π 0

g(s) ²ds <∞.

Hence the right-hand side of (5.1) is convergent inL²(Ω,A, P ).

We show now that E

^2π

0

g(s)dWj(s)−

n∈Z

g, f_n 2π 0

f_n(s)dWj(s) ²=0,

which implies (5.1). We have E

^2π

0

g(s)dWj(s)−

n∈Z

g, f_n 2π 0

f_n(s)dWj(s) ²=^E

^2π

0

g(s)dWj(s) ²+^E

n∈Z

g, f_n 2π 0

f_n(s)dWj(s) ²

−2 ReE 2π

0

g(s)dWj(s)

n∈Z

g, f_n 2π 0

f_n(s)dWj(s)

=:A₁+A₂−2 ReA₃. Then we get

A1=^E 2π 0

g(s) ²ds,

A2=

n∈Z E

g, fn 2π 0

fn(s)dWj(s) ²=

n∈Z

E g, fn ²=^E 2π 0

g(s) ²ds,

A₃=

n∈Z E

2π 0

g(s)dWj(s)g, f_n 2π 0

f_n(s)dWj(s)

.

Let us denote theσ-algebra generated by the process(g(s))_s_∈[_0,2π_]byF(g). Then we obtain A₃=

n∈Z E E

2π 0

g(s)dWj(s)g, f_n 2π 0

f_n(s)dWj(s) F(g)

=

n∈Z E

g, f_n^E 2π

0

g(s)dWj(s) 2π

0

f_n(s)dWj(s) F(g)