· Markov Processes Relat. Fields 17, 315–346 (2011) Markov MP &RF „‚ ”• Processes and...

Markov Processes Relat. Fields 17, 315–346 (2011) Markov MPRF&¹¸

º·ProcessesandRelated Fieldsc©Polymat, Moscow 2011

On the Equivalence Between Liggett

Duality of Markov Processes and the

Duality Relation Between Their

Generators

A. Voß-Bohme , W. Schenk and A.-K. KoellnerInst. Math. Stochastik, Technische Universitat Dresden, 01062 Dresden, Germany.E-mail: [email protected]

Received July 16, 2010, revised March 23, 2011

Abstract. A general result on duality for Feller processes is stated which ap-plies in particular to interacting particle systems. A rather natural conditionis presented which ensures that the duality relation between the generator of aconservative Feller process with compact state space and the rate matrix of aMarkov chain is equivalent to the according duality relation at the semigrouplevel. These findings are valid for general duality functions.

Keywords: interacting particle system, duality relation, dual Markov chain

AMS Subject Classification: 82C22, 60J27, 60J35

1. Introduction

The stochastic concept of duality plays an important role in the theory ofinteracting particle systems (IPS). It allows to relate the evolution of a giveninteracting particle system to the evolution of another Markov process whichis often easier to analyze. The duality technique goes back to Spitzer [14] whoused it for the characterization of the stationary distribution of the symmetricexclusion process. Duality for spin-flip processes was employed by Holley andStroock [6] for the study of the ergodic properties of such processes. Bothapproaches were systemized by Liggett [9]. There are many recent publicationssuch as [7, 8, 10,11] where duality is successfully applied.

The idea behind duality shall be illustrated within the following simplifiedsetting. Assume that (Tt)t≥0 and (Pt)t≥0 are the transition semigroups of two

316 A. Voß-Bohme, W. Schenk and A.-K. Koellner

time-continuous Markov processes on X and Y , respectively, where X and Yare finite sets. Let be H : X × Y → R. The two Markov chains are calledH-dual, if

TtH(·, B)(η) = PtH(η, ·)(B), η ∈ X , B ∈ Y , t ≥ 0. (1.1)

The infinitesimal characteristics of (Tt)t≥0 and (Pt)t≥0 shall be denotedby A and Q, respectively. The rate matrices A and Q are H-dual, if

AH(·, B)(η) = QH(η, ·)(B), η ∈ X , B ∈ Y , (1.2)

is fulfilled. One finds easily, that (1.1) implies (1.2). Indeed, one obtainsfrom (1.1) that for fixed η ∈ X , B ∈ Y ,

1t

(TtH(·, B)(η)−H(η,B)

)=

1t

(PtH(η, ·)(B)−H(η,B)

), t > 0.

The limit of the left-hand side as t → ∞ exists and is equal to AH(·, B)(η).The according limit of the right-hand side is equal to QH(η, ·)(B). Thus weget (1.2). To show that the reverse implication is also true, the following ideais pursued. Fix η ∈ X , B ∈ Y , and define

u(t, η, B) = TtH(·, B)(η), t ≥ 0,

andv(t, B, η) = PtH(η, ·)(B), t ≥ 0.

Then, as (d/dt)Pt = QPt, it holds that

d

dtv(t, B, η) = Qv(t, ·, η)(B), t > 0; v(0+, B, η) = H(η,B)

for each B ∈ Y , η ∈ X . Thus the function v(·, ·, η) solves for each η ∈ X theinitial value problem

d

dtv(t, B) = Qv(t, ·)(B), t > 0; v(0+, B) = h(B), (1.3)

where h = H(η, ·). As (d/dt)Tt = ATt = TtA, one obtains

d

dtu(t, η, B) = TtAH(·, B)(η), t > 0; u(0+, η, B) = H(η,B),

for each η ∈ X . Inserting the infinitesimal duality relation (1.2), one finds forthe term on the right-hand side that

TtAH(·, B)(η) = Tt

(ζ 7→ AH(B, ·)(ζ)

)(η) = Tt

(ζ 7→ QH(ζ, ·)(B)

)(η)

= Tt

( ∑

C∈Y

Q(B,C)H(·, C))(η) =

∑

C∈Y

Q(B,C)TtH(·, C)(η)

=∑

C∈Y

Q(B, C)u(t, η, C)

= Qu(t, η, ·)(B), t > 0, η ∈ X , B ∈ Y .

On the duality between IPS and Markov chains 317

Hence

d

dtu(t, η, B) = Qu(t, η, ·)(B), t > 0, u(0+, η, B) = H(η, B),

for each η ∈ X , B ∈ Y , that is u(·, η, ·) is for each η ∈ X a solution of (1.3).If this initial value problem has for each η ∈ X a unique solution, then thesemigroup duality relation (1.1) holds:

TtH(·, B)(η) = u(t, η, B) = v(t, η, B) = PtH(η, ·)(B)

for t ≥ 0, η ∈ X , B ∈ Y .In the context of IPS one wishes to replace A by the Markov generator A of

a Feller semigroup (Tt)t≥0 on some compact space (X , d). The state space Yof the dual chain is usually chosen to be countably infinite. The question thatshall be addressed in this study is under which conditions (1.1) and (1.2) areequivalent in this general situation.

In [9, §III.4 and §VIII.1], the equivalence of (1.1) and (1.2) was consideredfor specific duality functions and specific interacting particle systems, namelyspin-flip systems and symmetric exclusion processes. Essentially the same ar-guments as in [9] were used, for instance, in [7] for the voter-exclusion process,in [8] for an exclusion process with multiple interactions and in [10] for the sym-metric two-particle exclusion-eating process. However, the key issue of provingthat the initial value problem (1.3) has a unique bounded solution was nothandled in these works. This fact was claimed without proof or, in the caseof [7], with reference to [4, Thm.1.3]. The latter reference contains an onlyvaguely related statement about strongly continuous semigroups. Note that thesemigroup on the set of bounded functions generated by a rate matrix Q is,in general, not strongly continuous. In the special case of spin-flip processes,Holley and Stroock [6] employed the martingale problem to derive duality prop-erties. They considered the initial value problem (1.3), as well, and referred to‘obvious modifications’ of Lemma (4.1) and Lemma (4.2) in [15] to conclude itsunique solvability. However, the cited propositions are about diffusions on Rd.Therefore it is not clear in which way they apply within the IPS-duality con-text. Despite this weak point, many authors referred to [6] without giving moredetailed arguments there. For instance, Lopez and Sanz [11] considered spinprocess, where a finite number of states is allowed at each lattice site, andrather general duality functions. Although their setting is in many respectsmore general than that in [6], they referred to this source without presentingarguments that justify this.

In this paper, the equivalence of (1.1) and (1.2) is derived in full detail forgeneral Feller processes with compact state space and general duality functions.Provided that the generator A of the Feller process and some rate matrix Qsatisfy the infinitesimal duality relation (1.2) with respect to a suitable duality


function H, a rather natural condition (Q) on Q ensures that Q is non-explosiveand the Markov chain generated by Q is H-dual to the process correspondingto A, that is (1.1) holds. This result applies in particular to general interactingparticle systems in the sense of Liggett [9, §I.3]. It is valid without restric-tions on the structure of the duality function H. Thus a generalization of theabove-mentioned method is derived and the previously lacking details in theliterature are worked out within a more abstract setting. As a corollary of ourconsiderations, an additional theorem for Feller processes that are generated bya sum of operators is presented. The findings are applied to a couple of IPSexamples. Here the goal is to illustrate that the developed condition for dualityis manageable for a variety of different IPS.

The paper is organized as follows. After the main concepts on Markovsemigroups, interacting particle systems and Markov chains are introduced inSection 2, the equivalence of (1.1) and (1.2) under condition (Q) is proven inSection 3 for general conservative Feller processes on compact sets and generalduality functions (Theorem 3.1). This statement is based on Proposition 3.1where the unique solvability of the initial value problem (1.3) is shown. Pre-requisites are Lemmata 3.1–3.4 where the arguments that were outlined aboveare proven rigorously in the generalized setting. The additional theorem is pre-sented in Corollary 3.1. In Section 4, the results are specialized to apply tospecific interacting particle systems.

2. Setup and notations

2.1. Markov semigroups

Let be given a compact, separable metric space (X ; d). Define the spaceB(X ) of all bounded and Borel-measurable real functions on X equipped withthe supremum norm ‖ · ‖∞. The subspace of all continuous real functions isdenoted by C(X ).

In accordance with [9], a strongly continuous non-negative contraction semi-group (Tt)t≥0 on C(X ) which satisfies Tt1 = 11, t ≥ 0, is called a Markovsemigroup. Note that each Markov semigroup on C(X ) determines a uniqueMarkov process, which is even a Feller process [5, Thm.4.2.7].

From the Hille – Yosida Theorem it follows that each Markov semigroup onC(X ) has an infinitesimal generator which is a Markov generator in the senseof [9, Def.I.2.7]. Conversely, each Markov generator generates a unique Markovsemigroup on C(X ) [5, Thm.4.2.2]. Thus one can think of a Feller process onX as being given by its Markov generator. In the case that X is compact, it isoften suitable to construct the Markov generator from a Markov pregenerator. AMarkov pregenerator is a linear operator A : ϑ(A) → C(X ) with dense domainϑ(A) ⊂ C(X ) which satisfies the following conditions (G1) and (G2).

11(η) := 1, η ∈ X .


(G1) 1 ∈ ϑ(A), A1 = 0;

(G2) If f ∈ ϑ(A) and f(η0) = max{f(η) : η ∈ X } ≥ 0 for some η0 ∈ X , thenAf(η0) ≤ 0.

Condition (G2) is known as maximum principle. Note that a Markov pregener-ator is closable in C(X ) [9, Prop. I.2.5]. Its closure A is a Markov generator ifand only if A satisfies the following condition (G3) [5, §1.2].

(G3) There is a λ0 > 0 such that R(λ0I −A) = C(X ).

2.2. Generation of interacting particle systems

The Markov processes that will be considered in the following have statespaces (X , d) which are special function spaces. See the book of T. Liggett [9]for the detailed construction. Below the main terms are outlined.

Suppose that (W,ρ) is a compact metric space. Let be S a countable set.Frequently S is chosen as Zn, n ∈ N. There is a metric d on X := WS whichgenerates the product topology on X . Note that (X , d) is a compact metricspace. The elements of X are referred to as configurations on S.

In the context of this paper, a Markov process is named interacting particlesystem (IPS ), if it is a Feller process on X that is constructed in the wayoutlined below. An IPS models the random time changes of configurations.

According to Paragraph 2.1, each IPS is uniquely determined by a Markovgenerator and can be constructed from a Markov pregenerator. The latter isdetermined by the specification of local transition rates. In detail, let be T :={T ⊂ S : |T | < ∞, T 6= ∅}2 and T∅ := {T ⊂ S : |T | < ∞} = T ∪{∅}. Obviously,T is finite or countably infinite. For T ∈ T , define the local configuration spaceXT := WT and the corresponding projection

πT : X −→ XT : πT η := (η(x))x∈T .

Consider the map

τT : X ×XT −→ X : τT (η, v)(x) :=

{η(x), if x /∈ T,

v(x), if x ∈ T.

Given a configuration η ∈ X and a local configuration v ∈ XT , the trans-formation τT (η, v) replaces the local configuration πT η in T by v. A family(cT (·, ·))T∈T of functions cT : X × B(XT ) → [0,∞) is a family of transi-tion rate functions, if the functions cT satisfy, for each T ∈ T , the followingconditions (C1)–(C3).

(C1) The function η 7→ cT (η, Γ), η ∈ X , is measurable for each Γ ∈ B(XT ).2|T | stands for the cardinality of T


(C2) The map Γ 7→ cT (η, Γ), Γ ∈ B(XT ), is a finite measure on B(XT ) foreach η ∈ X .

(C3) The function η 7→ ∫XT

f(v) cT (η, dv), η ∈ X , is continuous for each f ∈C(XT ).

We define

cT (x) := sup{|cT (η, du)− cT (ζ, dv)| : πS\{x}η = πS\{x}ζ

}, x ∈ S, T ∈ T ,

andcT := sup

η∈X

∑

u∈XT

cT (η, du), T ∈ T .

A family c = (cT (·, ·))T∈T of transition rate functions is admissible, if

(C4) supx∈S

∑T3x cT < ∞,

and

(C5) supx∈S

∑T3x

∑z 6=x cT (z) < ∞.

Next some function spaces are specified which can serve as domains forthe pregenerator of the IPS that shall be constructed. A continuous functionf : X → R is a tame function, if there exist a set T ∈ T and a functionfT : XT → R such that the representation f = fT ◦ πT holds. In this case,the function f does not depend on the coordinates on S \ T , that is for anyη1, η2 ∈ T satisfying πT η1 = πT η2 one has f(η1) = f(η2). Let the set of alltame functions be denoted by T (X ). Note that T (X ) is a dense subset ofC(X ) (see, for instance, [9, §I.3]). Define further

D(X ) :={

f ∈ C(X ) :∑

x∈S

sup{|f(η)− f(ζ)| : η, ζ ∈ X , πS\{x}η = πS\{x}ζ

}< ∞

}.

Obviously T (X ) ⊂ D(X ) and therefore D(X ) is dense in C(X ).Suppose now that a family c = (cT (·, ·))T∈T of transition rate functions is

given. For T ∈ T , the operator AT : C(X ) → C(X ) defined by

AT f(η) :=∫

XT

(f(τT (η, ζ))− f(η)

)cT (η, dζ), f ∈ C(X ), η ∈ X ,

is a bounded linear operator. Indeed, the property (C3) ensures that the func-tion AT f is continuous, if f is continuous. The linearity of AT is obvious andthe boundedness of AT follows from

|AT f(η)| ≤ 2‖f‖∞ supζ∈X

cT (ζ, XT ) < ∞, f ∈ C(X ), η ∈ X .


The fact supζ∈X cT (ζ, XT ) < ∞ is a consequence of the continuity of cT (·, XT )on the compact set X . Moreover, it is easy to see that AT has the properties(G1) and (G2) of a Markov pregenerator. As AT is bounded, it is a Markovgenerator.

Next we define an operator A : D(X ) → C(X ) by

Af(η) :=∑

T∈T

AT f(η) (2.1)

=∑

T∈T

∫

XT

cT (η, dv)[f(τT (η, v))− f(η)

], η ∈ X , f ∈ D(X ).

Note that T is a directed set with respect to the set inclusion ⊂, thus the infinitesums above are to be understood in terms of net convergence. By [9, Prop. I.3.2],A is well-defined if c is admissible. Further, according to [9, Thm. I.3.9], theclosure of A is a Markov generator which generates a Markov semigroup (Tt)t≥0

on C(X ). The corresponding Markov process with cadlag trajectories is aninteracting particle system.

2.3. Generation of continuous-time conservative Markov chains

The set-up in the following corresponds essentially to [2, Ch. 8] and [13,Ch. 2]. Suppose that Y is a finite or countably infinite set. One says that fora : Y → R the sum ∑

B∈Y

a(B) =: a

exists and is equal to a, if for each monotonically increasing sequence (Yn)n∈Nwith Yn ⊂ Y , |Yn| < ∞ and Y = ∪n∈NYn it holds that

limn→∞

∑

B∈Yn

a(B) = a.

Define the space B(Y ) of all bounded real functions on Y equipped with thesupremum norm ‖ · ‖∞.

A function P : Y × Y → [0, 1] is called transition matrix or stochasticmatrix, if

∑C∈Y P(B,C) = 1 for each B ∈ Y . Each stochastic matrix P can

be understood as a contraction on B(Y ) by

Ph(B) :=∑

C∈Y

P(B,C)h(C), h ∈ B(Y ), B ∈ Y .

If a function h : Y → R is not bounded, but the sum on the right-hand sideabove converges for each B ∈ Y , the symbol Ph is used as well for the resultingfunction on Y .


A family (Pt)t≥0 of stochastic matrices is a conservative transition semigroupon Y , if P0 = I, where I is the identity matrix, and Pt+s = PtPs, t, s ≥ 0.A conservative transition semigroup (Pt)t≥0 is continuous, if, for each B ∈ Y ,lims↓0(1−Ps(B, B)) = 0. Note that the continuity of the transition semigroup(Pt)t≥0 implies that each function t 7→ Pth(B), t ≥ 0, is continuous, whereh ∈ B(Y ) and B ∈ Y are fixed. Indeed, by the semigroup property it issufficient to show right-continuity at zero. Choose h ∈ B(Y ) and B ∈ Y . Then

∣∣Pth(B)− h(B)∣∣ ≤ 2‖h‖∞

∑

C∈Y \{B}Pt(B, C)

= 2‖h‖∞(1− Pt(B, B)), B ∈ Y ,

hence the function t 7→ Pth(B), t ≥ 0, is right-continuous at 0.Each continuous, conservative transition semigroup (Pt)t≥0 is associated to

a unique Markov chain on Y .Suppose that we are given a continuous, conservative transition semigroup

(Pt)t≥0. Then it is well-known that the infinitesimal characteristics

Q(B) := limt↓0

1t(1− Pt(B,B)) ∈ [0;∞], B ∈ Y ,

Q(B, C) := limt↓0

1t(Pt(B, C)) ∈ [0;∞), B, C ∈ Y ,

exist [2, Thm. 2.1]. For convenience let

Q(B, B) := −Q(B), B ∈ Y .

Since the transition semigroup is conservative, one has∑

C∈Y \{B}Q(B, C) = −Q(B, B) < ∞, B ∈ Y .

Hence the infinitesimal characteristics Q of a continuous, conservative transitionsemigroup satisfies the conditions (R1)–(R3) below.

A continuous, conservative transition semigroup (Pt)t≥0 on Y with infinites-imal characteristics Q is said to satisfy condition (PQ) if

∑

C∈Y

Pt(B, C)Q(C) < ∞, B ∈ Y , t > 0. (PQ)

A matrix Q := (Q(B, C))B,C∈Y with

0 ≤ Q(B,C) < ∞, B, C ∈ Y , B 6= C, (R1)

and−∞ < Q(B, B) =: −Q(B) ≤ 0, B ∈ Y , (R2)


as well as ∑

C∈Y

Q(B, C) = 0, B ∈ Y , (R3)

is a rate matrix or Q-matrix. Each rate matrix Q := (Q(B,C))B,C∈Y can beunderstood as a linear operator on B(Y ) by

Qh(B) :=∑

C∈Y

Q(B, C)h(C)

=∑

C∈Y

Q(B,C)[h(C)− h(B)], h ∈ B(Y ), B ∈ Y .

This operator is bounded, if the condition

supB∈Y

Q(B) < ∞ (Q∗)

holds. If a function f : Y → R is nonnegative then

Qf(B) :=∑

C∈Y

Q(B, C)f(C) = −Q(B)f(B)+∑

C∈Y \{B}Q(B, C)f(C), B ∈ Y ,

is well-defined in [0,∞].Below it is typically assumed that a rate matrix satisfies the following con-

dition which is weaker than (Q∗).

(Q) There are a sequence (Yn)n∈N of subsets Yn ⊂ Y that increase to Y asn →∞, a nonnegative function ϕ on Y with limn→∞ infB /∈Yn

ϕ(B) = ∞and a real c such that the following conditions are satisfied.

supB∈Yn

Q(B) < ∞, n ∈ N; (Q0)

Qϕ(B) ≤ cϕ(B), B ∈ Y ; (Q1)

Q(B) ≤ cϕ(B), B ∈ Y . (Q2)

Condition (Q1) is sufficient for the existence of a unique conservative and contin-uous transition semigroup (Pt)t≥0 with given infinitesimal characteristics Q [1,Cor. 2.2.16]. According to [1, Prop. 2.2.13], (Q1) is equivalent to the statement

Ptϕ ≤ ectϕ(B), t ≥ 0, B ∈ Y . (P1)

If a rate matrix Q satisfies condition (Q), then it fulfills condition (PQ). Indeed,from (Q1) and (Q2) via (P1) one has

∑

C∈Y

Pt(B, C)Q(C) ≤ c∑

C∈Y

Pt(B, C)ϕ(C) ≤ cectϕ(B) < ∞,

where t ≥ 0, B ∈ Y . Note that the condition (Q) and thus condition (PQ) areimplied by the more restrictive condition (Q∗).


3. A duality criterion

Assume that (X ; d) is a compact metric space and let be (Tt)t≥0 a Markovsemigroup on C(X ) with associated Markov generator (A, ϑ(A)). Supposethat Y is an at most countably infinite set and (Pt)t≥0 is a continuous, conser-vative transition semigroup on Y with infinitesimal characteristics Q. Furtherassume that H : X × Y → R is a bounded function which satisfies

H(·, B) ∈ ϑ(A), B ∈ Y . (3.1)

The semigroups (Tt)t≥0 and (Pt)t≥0 and, accordingly, the correspondingMarkov processes are said to be in duality with respect to H (H-dual), if thefollowing condition (D-S) is satisfied.

TtH(·, B)(η) = PtH(η, ·)(B), η ∈ X , B ∈ Y , t ≥ 0. (D-S)

The generator A of a Markov semigroup (Tt)t≥0 and the infinitesimal character-istics Q of a continuous, conservative transition semigroup (Pt)t≥0 on Y satisfycondition (D-I), if

AH(·, B)(η) = QH(η, ·)(B), η ∈ X , B ∈ Y , (D-I)

is fulfilled.

Lemma 3.1. Suppose that f ∈ B([0,∞) × Y )3. If the limit limt↓0 f(t, B) =:f(0+, B) exists for each B ∈ Y , then

limt↓0

Ptf(t, B)− f(t, B)t

= Qf(0+, B), B ∈ Y .

Proof. Let B ∈ Y , f ∈ B([0,∞) × Y ) and define g(t, C) := f(t, C) − f(t, B),t ≥ 0, C ∈ Y . It is to show that

limt↓0

1t

( ∑

C∈Y

g(t, C)Pt(B, C))

=∑

C∈Y

g(0+, C)Q(B, C).

Let some sequence (Yn)n∈N be given, the Yn’s being finite subsets of Y whichincrease towards Y as n → ∞. There exists n0 ∈ N such that B ∈ Yn for alln > n0. Thus one obtains for n > n0, t > 0,

Ptg(t, B) =∑

C∈Y

g(t, C)Pt(B,C)

≥∑

C∈Yn

g(t, C)Pt(B, C)− ‖g‖∞∑

C∈Y \Yn

Pt(B, C)

=∑

C∈Yn

g(t, C)Pt(B, C)− ‖g‖∞((1− Pt(B, B))−

∑

C∈Yn\{B}Pt(B, C)

).

3For general measurable spaces E, the symbol B(E) denotes the space of real-valued,bounded and measurable functions on E.


Both sums at the right-hand side above are finite, thus one concludes

limt↓0

∑

C∈Yn

g(t, C)1tPt(B,C) =

∑

C∈Yn

g(0+, C)Q(B, C)

and

limt↓0

∑

C∈Yn\{B}

1tPt(B,C) =

∑

C∈Yn\{B}Q(B,C).

Consequently,

lim inft↓0

1tPtg(t, B)

≥∑

C∈Yn

g(0+, C)Q(B, C)− ‖g‖∞(Q(B, B)−

∑

C∈Yn\{B}Q(B,C)

)

for n > n0. The term Q(B,B) − ∑C∈Yn\{B}Q(B, C) tends to 0 as n → ∞,

thereforelim inf

t↓01tPtg(t, B) ≥

∑

C∈Y

g(0+, C)Q(B, C).

Analogously, the estimate

Ptg(t, B) =∑

C∈Y

g(t, C)Pt(B,C)

≤∑

C∈Yn

g(t, C)Pt(B, C) + ‖g‖∞∑

C∈Y \Yn

Pt(B, C)

yields

lim supt↓0

1tPtg(t, B) ≤

∑

C∈Y

g(0+, C)Q(B, C).

2

Lemma 3.2. If f ∈ B([0,∞)×Y ) is right-continuous with respect to the firstargument, then

lims→t

Psf(s, B) = Ptf(t, B), t > 0, B ∈ Y .

Proof. Fix B ∈ Y , f ∈ B([0,∞) × Y ) and choose a sequence (Yn)n∈N, theYn’s being finite subsets of Y which increase towards Y as n →∞. Then, forsufficiently large n such that B ∈ Yn,

Psf(s,B) ≥∑

C∈Yn

f(s, C)Ps(B, C)− ‖f‖∞(1−

∑

C∈Yn

Ps(B, C)),


which yields

lim infs→t

Psf(s,B) ≥∑

C∈Yn

f(t, C)Pt(B, C)− ‖f‖∞(1−

∑

C∈Yn

Pt(B, C)).

Taking the limit n →∞ on the right-hand side, one obtains

lim infs→t

Psf(s,B) ≥∑

C∈Y

f(t, C)Pt(B,C).

Analogously,lim sup

s→tPsf(s,B) ≤

∑

C∈Y

f(t, C)Pt(B, C).

2

Lemma 3.3. Let B ∈ Y and h ∈ B(Y ). Then the function t 7→ Pth(B),t > 0, is continuously differentiable and the derivative satisfies

(i)

d

dtPth(B) = Q(Pth)(B) =

∑

C∈Y

Q(B, C)[Pth(C)− Pth(B)], t > 0.

(ii) If the transition semigroup (Pt)t≥0 satisfies the condition (PQ), then

d

dtPth(B) = Pt(Qh)(B)

=∑

C∈Y

Pt(B, C)∑

V ∈Y

Q(C, V )[h(V )− h(C)], t > 0.

Remark 3.1. A proof of the above lemma can be found in [3, §13.5], where it wasused to derive an integration by parts formula for jump processes. Neverthelessan independent proof is given here to make the argumentation self-contained.

Proof.(i) Let t > 0, B ∈ Y and h ∈ B(Y ). The semigroup property of (Ps)s≥0

gives

1s

(Pt+sh(B)− Pth(B)

)=

1s

∑

C∈Y

Ps(B, C)[Pth(C)− Pth(B)], s > 0.

Defineuh,B(t, ·) := Pth(·)− Pth(B).


Clearly, uh,B(t, ·) ∈ B(Y ) and uh,B(t, B)=0. Applying Lemma 3.1 to uh,B(t, ·),one obtains

lims↓0

1s

∑

C∈Y

Ps(B,C)[Pth(C)− Pth(B)]

= lims↓0

1sPsuh,B(t, ·)(B) =

∑

C∈Y

Q(B, C)uh,B(t, C)

=∑

C∈Y

Q(B,C)[Pth(C)− Pth(B)].

Hence

lims↓0

1s(Pt+sh(B)− Pth(B)) =

∑

C∈Y

Q(B, C)[Pth(C)− Pth(B)].

This shows that the continuous function t 7→ Pth(B), t > 0, has at each t > 0the right-hand side derivative

∑

C∈Y

Q(B, C)uh,B(t, C).

As uh,B(·, C) is continuous and uniformly bounded by 2‖h‖∞ and because of∑

C∈Y \{B}Q(B, C) = Q(B) < ∞,

this right-hand side derivative of t 7→ Pth(B), t > 0 is continuous. Each con-tinuous function with continuous right-hand side derivative is differentiable andits derivative agrees with its right-hand side derivative. This proves (i).

(ii) Suppose that (Pt)t≥0 satisfies (PQ). For B ∈ Y and h ∈ B(Y ), oneobtains from the semigroup property of (Pt)t≥0 that

(Pt+s h(B)− Pth(B))

= Pt(Psh(·)− h(·))(B) =∑

C∈Y

Pt(B, C)(Psh(C)− h(C))

=∑

C∈Y

Pt(B, C)∑

V ∈Y

Ps(C, V )[h(V )− h(C)], s, t > 0.

Obviously∣∣∣

∑

V ∈Y

Ps(C, V )[h(V )− h(C)]∣∣∣ ≤ 2‖h‖∞

∑

V ∈Y \{C}Ps(C, V )

= 2‖h‖∞(1− Ps(C, C)), s > 0, C ∈ Y .


Each continuous conservative transition semigroup (Pt)t≥0 with infinitesimalcharacteristics Q satisfies

1s(1− Ps(C,C)) ≤ Q(C), s > 0, C ∈ Y ,

see, for instance [2, 8.(2.15)]. Hence

1s

∣∣∣∑

V ∈Y

Ps(C, V )[h(V )− h(C)]∣∣∣ ≤ 2‖h‖∞Q(C), s > 0, C ∈ Y .

By (PQ), it follows that

1s

∑

C∈Y

Pt(B, C)∣∣∣

∑

V ∈Y

Ps(C, V )[h(V )− h(C)]∣∣∣

≤ 2‖h‖∞∑

C∈Y

Pt(B, C)Q(C) < ∞

for s, t > 0. According to Lemma 3.1, it holds for each C ∈ Y that

lims↓0

(1s

∑

V ∈Y

Ps(C, V )[h(V )− h(C)])

=∑

V ∈Y

Q(C, V )[h(V )− h(C)].

Hence

lims↓0

1s

∑

C∈Y

Pt(B,C)∑

V ∈Y \{C}Ps(C, V )[h(V )− h(C)]

=∑

C∈Y

Pt(B,C) lims↓0

1s

∑

V ∈Y \{C}Ps(C, V )[h(V )− h(C)]

=∑

C∈Y

Pt(B,C)∑

V ∈Y

Q(C, V )[h(V )− h(C)], t > 0.

Thus the right-hand side derivative of the function t 7→ Pth(B), t > 0, admitsfor each t > 0 the representation

lims↓0

1s(Pt+sh(B)− Pth(B)) =

∑

C∈Y

Pt(B,C)∑

V ∈Y

Q(C, V )[h(V )− h(C)].

It is argued in the proof of (i) that the derivative of the function t 7→ Pth(B),t > 0, exists and agrees with its right-hand side derivative. Hence

d

dtPth(B) =

∑

C∈Y

Pt(B, C)∑

V ∈Y

Q(C, V )[h(V )− h(C)], t > 0.

2


If a function u ∈ B((0,∞) × Y ) is differentiable with respect to the firstargument, then this partial derivative shall be denoted by

u′(t, B) :=d

dtu(t, B), B ∈ Y .

Lemma 3.4. Let Q be a rate matrix with associated transition semigroup(Pt)t≥0. Suppose that (PQ) holds. If a function u ∈ B((0,∞) × Y ) is con-tinuously differentiable with respect to the first argument and satisfies

|u′(t, B)| ≤ KQ(B), t > 0, B ∈ Y , (3.2)

for some constant K > 0, then for each B ∈ Y the function t 7→ Ptu(t, B),t > 0, is differentiable and its derivative satisfies

d

dt

(Ptu(t, ·)(B)

)=

d

ds

(Psu(t, ·)(B)

)∣∣∣s=t

+ Ptu′(t, ·)(B), t > 0.

Proof. Fix t > 0 and consider a function u ∈ B([0,∞)× Y ) that satisfies (3.2)for some constant K > 0. Choose δ > 0, v ∈ R with |v| < δ < t. One has

Pt+v u(t + v, ·)− Ptu(t, ·)= Pt(u(t + v, ·)− u(t, ·)) + (Pt+vu(t + v, ·)− Ptu(t + v, ·)).

Fix B ∈ Y and consider

1vPt(u(t + v, ·)− u(t, ·)) =

∑

C∈Y

Pt(B,C)u(t + v, C)− u(t, C)

v. (3.3)

One finds that

∣∣∣u(t + v, C)− u(t, C)v

∣∣∣ ≤ 1v

t+v∫

t

|u′(s, C)| ds ≤ KQ(C), C ∈ Y ,

where∑

C∈Y Pv(B, C)Q(C) < ∞, B ∈ Y , according to (PQ). Thus the Dom-inated Convergence Theorem applies for the limit v ↓ 0 in (3.3). One concludes

limv↓0

1v

∑

C∈Y

Pt(B, C)u(t + v, D)− u(t, D)

v=

∑

C∈Y

Pt(B, C)u′(t, C).

One verifies easily, that the function [0,∞) × Y 3 (v, C) 7→ Ptu(t + v, ·)(C)is bounded and satisfies limv→0Ptu(t + v, ·)(C) = Ptu(t, ·)(C), C ∈ Y , by theDominated Convergence Theorem. Therefore, the limit

limv↓0

1v

(PvPtu(t + v, ·)− Ptu(t + v, ·))(B)

= Q(Ptu(t, ·))(B) =d

ds

(Psu(t, ·)(B)

)∣∣∣s=t


exists for each B ∈ Y by Lemma 3.1 and Lemma 3.3 (i). For v < 0, observethat

Pt+vu(t + v, ·)− Ptu(t + v, ·) = −(P−vPt+vu(t + v, ·)− Pt+vu(t + v, ·)).

Since limv→0Pt+vu(t + v, ·)(B) = Ptu(t, ·)(B), B ∈ Y , by Lemma 3.2, onecan conclude from Lemma 3.1 applied to the bounded function [0,∞) × Y 3(v, C) 7→ Pt+vu(t + v, ·)(C) that

limv↑0

1v

(P−vPt+vu(t + v, ·)− Pt+vu(t + v, ·))(B) = Q(Ptu(t, ·))(B), B ∈ Y .

Consequently,

limv→0

1v

(Pt+vu(t + v, ·)− Ptu(t + v, ·))(B) =

d

ds

(Psu(t, ·)(B)

)∣∣∣s=t

, B ∈ Y .

2

Proposition 3.1. Let Q be a rate matrix with associated transition semigroup(Pt)t≥0 and let the condition (PQ) be satisfied. Let h ∈ B(Y ) be fixed. Thenthe initial value problem

d

dtu(t, B) =

∑

C∈Y

Q(B, C)[u(t, C)− u(t, B)], (IVP)

u(0+;B) = h(B), B ∈ Y , t > 0,

has a unique solution in B([0,∞)× Y ) which is given by

u(t, B) = Pth(B), t ≥ 0, B ∈ Y .

Proof. Fix h ∈ B(Y ). The function u(t, B) := Pth(B), t ≥ 0, B ∈ Y , sat-isfies (IVP) by Lemma 3.3 (i). Since the semigroup (Pt)t≥0 is contractive,u ∈ B([0,∞)× Y ).

Suppose that v ∈ B([0,∞) × Y ) solves (IVP). Then the function v(·, B)is differentiable for each B ∈ Y . Its derivative is continuous, since it admits arepresentation as the uniform limit of continuous functions. Indeed, choosing asequence (Yn)n∈N of finite subsets of Y which increase towards Y as n → ∞,one finds that

u′(·, B) =∑

C∈Y

Q(B,C)[u(·, C)−u(·, B)] = limn→∞

∑

C∈Yn

Q(B, C)[u(·, C)−u(·, B)].

In addition, it holds that

|v′(t, B)| ≤∑

C∈Y \{B}Q(B, C)|v(t, C)− v(t, B)| ≤ 2‖v‖∞Q(B), t > 0, B ∈ Y .


Hence Lemma 3.4 applies. One finds that for fixed B ∈ Y , t > 0, the functions → Pt−sv(s,B), 0 < s < t, is differentiable and its derivative satisfies

d

ds

(Pt−sv(s, ·)(B)

)= − d

dr

(Prv(s, ·)(B)

)∣∣∣r=t−s

+ Pt−sv′(s, ·)(B).

By (IVP) and Lemma 3.3 (ii),

Pt−s

( d

dsv(s, ·)

)(B) = Pt−s(Qv(s, ·))(B)

=∑

C∈Y

∑

V ∈Y

Pt−s(B, C)Q(C, V )[v(s, V )− v(s, C)]

=d

dr

(Prv(s, ·)(B)

)∣∣∣r=t−s

.

Thusd

ds

(Pt−sv(s, ·)(B)

)= 0,

which implies that the map s 7→ Pt−sv(s, ·)(B) equals a constant on (0, t). Itfollows that

Ptv(0+, ·)(B) = lims↓0Pt−sv(s, ·)(B) = lim

s↑tPt−sv(s, ·)(B) = v(t, B),

where the first and last equality are a consequence of Lemma 3.2. Because ofv(0+, ·) = h one obtains v(t, B) = Pth(B). 2

Theorem 3.1. Let (Tt)t≥0 be a Markov semigroup on C(X ) with infinitesimalgenerator (A, ϑ(A)) and let H be a real-valued bounded function on X × Ysatisfying the condition

H(·, B) ∈ ϑ(A), B ∈ Y .

(a) Suppose that (Pt)t≥0 is a continuous conservative transition semigroupon Y with infinitesimal characteristics Q. If (Tt)t≥0 and (Pt)t≥0 are induality with respect to H, then

AH(·, B)(η) = QH(η, ·)(B), η ∈ X , B ∈ Y , (D-I)

is satisfied.

(b) Suppose that Q := (Q(B, C))B,C∈Y is a rate matrix satisfying (Q). IfQ and A fulfill condition (D-I), then the continuous conservative transi-tion semigroup (Pt)t≥0 on Y which is associated to Q and the Markovsemigroup (Tt)t≥0 are in duality with respect to H. This is, the condition

TtH(·, B)(η) = PtH(η, ·)(B), η ∈ X , B ∈ Y , (D-S)

holds.


Proof.(a) By the definition of the infinitesimal generator, the limit

limt↓0

1t

(TtH(·, B)(η)−H(η, B)

)= AH(·, B)(η)

exists for each B ∈ Y uniformly with respect to η ∈ X . Hence it follows from(D-S) that

limt↓0

1t

∑

C∈Y

Pt(B,C)(H(η, C)−H(η, B)

)= AH(·, B)(η) (3.4)

exists for each B ∈ Y uniformly with respect to η ∈ X . Applying Lemma 3.1with h(·) := H(η, ·)−H(η,B), where B ∈ Y is fixed, one obtains (D-I).

(b) If the condition (Q) holds for the rate matrix Q, there exists a uniquelydetermined continuous conservative transition semigroup (Pt)t≥0 on Y withinfinitesimal characteristics Q. Let (D-I) be satisfied. Consider the functionv : [0,∞)× Y ×X → R defined by

v(t, V ; η) := PtH(η, ·)(V ), t ≥ 0, V ∈ Y , η ∈ X .

Since (Pt)t≥0 is contractive and H ∈ B(X × Y ), one finds that

|v(t, V ; η)| ≤ ‖H‖∞, t ≥ 0, V ∈ Y , η ∈ X ,

therefore v(·, ·; η) ∈ B([0,∞) × Y ) for each η ∈ X . By Proposition 3.1, thefunction v(·, ·; η) is for fixed η ∈ X the unique solution in B([0,∞)×Y ) of theinitial value problem

d

dtv(t, V ) =

∑

C∈Y

Q(V, C)[v(t, C)− v(t, V )], V, C ∈ Y , t > 0,

v(0+; V ) = H(η, V ), V ∈ Y . (3.5)

Define a function u : [0,∞)×X × Y → R by

u(t, η; B) := TtH(·, B)(η), t ≥ 0, η ∈ X , B ∈ Y .

Since (Tt)t≥0 is contractive and H ∈ B(X × Y ), one finds that

|u(t, η; B)| ≤ ‖H‖∞, t ≥ 0, η ∈ X , B ∈ Y ,

therefore u(·, ·; η) ∈ B([0,∞)×Y ) for each η ∈ X . Consider u(·, ·; B) for fixedB ∈ Y . For the Markov semigroup (Tt)t≥0 and its infinitesimal generator Athe following equation is valid on ϑ(A)

d

dtTt = TtA, t ≥ 0,


see, for instance, [5]. Since H(·, B) ∈ ϑ(A), this yields

d

dtu(t, η;B) = TtAH(·, B)(η), t ≥ 0, η ∈ X , B ∈ Y .

It follows from (D-I) that, for each B ∈ Y , the function η 7→ QH(η, ·)(B) iscontinuous on X , because AH(·, B) ∈ C(X ). Consequently, applying (D-I) tothe above equation yields

d

dtu(t, η;B) = Tt

( ∑

C∈Y

Q(B, C)[H(·, C)−H(·, B)])(η), t ≥ 0, η ∈ X , B ∈ Y .

Now choose some sequence (Yn)n∈N, the Yn’s being finite subsets of Y whichsatisfy Yn ↑ Y for n →∞. As Tt is bounded, one finds

d

dtu(t, η; B) = Tt

( ∑

C∈Y

Q(B,C)[H(·, C)−H(·, B)])(η)

= Tt

(lim

n→∞

∑

C∈Yn

Q(B,C)[H(·, C)−H(·, B)])(η)

= limn→∞

∑

C∈Yn

Q(B, C)[TtH(·, C)(η)−H(η, B)]

= limn→∞

∑

C∈Yn

Q(B, C)[u(t, η;C)− u(t, η; B)]

=∑

C∈Y

Q(B, C)[u(t, η;C)− u(t, η; B)], t ≥ 0, η ∈ X .

Observe further that u(0+, η; B) = H(η, B), η ∈ X . Since B ∈ Y can bechosen arbitrarily, we obtain that the function u(·, η; ·) satisfies the initial valueproblem (IVP) for each η ∈ X . As the solution of this initial value problem isunique in B([0,∞)× Y ), it follows that

v(t, B; η) = u(t, η;B), t ≥ 0, η ∈ X , B ∈ Y .

This is (D-S). 2

Remark 3.2. Actually, it is enough to require in Theorem 3.1 (b) that the ratematrix Q and the semigroup (Pt)t≥0 generated by Q satisfy condition (PQ).However, if only Q is given explicitly, it is often easier to verify that the condi-tion (Q) is met.

Corollary 3.1. Suppose that (Ai, ϑ(Ai)), i ∈ I, are Markov generators on X ,where I is a countable index set. Assume that there is a set D ⊂ ∩i∈Iϑ(Ai)


with 1 ∈ D which is a core4 for each operator Ai, i ∈ I, such that the operator

A : D −→ C(X ) : Af :=∑

i∈I

Aif5

is a Markov pregenerator whose closure is a Markov generator.Let H be a real-valued bounded measurable function on X × Y satisfying

the conditionH(·, B) ∈ D, B ∈ Y .

Assume further that there are Markov chains with infinitesimal characteris-tics Qi which are H-dual to the Markov processes generated by Ai, i ∈ I. If

−∞ < Q(B, C) :=∑

i∈I

Qi(B; C) < ∞, B,C ∈ Y , (3.6)

and the corresponding Q-matrix6 Q = (Q(B, C)B,C∈Y satisfies (Q), then theMarkov process corresponding to A is H-dual to the Markov chain on Y withinfinitesimal characteristics Q.

Remark 3.3.

(1) Since each (Ai, D), i ∈ I is a Markov pregenerator, the operator A : D →C(X ) : Af :=

∑i∈I Aif satisfies the conditions (G1) and (G2) of § 2.1

and is therefore a Markov pregenerator. It is not immediate, in general,that (A, D) satisfies the condition (G3), as well.

(2) Suppose that A1 and A2 are generators of IPS as defined in § 2.2 with cor-responding families of transition rates (c(1)

T (·, ·))T∈T and (c(2)T (·, ·))T∈T ,

respectively. Then D(X ) ⊂ ϑ(A1) ∩ ϑ(A2) is a core for Ai, i = 1, 2and the operator (A, D(X )) is the pregenerator of an IPS with transitionfamily (cT (·, ·))T∈T satisfying cT (·, ·) = c

(1)T (·, ·) + c

(2)T (·, ·). It is easily

checked that the family (cT (·, ·))T∈T is admissible, if (c(1)T (·, ·))T∈T and

(c(2)T (·, ·))T∈T are admissible. Thus one obtains by following the argu-

ments in § 2.2 that the closure of (A,D(X )) is a Markov generator.

(3) Suppose that A is the generator of an IPS as defined in § 2.2 with corre-sponding family of transition rates (cT (·, ·))T∈T . According to (2.1), itholds that Af =

∑T∈T AT f , f ∈ D(X ), where AT : C(X ) → C(X ),

4See [5, §1.3] for the definition of a core.5The convergence is with respect to the supremum norm in C(X ).6Q(B) :=

PC 6=B Q(B, C) =

PC 6=B

Pi∈I Qi(B, C) =

Pi∈I

PC 6=B Qi(B, C) =P

i∈I −Qi(B, B) = −Q(B, B) < ∞, B ∈ Y , where the rearrangement in the order of sum-mation is allowed because all summands are non-negative.


T ∈ T are bounded Markov generators that describe the local transi-tions. Thus D(X ) ⊂ ⋃

T∈T ϑ(AT ) is a core for each AT , T ∈ T , andthe operator (A, D(X )) is a pregenerator of a Markov process. If thelocal mechanisms AT are H-dual to Markov chains on Y with infinites-imal characteristics QT , T ∈ T , then the H-dual of A exists and hasinfinitesimal characteristics Q =

∑T∈T QT , if Q is well-defined in the

sense of (3.6) and satisfies condition (Q).

Proof of Corollary 3.1. By Theorem 3.1, the infinitesimal characteristics Qi ofthe H-dual chains satisfy

AiH(·, B)(η) = QiH(η, ·)(B), i ∈ I, η ∈ X , B ∈ Y .

Hence

AH(·, B)(η) =∑

i∈I

AiH(·, B)(η) =∑

i∈I

QiH(η, ·)(B)

=∑

i∈I

∑

C 6=B

Qi(B,C)(H(η, C)−H(η, B)

)

=∑

C 6=B

∑

i∈I

Qi(B,C)(H(η, C)−H(η, B)

)

= QH(η, ·)(B), η ∈ X , B ∈ Y .

The change in the order of summation is allowed since the series is absolutelyconvergent. Indeed,

∑

i∈I

∑

C 6=B

∣∣Qi(B, C)(H(η, C)−H(η, B)

)∣∣

≤∑

i∈I

∑

C 6=B

Qi(B,C) |H(η, C)−H(η,B)|︸︷︷︸≤2

≤ 2∑

i∈I

Qi(B) < ∞.

Since the matrix Q satisfies (Q), there exists a uniquely determined continuousand conservative transition semigroup on Y with infinitesimal characteristics Q.By Theorem 3.1, the corresponding Markov chain is H-dual to the IPS generatedby A. 2

4. Duals of IPS

Now the results of Section 3 are applied to special IPS and special dualityfunctions H. Thereby it is illustrated that the criterion for duality which hasbeen developed above is indeed manageable for a variety of different interactingparticle systems. Note that the duality relations that will be derived in theexamples below have been asserted in the literature already. However the proof


of these relations was either only scheduled or even omitted in the originalpublication.

In § 4.1 to § 4.3, IPS with state space X = WS are considered, where W ={0, 1} and S = Zd with d ∈ N. Define

T∅ := {T ⊂ S : |T | < ∞}, T := T∅\{∅}.For the duality function the following map H : X ×T∅ → {0, 1} is chosen

H(η,B) :=∏

x∈B

η(x), η ∈ X , B ∈ T , H(·, ∅) ≡ 1. (4.1)

Hence the state space of H-dual Markov chains is Y := T∅. Obviously, H(·, B)∈T (X ) ⊆ D(X ) for each B ∈ T∅. Note that

H(η, B) =∏

x∈B

η(x) = 1 ⇐⇒ η(x) = 1, x ∈ B.

In § 4.4–§ 4.5, the local state space W as well as the duality function H arechosen to be more general.

4.1. Spin-flip systems

Spin-flip systems are IPS with single site space W = {0, 1} and admissibletransition rates c = (cT (·, ·))T∈T satisfying cT (η, {v}) = 0, T ∈ T , η ∈ X ,v ∈ XT unless T = {x} for some x ∈ S. For convenience, denote for η ∈ X ,x ∈ S,

ηx(z) :=

{η(z), z 6= x,

1− η(x), z = x,z ∈ S,

andc(x, η) := c{x}

(η, {1− η(x)}).

The corresponding Markov pregenerator A : D(X ) → C(X ) takes the form

Af(η) =∑

x∈S

c(x, η)(f(ηx)− f(η)), f ∈ D(X ), η ∈ X .

As above, the Markov semigroup that is generated by A is denoted by (Tt)t≥0.The results of Section 3 shall be applied to spin-flip systems where the

transition rates have a special structure, compare [9, § III.4]. In detail, letp : S ×T∅ → [0, 1] be a map that satisfies

∑

F∈T∅

p(x, F ) = 1, x ∈ S, (4.2)

supx∈S

∑

F∈T∅

p(x, F )|F | =: κ < ∞.


It is supposed that the transition rates c of the spin-flip system take the form

c(x, η) := η(x) + (1− 2η(x))∑

F∈T∅

p(x, F )H(η, F ), η ∈ X , x ∈ S. (4.3)

Note that (4.2) guarantees that the family c = (c(x, ·))x∈S is admissible [9,§ III.4]. A rate matrix Q on T∅ is specified via7,8

Q(B, C) :=∑

x∈B

∑

F∈T∅

p(x, F )δF∪(B\{x})(C), B ∈ T , C ∈ T∅, B 6= C, (4.4)

andQ(B, B) := −Q(B) := −

∑

C∈T∅\{B}Q(B,C), B ∈ T∅.

One finds

Q(B) =∑

C∈T∅,C 6=B

Q(B, C) =∑

C∈T∅,C 6=B

∑

x∈B

∑

F∈T∅

p(x, F )δF∪(B\{x})(C)

≤∑

x∈B

∑

F∈T∅

p(x, F ) = |B| < ∞, B ∈ T∅,

thus the matrix Q is actually well-defined. In addition, following [9, § III.4], oneobtains

∑

C∈T∅,C 6=B

Q(B, C)[|C| − |B|]

=∑

x∈B

∑

F∈T∅

p(x, F )[|(B \ {x}) ∪ F | − |B|]

≤∑

x∈B

∑

F∈T∅

p(x, F )(|F | − 1) ≤ κ|B|, B ∈ T∅.

Choosing Tn := {T ∈ T∅ : |T | ≤ n}, n ∈ N, and ϕ(B) := |B|, B ∈ T∅,the condition (Q) of § 2.3 is satisfied. Hence the matrix Q generates a uniquelydetermined continuous conservative transition semigroup (Pt)t≥0 on T∅. Oneeasily verifies that A and Q satisfy condition (D-I). Indeed, with

H(ηx, B)−H(η,B) =

{(1− 2η(x))H(η, B \ {x}) for x ∈ B,

0 for x /∈ B,

andH(η, C)H(η, D) = H(η, C ∪D), C, D ∈ T∅,

7The Kronecker symbol satisfies δB(C) = 1 if C = B and δB(C) = 0 otherwise.8By convention, a sum taken over the empty set is zero.


it follows from (4.3) that

AH(·, B)(η)

=∑

x∈B

c(x, η)[H(ηx, B)−H(η,B)] =∑

x∈B

c(x, η)(1− 2η(x))H(η, B \ {x})

=∑

x∈B

{(η(x) + (1− 2η(x))

∑

F∈T∅

p(x, F )H(η, F ))(1− 2η(x))H(η,B \ {x})

}

=∑

x∈B

η(x)(1− 2η(x))H(η,B \ {x})

+∑

x∈B

(1− 2η(x))2∑

F∈T∅

p(x, F )H(η, F )H(η,B \ {x})

=∑

x∈B

∑

F∈T∅

p(x, F )[H(η, F ∪ (B \ {x})−H(η, B)

]

=∑

C∈T∅

Q(B,C)[H(η, C)−H(η, B)], η ∈ X , B ∈ T∅.

Hence the IPS generated by A and the Markov chain corresponding to Q are induality w.r.t. H.

Example 4.1. If one specializes

p(x, F ) :=

{0, if |F | 6= 1pv(x, y), if F = {y} ⊂ S,

where pv is an irreducible stochastic matrix on S, then the rates in (4.3) takethe form

cv(x, η) =∑

y∈S

pv(x, y)(η(y)− η(x)

)2, x ∈ S, η ∈ X . (4.5)

These are the transition rates of a so-called linear voter system. The infinitesimalcharacteristics of its H-dual Markov chain on T∅ are given by

Qv(B, C) =∑

x∈B

∑

y∈C

p(x, y)δ{y}∪(B\{x})(C), B ∈ T , C ∈ T∅, B 6= C. (4.6)

4.2. Symmetric exclusion processes

Symmetric exclusion processes fall into the class of spin-exchange processes.The latter processes are IPS with W = {0, 1} and admissible transition ratesc = (cT (·, ·))T∈T satisfying cT (η, {v}) = 0, T ∈ T , η ∈ X , v ∈ XT unless


T = {x, y} for some x, y ∈ S, x 6= y and v(x) = η(y), v(y) = η(x). Forconvenience, denote for η ∈ X , x, y ∈ S, x 6= y,

ηxy(z) :=

η(z), z 6= x, y,

η(y), z = x,

η(x), z = y,

z ∈ S,

and

c(x, y, η) := c{x,y}(η, {v}), where v ∈ X{x,y} with v(x) := η(y), v(y) := η(x).

If the transition rates c(x, y, η) admit a representation c(x, y, η) = ce(x, y, η)with

ce(x, y, η) = pe(x, y)η(x)(1− η(y)), x, y ∈ S, x 6= y, η ∈ X , (4.7)

where pe = (pe(x, y))x,y∈S is an irreducible and symmetric stochastic matrixon S with pe(x, x) = 0, x ∈ S, then the corresponding IPS is a symmetricexclusion process. Note that the family of transition rates (c(x, y, ·))x,y∈S is ad-missible, since, by the symmetry of pe, the condition supy∈S

∑x∈S pe(x, y) < ∞

is satisfied [9, Ch.VIII]. The corresponding Markov pregenerator Ae : D(X ) →C(X ) takes the form

Aef(η) =∑

x,y∈S

p(x, y)η(x)(1− η(y))(f(ηxy)− f(η)), f ∈ D(X ), η ∈ X .

(4.8)Let

Qe(B, C) :=∑

x∈B

∑

y∈C\Bpe(x, y)δ(B\{x})∪{y}(C), B, C ∈ T∅, B 6= C,

Qe(B,B) := −∑

C∈T∅\{B}Qe(B, C), B ∈ T∅. (4.9)

Then one has for B ∈ T∅, C ∈ T∅, B 6= C, that

Qe(B,C) ≤∑

x∈B

∑

y∈C

pe(x, y)δ{y}∪(B\{x})(C) = Qv(B, C)

andQe(B) ≤ Qv(B),

where Qv is the rate matrix of a linear voter system, see (4.6). This means thatQe is a well-defined rate matrix which satisfies condition (Q2) for ϕ(B) = |B|,B ∈ T∅. It satisfies the condition (Q1), as well, since one finds that

δ(B\{x})∪{y}(C)[|C| − |B|] = 0 ≤ |B|,


for B, C ∈ T∅ and x, y ∈ S, x ∈ B, y /∈ B. Now it shall be verified that Ae

and Qe are H-dual, where H is the defined in (4.1). Using that

H(ηxy, B) = H(η, (B \ {x}) ∪ {y}) for x ∈ B, y /∈ B,

and applying the symmetry of pe, one finds, following [9, §VIII.1],

AeH(·, B)(η) =∑

x∈S

∑

y∈S

pe(x, y)η(x)(1− η(y))[H(ηxy, B)−H(η, B)]

=∑

x∈B

∑

y 6∈B

pe(x, y)[η(x)(1− η(y)) + η(y)(1− η(x))

]

× [H(η, (B \ {x}) ∪ {y})−H(η, B)

]

=∑

x∈B

∑

y 6∈B

pe(x, y)[H(η, (B \ {x}) ∪ {y})−H(η,B)

]

=∑

C∈T∅

Qe(B, C)[H(η, C)−H(η, B)], η ∈ X , B ∈ T∅.

The third equality follows from the fact that[η(x)(1−η(y))+η(y)(1−η(x))

]= 1

if η(x) 6= η(y). Thus Ae and Qe satisfy (D-I) w.r.t. H. Consequently, byTheorem 3.1, the symmetric exclusion process has an H-dual Markov chainon T∅ with infinitesimal characteristics Qe.

4.3. Voter-exclusion processes

Suppose that (cv(x, ·))x∈S and (ce(x, y, ·))x,y∈S are the transition rates ofa linear voter system w.r.t. the irreducible stochastic matrix pv on S and of aspin-exchange process w.r.t. the symmetric and irreducible stochastic matrix pe

on S, respectively, as defined in (4.5) and (4.7). The corresponding Markovpregenerators are given by

Avf(η) =∑

x∈S

∑

y∈S

pv(x, y)(η(y)− η(x))2(f(ηx)− f(η)), η ∈ X , f ∈ D(X ),

and

Aef(η) =∑

x∈S

∑

y/∈S

pe(x, y)η(x)(1− η(y))(f(ηxy)− f(η)), η ∈ X , f ∈ D(X ).

The IPS corresponding to A := Av + Ae on D(X ) is called voter-exclusionprocess9.

Let Qv and Qe be the infinitesimal characteristics of the H-dual chainscorresponding to Av and Ae, respectively, as given in (4.6) and (4.9). Both

9see [7].


Av,Qv and Ae, Qe satisfy (D-I) with respect to the duality function H givenin (4.1). Since Qv and Qe meet condition (Q) w.r.t. the same function ϕ = | · |,one finds that Q := Qv +Qe satisfies (Q). Thus Corollary 3.1 applies. It followsthat A and Q satisfy (D-I) with respect to H. Hence the Markov chain whichis H-dual to the voter-exclusion process has the infinitesimal characteristics

Q(B,C) =∑

x∈B

∑

y∈C

(pv(x, y) + pe(x, y)1Bc(y)

)δ{(B\{x})∪{y}(C),

for B ∈ T∅, C ∈ T∅, B 6= C.

4.4. Symmetric two-particle exclusion-eating process

The symmetric two-particle exclusion-eating process was considered in [10].It is an IPS on X = WS with W := {0, 1, 2} and S a countable set, where onlytransitions η → ηx→y, x, y ∈ S, x 6= y, have positive rates. These transitionsare defined for x, y ∈ S, x 6= y by

ηx→y(z) :=

η(z), if z /∈ {x, y},η(x), if z = y,

min {2η(y), η(x)}, if z = x,

z ∈ S.

Suppose that p is a symmetric irreducible stochastic matrix on S. Then thegenerator of the two-particle exclusion-eating process is given by the closure ofthe operator

Af(η) =∑

x∈S

∑

y∈S

(1− δ0(η(x))

)(1− δη(x)(η(y))

)p(x, y)(f(ηx→y)− f(η)),

where f ∈ D(X ), η ∈ X . For η ∈ X and x, y ∈ S, x 6= y with η(x) 6= 0,η(y) 6= η(x), it holds that

ηx→y =

{ηxy, if η(y) = 0,

ηx, if η(y) 6= 0,

with exclusion transformation

ηxy(z) :=

η(z), if z /∈ {x, y},η(x), if z = y,

η(y), if z = x,

z ∈ S,

and spin-flip transformation

ηx(z) :=

{η(z), if z 6= x,

3− η(x), if z = x,z ∈ S.


Consequently, the operator A can be decomposed into the sum A = A1 + A2,where

A1f(η) =∑

x∈S

∑

y∈S

(1− δ0(η(x))

)δ0(η(y))p(x, y)(f(ηxy)− f(η))

and

A2f(η) =∑

x∈S

∑

y∈S

δ0(η(x)η(y))p(x, y)(f(ηx)− f(η))

for η ∈ X , f ∈ D(X ). It is easily verified that both A1 and A2 are IPS-pregenerators of the form (2.1), each one with admissible transition rates. There-fore, compare Remark 3.3 (2), the closure of A is indeed a Markov generatorin C(X ).

Now choose Y := {C = (C1, C2) ∈ T∅×T∅ : C1 ⊆ C2} to be the state spaceof the dual Markov chain. The duality function H|X × Y → {0, 1} shall begiven by

H(η,B) :=∏

x∈B1

δ2(η(x))∏

y∈B2

(1− δ0(η(y)), η ∈ X , B = (B1, B2) ∈ Y .

(4.10)Obviously, H(·,B) ∈ T (X ), B ∈ Y . Two rate matrices Q1, Q2 on Y aredefined by

Q1(B,C) :=∑

x∈B1

∑

y/∈B2

p(x, y)δB1\{x}∪{y}(C1)δB2\{x}∪{y}(C2)

and

Q2(B,C) :=∑

x∈B2\B1

∑

y/∈B2

p(x, y)δB1(C1)δB2\{x}∪{y}(C2),

where B = (B1, B2), C = (C1, C2) ∈ Y , B 6= C.From [10, Lemma 2.1] one has that (D0) holds for a symmetric two-particle

exclusion-eating process and a Markov chain on Y having the infinitesimalcharacteristics Q := Q1 +Q2, that is

(A1 + A2)H(·,B)(η) =∑

C∈Y

(Q1 +Q2)(B,C)[H(η,C)−H(η,B)],

for η ∈ X , B,C ∈ Y . To conclude that the semigroup duality relation (D-S)holds, one has to show that Q satisfies the condition (Q). To prove this, con-sider Q1 and Q2 separately. With Qi(B) :=

∑C6=BQ(B,C), B ∈ Y , i = 1, 2,

one finds thatQ1(B) ≤

∑

x∈B1

∑

y/∈B2

p(x, y) ≤ |B1|,


andQ2(B) ≤

∑

x∈B2\B1

∑

y/∈B2

p(x, y) ≤ |B2| − |B1|,

for B = (B1, B2) ∈ Y . Therefore Q(B) = Q1(B) + Q2(B) ≤ |B2|, B =(B1, B2) ∈ Y . Because of |B \ {x} ∪ {y}| = |B| for B ∈ T∅, x ∈ B, y /∈ B, itholds for i = 1, 2 that Qi(B,C) = 0 for B = (B1, B2), C = (C1, C2) ∈ Y with|C2| 6= |B2|. Hence

∑

C∈Y

Q(B,C)(|C2| − |B2|) = 0, B ∈ Y .

Choosing now Yn := {C = (C1, C2) ∈ Y : |C2| ≤ n}, n ∈ N, and ϕ : Y →[0,∞) : ϕ(B) := |B2|, B = (B1, B2) ∈ Y , one finds that (Q) is satisfied.

This shows that the Markov chain generated by Q is H-dual to the symmet-ric two-particle exclusion-eating process. These arguments complete the proofof [10, Theorem 2.7].

4.5. Lattice gas with energy

In [12], Nagahata studies IPS in X := WZd

, W := {0, 1, . . ., M}, M, d ∈ N,where transitions η → ηx→y and η → ηx,y are considered that are given by

ηx→y(z) :=

η(x)− 1, z = x,

η(y) + 1, z = y,

η(z), z /∈ {x, y},and

ηx,y(z) :=

η(y), z = x,

η(x), z = y,

η(z), z /∈ {x, y},x, y ∈ Zd, x 6= y. These transitions shall occur with rates Cge(η(x)) andCex(η(x)), if x and y are neighboring lattice sites, that is |x − y| = 110. Thetransition rates Cge, Cex : Zd → [0,∞) are given by

Cge(k) := cge(k)1[2,M ](k), Cex(k) := cex(k)(1− δ0(k)), k ∈ W,

where cge, cex : W → (0,∞) are specified in advance.It is easily checked that the linear operators Age, Aex : T(X ) → C(X )

defined by

Agef(η) :=∑

x,y∈Zd

δ1(|x− y|)Cge(η(x))(η(x))1[1,M−1](η(y))(f(ηx→y)− f(η)),

10Here |x| :=qPd

n=1 x2n, x ∈ Zd.


and

Aexf(η) :=∑

x,y∈Zd

δ1(|x− y|)Cex(η(x))δ0(η(y))(f(ηx,y)− f(η)),

f ∈ T(X ), η ∈ X , as well as the sum A := Age +Aex are Markov pregeneratorsin C(X ) whose closures in C(X ) are Markov generators.

According to Nagahata [12], the associated Feller process is called a latticegas with energy. It is thought as a model of the time evolution of a gas con-sisting of particles with several energy levels. For η ∈ X , x ∈ Zd, η(x) = 0 isinterpreted as ‘site x is vacant’ and η(x) 6= 0 as ‘there is a particle on the sitex with energy level η(x)’. A particle at site x moves with rate cex(η(x)) to anearest neighbor site y if y is vacant. A unit of energy of the particle at sitex is transferred to a nearest neighbor particle at site y with rate cge(η(x)), ifη(y) < M .

The dual chain shall act on the state space Y of T M∅ defined by

Y :={(B1, . . ., BM ) ∈ T M

∅ : Bj ∩Bk = ∅, j 6= k, j, k = 1, . . ., M}.

Define H : X × Y → {0, 1} by

H(η,B) :=M∏

i=1

∏

x∈Bi

δi(η(x)), η ∈ X , B = (B1, . . ., BM ) ∈ Y .

Obviously, for each B ∈ Y , it holds that H(·,B) ∈ T(X ) ⊂ C(X ).One checks easily that

H(ηx→y,B) = H(η,By→x) and H(ηx,y,B) = H(η,Bx,y), η ∈ X , B ∈ Y ,

where, for B ∈ Y ,

Bx→y :=(B1, . . ., Bi−1 ∪ {x}, Bi \ {x}, . . ., Bj \ {y}, Bj+1 ∪ {y}, . . ., BM

)

if x ∈ Bi, y ∈ Bj , 2 ≤ i ≤ M , 1 ≤ j ≤ M − 1, and Bx→y := B otherwise; and

Bx,y :=(B1, . . ., Bi \ {x} ∪ {y}, Bi+1, . . ., BM

)

if x ∈ Bi, y /∈ ∪Mi=1Bi and Bx,y := B otherwise. It follows that

AgeH(·,B)(η) = QgeH(η, ·)(B), and AexH(·,B)(η) = QexH(η, ·)(B),(4.11)

η ∈ X , B ∈ Y , if one chooses

Qge(B,C) :=M∑

i=2

∑

x∈Bi

M−1∑

j=1

∑

y∈Bj

δ1(|x− y|)Cge(i)δBy→x(C)


and

Qex(B,C) :=M∑

i=1

∑

x∈Bi

∑

y∈∪Mk=1(Ck\Bk)

δ1(|x− y|)Cex(i)δBx,y (C),

B,C ∈ Y , B 6= C. Let

Qge(B) := −Qge(B,B) :=∑

C∈Y

Qge(B,C), B ∈ Y ,

andQex(B) := −Qex(B,B) :=

∑

C∈Y

Qex(B,C), B ∈ Y .

With

|B| :=M∑

k=1

|Bk|, B = (B1, . . ., BM ) ∈ Y ,

one finds|Bx→y| = |Bx,y| = |B|. (4.12)

Therefore

0 ≤ Qge(B) ≤M∑

i=1

Cge(i)∑

x∈Bi

M∑

j=1

∑

y∈Bj

δ1(|x− y|)

≤ 2dM

M∑

i=1

Cge(i)|Bi| ≤ 2dMC0|B|, (4.13)

where C0 := max {Cge(i), 1 ≤ i ≤ M}. In a similar way, one obtains

0 ≤ Qge(B) ≤ 2dC1|B|, (4.14)

where C1 := max {Cex(i), 1 ≤ i ≤ M}. Hence Qge and Qex are rate matrices.Because of (4.11), Qge, Qex and the sum Q := Qge + Qex are H-dual to

Age, Aex and A, respectively. In particular, it holds that

AH(·,B)(η) = QH(η, ·)(B), η ∈ X , B ∈ Y .

Both Qge and Qex satisfy the condition (Q) with respect to the same functionϕ : Y → R. Indeed, let Yn := {B ∈ Y : |Bi| ≤ n, 1 ≤ i ≤ M}, n ∈ N, anddefine ϕ(B) := |B|, B ∈ Y . Then the sets Yn increase to Y with n → ∞and (Q0) is satisfied. Condition (Q1) is easily checked using (4.12) while (Q2)follows from (4.13) and (4.14). Thus, by Corollary 3.1, the Markov chain definedby Q is non-exploding and H-dual to the IPS generated by A.


References

[1] W. Anderson (1991) Continuous-Time Markov chains. Springer.

[2] P. Bremaud (1999) Markov Chains. Gibbs Fields, Monte Carlo Simulation andQueues. Springer.

[3] M.F. Chen (1992) From Markov Chains to Non-equilibrium Particle Systems.World Scientific.

[4] E.B. Dynkin (1965) Markov Processes. Springer.

[5] S.N. Ethier and T.G. Kurtz (1986) Markov Processes: Characterization andConvergence. Wiley.

[6] R.A. Holley and D.W. Stroock (1979) Dual processes and their applicationto infinite interacting systems. Adv. Math. 32, 149–174.

[7] P. Jung (2005) The noisy voter-exclusion process. Stoch. Process. Appl. 115(12), 1979–2005.

[8] Y. Kovchegov (2005) Exclusion processes with multiple interactions. Stoch.Process. Appl. 115 (7), 1233–1256.

[9] T.M. Liggett (1985) Interacting Particle Systems. Springer.

[10] X. Liu (1995) Symmetric two-particle exclusion-eating processes. Ann. Prob. 23(3), 1439–1455.

[11] F.J. Lopez and G. Sanz (2000) Duality for general interacting particle systems.Markov Processes Relat. Fields 6 (3), 305–328.

[12] Y. Nagahata (2005) Regularity of the diffusion coefficient matrix for the latticegas with energy. Ann. Inst. H. Poincare 41 (1), 45–67.

[13] J.R. Norris (1997) Markov Chains. Cambridge University Press.

[14] F. Spitzer (1970) Interaction of Markov processes. Adv. Math. 5 (2), 246–290.

[15] D.W. Stroock and S.R.S. Varadhan (1972) On degenerate elliptic-parabolicoperators of second order and their associated diffusions. Comm. Pure and Appl.Math. 25, 651–713.

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