Q-matrices as pseudo-differential operators with negative definite symbols

Math. Nachr., 1 – 10 (2012) / DOI 10.1002/mana.201100217

Q-matrices as pseudo-differential operators with negative definitesymbols

Kristian Evans∗ and Niels Jacob∗∗

Department of Mathematics, Swansea University, Swansea. SA2 8PP, UK

Received 23 August 2011, revised 3 February 2012, accepted 21 February 2012Published online 30 May 2012

Key words Q-matrices, continuous time Markov chains, positive maximum principle, pseudo-differential op-erators with negative definite symbols on Zn , estimates for transition functions

MSC (2010) 47006, 47007, 47G30, 35S05, 60J35

Dedicated to Professor Hans Triebel on the occasion of his 75th birthday

Operators induced by Q-matrices on lp (Zm ) are shown to satisfy the positive maximum principle and to havea representation as a pseudo-differential operator with symbol q : Z

m × T m → C which is with respect tothe co-variable negative definite (in the sense of Schoenberg). This observation leads already towards a moregeometric interpretation of the transition matrix of the associated Markov chain.

c© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Fourier analysis is a powerful tool in the analysis of (not necessarily linear) partial differential operators, or inthe analysis of “periodic” problems, for example problems related to signal processing. In addition it has manycentral applications in modern probability theory, and of course this list is by no means exhaustive. Commutativeharmonic analysis extends the original ideas of Fourier analysis from R

n and Tm , the m-dimensional torusconsidered as R

m /Zm , to locally compact Abelian groups.

A non-classical application of Fourier analysis to probability theory is the observation that generators of manyFeller processes with state space R

n are (extensions of ) pseudo-differential operators with negative definite sym-bols, i.e., ξ �→ q(x, ξ) has for each x ∈ R

n a Levy-Kinchin representation, compare Ph. Courrege [3] and inparticular [7]–[9]. This symbol generalises the characteristic exponent of a Levy process, i.e., a process withindependent and stationary increments, and as the characteristic exponent the symbol is very useful to study theprocess. For example in [17]–[19] Schilling has shown using the symbol that certain weighted Besov spaces andTriebel- Lizorkin spaces are best suited to describe path properties of the corresponding process. This gave a niceand natural application to the work of Haroske and Triebel [5]–[6] discussing these spaces also for p < 1, thecase needed in Schilling’s considerations.

Recently interest grew in pseudo-differential operators acting on “functions” on more general domains, forexample groups, including discrete groups where functions are often considered as sequences. We refer to themonograph [15] by Ruzhansky and Turunen or the recent monograph [21] by Wong. Even pseudo-differentialoperatos on Z are meanwhile of some interest, see for example Molahajloo [12], and in particular the paper [16]by Ruzhansky and Turunen.

Markov processes with state space Zm are Markov chains and we refer to Chen [2], Norris [13] or Stroock [20]

for a readable introduction. Clearly, many analytic tools have been used to handle Markov chains. The basicobjects corresponding to generators are Q-matrices. In this short note we aim to add an observation to the analysisof Q-matrices, namely that they can be viewed as pseudo-differential operators acting on functions belonging tolp(Zm ), i.e., sequences k �→ ak . We restrict ourselves to Q-matrices inducing bounded operators Qop on lp(Zm )

∗ e-mail: [email protected], Phone: +44 1792 292905∗∗ Corresponding author: e-mail: [email protected], Phone: +44 1792 295461

c© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 K. Evans and N. Jacob: Q-matrices

into itself. The structural result of the statement is in our interest, not yet its most generality. The definition ofa Q-matrix implies that the corresponding operator must satisfy an analogous form of the positive maximumprinciple (an observation which was not found to be made elsewhere before), and this fact can be used to givea representation of the operator Qop as a pseudo-differential operator with symbol defined on Z

m × Tm , Tm

considered as the dual group of Zm . This symbol turns out to be with respect to the co-variable a continuous

negative definite function.With Q we can also associate a semigroup p(t) = etQo p which has matrix elements pkl(t). For simplicity we

now restrict ourselves to the case m = 1. In the case of a translation invariant operator Qop we can express pt asa convolution

(ptu)(k) =∑l∈Z

pt(k − l)ul, (ul)l∈Z ∈ lp(Z),

where

pt(k) =∫

T 1e−iωk etq(ω )τ(dω).

Here τ is the Haar measure on T 1 and −q is the symbol of the generator. In the recent paper [10] it wasproposed for symmetric Levy processes to express the transition density as a “Gaussian in disguise”. Using someideas from [10], see also Knopova and Schilling [11], we can express now pt(0) in terms of the volume of asubset of T 1 determined by the metric dQ induced from the symbol of Qop , i.e., q. Thus the first step to expresspt as a Gaussian in disguise is possible, i.e., in searching for a representation of the type

pt(k) = pt(0)e−δ 2t (k,0) = cτ

(BdQ

(0; 1/

√t))

e−δ 2t (k,0)

with a suitable metric δt on Zm we can already determine the volume term, hence the diagonal behaviour of pt ,

where BdQ(0; 1√

t

)denotes the ball with respect to the metric dQ , centre 0 and radius 1√

t, compare Section 5,

formula (5.12).Our main objective is to achieve a uniformization of basic analytic concepts used in the theory of stochastic

processes, i.e., we long for more structural clarity, not necessarily to discover new probabilistic applications.It gives the second named author great pleasure to express his gratitude to Professor Hans Triebel by dedicating

this paper to him on the occasion of his 75th birthday.

2 Q-matrices

Continuous time Markov chains with state space Zm , m ∈ N, compare J. Norris [13] or D. Stroock [20]

for a definition, are characterised by Q-matrices. In our considerations we will restrict ourselves to boundedQ-matrices.

Definition 2.1 A real matrix Q = (Qkl)k,l∈Zm is called a bounded Q-matrix if

Qkl ≥ 0 for all k, l ∈ Zm , k �= l, (2.1)∑

k �= l

Qkl = −Qkk < ∞ for all k ∈ Zm . (2.2)

Thus for a bounded Q-matrix we have∑l∈Zm

Qkl = 0 for all k ∈ Zm . (2.3)

Since Qkk = −∑

l �=k Qkl it follows that∑

l �=k Qkl converges, hence∑

l∈Zm |Qkl | converges (using (2.1) and|Qkk | < ∞). Furthermore, since∑

k∈Zm

maxl∈Zm

|Qkl | ≤∑

k∈Zm

∑l∈Zm

|Qkl | = 2∑

k∈Zm

|Qkk |

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Math. Nachr. (2012) / www.mn-journal.com 3

as well as

maxk∈Zm

∑l∈Zm

|Qkl | = maxk∈Zm

2|Qkk | ≤ 2∑

k∈Zm

|Qkk |

we conclude

Lemma 2.2 Let Q = (Qkl)k,l∈Zm be a bounded Q-matrix and define for u = (ul)l∈Zm the operator

Qop(u)(k) :=∑

l∈Zm

Qklul , k ∈ Zm . (2.4)

If∑

k∈Zm |Qkk | < ∞ then Qop : lp(Zm ) → lp(Zm ) is for 1 ≤ p ≤ ∞ a bounded linear operator.

Since the aim of our paper is to discuss some ideas we do not long for optimal conditions. Therefore, through-out the paper we assume∑

k∈Zm

|Qkk | < ∞. (2.5)

As a bounded operator on lp(Zm ) every Q-matrix can be considered as the generator of a one-parameter semi-group (Pt)t≥0 of operators on lp(Zm ) defined by taking the exponential, i.e.,

P (t) = etQ : lp(Zm ) −→ lp(Zm ), t ≥ 0. (2.6)

For t > 0 fixed P (t) has a representation as an infinite matrix (Pkl(t))k,l∈Zm and it is a well-known fact that eachof the matrices (Pkl(t))k,l∈Zm , t > 0, is a (sub-)stochastic matrix, i.e.,

Pkl(t) ≥ 0 and∑

l∈Zm

Pkl(t) ≤ 1. (2.7)

Of some particular interest is the case when Qop is of convolution type, i.e.,

Qkl = qk−l . (2.8)

In order that (Qkl)k,l∈Zm is a Q-matrix and that the corresponding operator is bounded we have to assume thatq = (qk )k∈Zm ∈ l1(Zm ) and

qk ≥ 0 for k �= 0 and q0 = −∑k �=0

qk . (2.9)

In this case we can write Qop as

Qop(u)(k) =∑

l∈Zm

qk−lul , k ∈ Zm , (2.10)

and the boundedness of Qop can be shown by Young’s inequality inducing conditions on the sequence (qk )k∈Z

depending on p.

3 Some remarks about harmonic analysis on Zm

As pointed out in the introduction we are interested in studying Qop (and Pt, t ≥ 0) from a Fourier analyticpoint of view. Here we summarise some basic properties of the Fourier transform on Z

m . Our standard referenceis Rudin [14]. We consider Z

m as a discrete, hence locally compact, Abelian group. The Haar measure on Zm is

given by

ζ =∑

m∈Zm

εm (3.1)

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where εm is the unit mass at m ∈ Zm . The dual group of Z

m is the m-dimensional torus Tm on which weconsider the normalised restriction of the Lebesgue measure as the Haar measure τ , i.e.,

τ =m⊗

j=1

σj (3.2)

where σj = σ = 12π s, s being the surface measure on S1 . When working with functions on Tm it is most

convenient to identify Tm by Rm /2πZ

m and to work with periodic functions. For ω, ω′ ∈ Tm we write ω + ω′

and ω−ω′ for the sum and difference respectively which of course is addition in Rm modulo 2πZ

m . The Fouriertransform is defined for sequences u = (ul)l∈Zm by

u(ω) =∑

l∈Zm

uleiω l , (3.3)

i.e., u : Tm → C, and it is defined for example for u ∈ l1(Zm ). The Fourier transform for functions on Tm isgiven by

f(k) :=∫

T m

e−ikω f(ω) τ(dω), (3.4)

and for suitable functions u defined on Zm and f defined on Tm the inversion formulae hold, i.e.,

uk =∫

T m

e−ikω

( ∑l∈Zm

uleiω l

)τ(dω) (3.5)

and

f(ω) =∑

k∈Zm

(∫T m

e−ikω ′f(ω′) τ

(dω′)) eiωk . (3.6)

With the normalisation we have chosen, Plancherel’s theorem reads as

‖u‖l2 (Zm ) = ‖u‖L2 (T m ) , (3.7)

and the convolution theorem states

(u ∗ v)(ω) = (u · v)(ω), ω ∈ Tm , (3.8)

as well as

(f ∗ g)(k) =(f ∗ g

)(k), k ∈ Z

m . (3.9)

Here we may allow u and v to be bounded, signed measures on Zm and f and g to be bounded, signed measures

on Tm .On both Z

m and Tm the notions of positive definite and negative definite functions are well-defined. In thefollowing discussion we rely on Chr. Berg and Forst [1]. On every locally compact Abelian group G we call φ :G → C positive definite (in the sense of Bochner) if for every n ∈ N and every choice of elements ξ1 , . . . , ξn ∈ Gthe matrix

(φ(ξk − ξl

))k,l=1,...,n

is positive Hermitian, i.e.,

n∑k,l=1

φ(ξk − ξl

)zk zl ≥ 0 (3.10)

for all z1 , . . . , zn ∈ C. Characters of G are always positive definite functions. A function ψ : G → C is callednegative definite if

ψ(0) ≥ 0 and ξ −→ e−tψ (ξ) (3.11)



is positive definite for all t > 0. This condition is equivalent to the fact that for every n ∈ N and any choice ofξ1 , . . . , ξn ∈ G the matrix

(ψ

(ξk

)+ ψ ¯(

ξl)− ψ

(ξk − ξl

))k,l=1,...,n

(3.12)

is positive Hermitian. Many properties of (continuous) negative definite functions are discussed in [7]. In partic-ular it is known that the set of all (continuous) negative definite functions is a convex cone which is closed under(locally uniform) pointwise convergence. With ψ the function ψ and �ψ ≥ 0 are negative definite too as are allnon-negative constants. Moreover we have ψ(−ξ) = ψ(ξ). Furthermore, if φ : G → C is a positive definitefunction then ξ → φ(0) − φ(ξ) is a negative definite function. Thus we have

Remark 3.1 If χ is a character of G then ξ → 1 − χ(ξ) is a negative definite function.

If ψ is negative definite, then√

|ψ| is sub-additive, i.e.,

√|ψ(ξ + η)| ≤

√|ψ(ξ)| +

√|ψ(η)|. (3.13)

It follows that for a real-valued negative definite function ψ we have for ψ12 the triangle inequality

ψ12 (ξ − η) ≤ ψ

12 (ξ − ρ) + ψ

12 (ρ − η). (3.14)

4 Q-matrices, the positive maximum principle and pseudo-differentialoperators with negative definite symbols

The positive maximum principle is a well understood condition in the study of Feller semigroups and their gen-erators. We give here a definition appropriate for our considerations.

Definition 4.1 We say that a linear bounded operator T : lp(Zm ) → lp(Zm ) satisfies the positive maximumprinciple if for u = (ul)l∈Zm ∈ lp(Zm ) it follows that supl∈Zm ul = uk0 ≥ 0 implies (Tu)(k0) ≤ 0.

The following proposition is an easy consequence of the definition of a Q-matrix, however, we have never seenits formulation before nor has it been used in the theory of Markov chains.

Proposition 4.2 Let Q be a Q-matrix and suppose that Qop maps lp(Zm ) continuously into itself. Then Qop

satisfies the positive maximum principle.

P r o o f. For u = (ul)l∈Zm ∈ lp(Zm ) we find

Qop(u)(k) =∑

l∈Zm

Qklul

= Qkkuk +∑l �=k

Qklul

(4.1)= −

∑l �=k

Qkluk +∑l �=k

Qklul

= −∑l �=k

Qkl(uk − ul).

Now, if supl∈Zm ul = uk0 ≥ 0 it follows that uk0 −ul ≥ 0 and since Qkl ≥ 0 for all k �= l the results is obtainedby (4.1).

Corollary 4.3 If Q satisfies (2.5) then Qop satisfies the positive maximum principle on every lp(Zm ), 1 ≤p ≤ ∞.

Note that from Proposition 4.2 it follows, as in Ethier and Kurtz [4], the proof of Theorem 2.2 on pages165–166, that the one-parameter semigroup Pt = etQ generated by Q is positivity preserving and a contractionon lp(Zm ), 1 ≤ p < ∞. (Recall that since Z

m is discrete all functions u : Zm → C are continuous therefore



lp(Zm ) ⊂ C∞(Zm ) for 1 ≤ p < ∞.) The next result will prepare a more general statement at the end of thissection as well as Section 5. Assume that Qop is of convolution type, i.e., Qkl = qk−l as in (2.8). We define

q(ω) :=∑

k∈Zm

qkeiωk (4.2)

to find

(Qopu)(ω) = (q ∗ u)(ω) = q(ω)u(ω), (4.3)

or

Qop(u)(k) =∫

T m

e−ikω q(ω)u(ω) τ(dω). (4.4)

Clearly, in the representation (4.4) we identify Qop as a pseudo-differential operator. We claim that the symbolq : Tm → C has the property that −q is a continuous negative definite function. Indeed we have

−q(ω) = −∑

k∈Zm

qkeiωk

= −q0 −∑k �=0

qkeiωk

=∑k �=0

qk −∑k �=0

qkeiωk

=∑k �=0

(1 − eiωk

)qk ,

where we used that∑

k �=0 qk + q0 = 0. Since qk ≥ 0 for k �= 0, and since by our previous remark in Section 3the functions ω → 1 − eiωk are continuous negative definite functions on Tm , it follows thatω →

∑k �=0

(1 − eiωk

)qk is a continuous negative definite function too (recall that (qk )k∈Zm belongs to l1(Zm )

implying uniform convergence).Corollary 4.4 The positivity preserving contraction semigroup (Tt)t≥0 generated by (4.4) on lp(Zm ), 1 ≤

p < ∞, is generated by a pseudo-differential operator with symbol q where −q is a negative definite symbol.

We want to extend Corollary (4.4) to the general case of an operator Qop with Q satisfying (2.5). For this weneed some auxiliary considerations first.

Definition 4.5 Let q : Zm × Tm → C be a function such that for k ∈ Z

m fixed the function ω → q(k, ω) iscontinuous. We call an operator qop (on its domain) a pseudo-differential operator on lp(Zm ) if it is of the type

(qopu)(k) =∫

T m

e−iωk q(k, ω)u(ω) τ(dω). (4.5)

In the case where ω → q(k, ω) is a continuous negative definite function, we call qop a pseudo-differentialoperator with negative definite symbol. For convenience we will sometimes write

σ(qop)(k, ω) = q(k, ω). (4.6)

Theorem 4.6 Let Q be a Q-matrix satisfying (2.5). On l1(Zm ) the operator −Qop is a pseudo-differentialoperator with negative definite symbol

σ(−Qop)(k, ω) =∑l �=k

Qkl

(1 − eiω (k−l)), (4.7)

i.e., Qop is a pseudo-differential operator and its symbol σ(Qop)(k, ω) has the property that ω → −σ(Qop(k, ω))is a continuous negative definite function.

P r o o f. First we prove that

ω −→∑l �=k

Qkl

(1 − eiω (k−l)) (4.8)



is a negative definite function for k ∈ Zm fixed. Since ω → eiω (k−l) is a character of Tm it follows that ω →(

1−eiω (k−l))

is a continuous negative definite function and Qkl ≥ 0 for k �= l implies that ω → Qkl

(1−eiω (k−l)

)is negative definite too. By assumption

∑l �=k Qkl converges hence

∑l �=k Qkl

(1−eiω (k−l)

)converges uniformly

with respect to ω ∈ Tm , i.e., (4.8) is a continuous function which is negative definite. Now, for u = (ul)l∈Zm wefind ∫

T m

e−iωk∑l �=k

Qkl

(1 − eiω (k−l))u(ω) τ(dω)

=∫

T m

∑l �=k

Qkl

(e−iωk − e−iω l

)u(ω) τ(dω)

=∑l �=k

Qkl

∫T m

(e−iωk − e−iω l

)u(ω) τ(dω)

=∑l �=k

Qkl(uk − ul) = uk

∑l �=k

Qkl −∑l �=k

Qklul

= −Qkkuk −∑l �=k

Qklul

= −∑

l∈Zm

Qklul = −(Qopu)(k),

proving the theorem.

5 Some geometric considerations on transition functions

In the following considerations we restrict for convenience to the case m = 1 and to convolution operators, i.e.,(2.8) holds. Further we assume symmetry, i.e., Qkl = Qlk implying that qk−l = ql−k . Thus we may start witha sequence (qk )k∈N ∈ l1(N), qk ≥ 0, and extend this sequence to Z according to q−k = qk for k ∈ N andq0 := −2

∑∞k=1 qk . For the corresponding symbol q of Qop we now find

−q(ω) =∑k �= 0k ∈ Z

(1 − eiωk

)qk

= −q0 −∞∑

k=1

qk

(eiωk + e−iωk

),

i.e.,

q(ω) = q0 + 2∞∑

k=1

qk cos ωk = −2∞∑

k=1

qk (1 − cos ωk). (5.1)

It is easy to see that in the case under consideration we find for the associated one-parameter semigroup (Pt)t≥0

(Ptu)(k) =∫

T 1e−iωk e+tq(ω ) u(ω) τ(dω) =

∑l∈Z

pt(k − l)ul (5.2)

with

pt(k) =∫

T 1e−iωk e+tq(ω ) τ(dω), (5.3)

where now τ denotes the Haar measure on T 1 . In the case of Rn and the symmetric Levy process (Xt)t≥0 for

the analogous object, i.e., the transition density

pt(x) =∫

Rn

eixξ e−tψ (ξ)dξ, (5.4)



where the continuous negative definite function ψ : Rn → R

n is the characteristic exponent of (Xt)t≥0 , ageometric interpretation of pt has been discussed in [10]. In particular, in the case where ψ

12 (x − y) is a metric

on Rn (with some nice properties) then

pt(0) =∫ ∞

0λ(n)

(Bψ

12(0;

√r/t

))e−r dr, (5.5)

where λ(n) denotes the Lebesque measure in Rn and

Bψ12 (y;R) =

{x ∈ R

n |ψ 12 (x − y) ≤ R

}, (5.6)

compare Knopova and Schilling [11] and in particular [10]. Further, in many cases one can write

pt(x) = pt(0)e−δ 2ψ (x,0) (5.7)

with a further metric δψ . We refer again to [10]. Here we aim to get a result analogous to (5.5) for pt given by(5.3). Since for k = 0 we have

pt(0) =∫

T 1etq(ω ) τ(dω) (5.8)

we need to discuss the integral on the right-hand side in (5.8). By our general assumption in this section q : T 1 →R is a real-valued continuous negative definite function. Now we want to discuss when (ω, ω′) → q

12 (ω − ω′)

gives a metric on T 1 . From (5.1) it follows that q(0) = 0, the symmetry follows by our assumptions and (3.14)yields the triangle inequality. Thus we need to clarify when q(ω) = 0 implies ω = 0, recall that we consider T 1

as R/2πZ. The problem to discuss is: when does

∞∑k=1

qk (1 − cos ωk) = 0, ω ∈ R/2πZ, (5.9)

imply ω = 0? Since qk ≥ 0 and (1 − cos ωk) ≥ 0 for all k and ω, (5.9) yields

qk (1 − cos ωk) = 0 (5.10)

for all k ∈ Z. This already implies that if q1 �= 0 then q12 induces a metric. Let k0 ≥ 1 be the smallest k ∈ N

such that qk0 �= 0. Then we need 1 − cos ωk0 = 0, i.e., it must hold that ω = 2πjk0

for some j < k0 . However

in this case cos 2πjk0

(k0 + 1) = cos(2πj + 2πj

k0

)= cos

( 2πjk0

)�= 0, hence we obtain additional conditions on the

coefficients qk . We do not intend to work out this “combinatorial” problem here. Instead our assumption fromnow on is that q

12 induces a metric on T 1 , in fact we make

Assumption 5.1 We require that this metric also induces the standard topology on T 1 .

We denote this metric by dQ , i.e.,

dQ

(ω, ω′) = (−q)

12(ω − ω′). (5.11)

Further we set

BdQ (ω;R) :={ω′ ∈ T 1 |dQ

(ω, ω′) ≤ R

}. (5.12)

Now we may proceed as in [10], compare also [11], to find

pt(0) =∫

T 1etq(ω )τ(dω) =

∫T 1

e−td2Q (0,ω )τ(dω)

= t

∫ ∞

0τ(BdQ

(0;√

ρ))

e−tρdρ (5.13)

=∫ ∞

0τ(BdQ

(0;

√r/t

))e−r dr.



Furthermore it follows that

pt(0) =∫ ∞

0τ(BdQ

(0;

√r/t

))e−r dr

≥∫ ∞

1τ(BdQ

(0; 1/

√t))

e−r dr (5.14)

=1e

τ(BdQ

(0; 1/

√t))

.

Note that if we have the volume doubling property, i.e.,

τ(BdQ (0, cr)

)≤ γ0(c)τ

(BdQ (0, r)

), c > 1, r > 0 (5.15)

and γ0(c) ≤ γ0(1)cα , c ≥ 1, we find also the upper bound (again compare [10])

pt(0) =∫ 1

0τ(BdQ

(0;

√r/t

))e−r dr +

∫ ∞

1τ(BdQ

(0;

√r/t

))e−r dr

≤(1 + e−1)τ(

BdQ(0; 1/

√t))

+ τ(BdQ

(0; 1/

√t)) ∫ ∞

1γ0(1)r

α2 e−r dr (5.16)

= κτ(BdQ

(0; 1/

√t))

.

Note that in the case of Rn the asymptotic behaviour of ψ determines whether or not the volume doubling property

holds. For the case of a torus it is not yet clear. Obviously periodicity must be excluded.A more thorough study of the metric measure space

(T 1 , dQ , τ

)or more generally

(Tm , dQ , τ (m )

)will be

given in a forthcoming paper. We want to illustrate our investigations by an example.

Example 5.2 Consider the sequence (qk )k∈N, qk = 1k 2 . Since

∑∞k=1

1πk 2 = π

6 we can construct a Q-matrixQkl = qk−l as follows

qk =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1πk2 , k ∈ N,

−π

3, k = 0,

1πk2 , −k ∈ N.

(5.17)

The function

−q(ω) = 2∞∑

k=1

1πk2 (1 − cos ωk) (5.18)

is a continuous negative definite function on T 1 and since q1 = 1π we can deduce that (−q)

12 induces a metric on

T 1 . In fact for ω ∈ R/2πZ we have

2

( ∞∑k=1

1πk2 (1 − cos ωk)

)=

(ω − ω2

2π

). (5.19)

Thus the diagonal behaviour of the heat kernel of the transition function associated with Qkl = qk−l is controlledby

τ

{ω ∈ T 1 |

(ω − ω2

2π

) 12

≤ 1√t

}

= τ

⎧⎨⎩ω ∈ T 1 |

√2

( ∞∑k=1

1πk2 (1 − cos ωk)

) 12

≤ 1√t

⎫⎬⎭ .

Off-diagonal estimates are clearly much more involved and more difficult to obtain. For the case G = R (orR

n in some cases) we refer to [10]. The case of Zm is a topic for further studies.



References

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Q-matrices as pseudo-differential operators with negative definite symbols