www.imstat.org/aihp 2010, Vol. 46, No. 3, 822–852
DOI:10.1214/10-AIHP370
© Association des Publications de l’Institut Henri Poincaré, 2010
Markovian perturbation, response and fluctuation dissipation theorem
Amir Dembo
a,1and Jean-Dominique Deuschel
b,2aDepartment of Statistics and Department of Mathematics, Stanford University, Stanford, CA 94305, USA. E-mail:amir@math.stanford.edu bInstitut für Mathematik, Sekr. MA 7-4, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.
E-mail:deuschel@math.tu-berlin.de
Received 15 July 2008; revised 26 February 2010; accepted 23 April 2010
Abstract. We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of “linear response function” in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov processX(s)but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is, if the initial measureνis invariant for the given Markov semi-group, then for any pair of timess < tand nice functionsf, g, the dissipation, that is, the derivative ins of the covariance ofg(X(t))andf (X(s))equals the infinitesimal response at timet and directiongto any Markovian perturbation that alters the invariant measure ofX(·)in the direction off at times. The same applies in the so-called FDT regime near equilibrium, i.e. in the limits→ ∞witht−sfixed, providedX(s)converges in law to an invariant measure for its dynamics. We provide the response function of two generic Markovian perturbations which we then compare and contrast for pure jump processes on a discrete space, for finite-dimensional diffusion processes, and for stochastic spin systems.
Résumé. Nous considérons le théorème de fluctuation-dissipation de la mechanique statistique dans une approche mathématique.
Nous donnons un concept formel de la réponse linéaire dans le cadre général de la théorie des processus de Markov. Nous dé- montrons que pour un processus hors d’équilibre celle ci dépend non seulement du processus de MarkovX(s)mais aussi de la perturbation choisie. Nous charactérisons l’ensemble de toutes les réponses possibles pour un processus de Markov donné et dé- montrons qu’à l’équilibre elles satisfassent toutes le théorème de fluctuation-dissipation. C’est à dire, si une mesureνest invariante pour un semigroupe markovien donné, alors pour tout tempss < t et functions régulièresf, g, la dissipation, definie comme la dérivée ensde la covariance deg(X(t))et def (X(s))est égale à la réponse infinitésimale au tempsten direction degpour toute perturbation markovienne qui modifie la mesure invarianteνen direction defau tempss. Ce résultat s’étend au régime proche de l’équilibre, c.-à.-d. dans la limites→ ∞avect−sfixe, en supposant queX(s)converge en loi vers sa mesure invariante. Nous donnons la réponse pour deux perturbations markoviennes génériques, que nous comparons ensuite pour des processus de sauts dans un espace discret, pour des diffusions à dimension finie et pour une dynamique stochastique de spins.
MSC:Primary 60J25; 82C05; secondary 82C31; 60J75; 60J60; 60K35
Keywords:Markov processes; Out of equilibrium statistical physics; Langevin dynamics; Dirichlet forms; Fluctuation Dissipation Theorem
1Research partially supported by NSF Grants #DMS-0806211, #DMS-0406042 and #DMS-FRG-0244323.
2Research partially supported by DFG Grant #663/2-3.
1. Introduction and outline
One of the fundamental premises of statistical physics, the Fluctuation Dissipation Theorem (FDT), follows from the assumption that the response of a system in thermodynamic equilibrium to a small external perturbation is the same as its relaxation after a spontaneous fluctuation. The FDT provides an explicit relationship between the equilibrium fluctuation properties of the thermodynamic system and its linear response (e.g., susceptibility), which involve out-of- equilibrium quantities. As such it relates the dissipation of dynamics at thermal equilibrium of molecular scale (i.e., microscopic) models, with observable macroscopic response to external perturbations, allowing the use of microscopic models for predicting material properties (in the context of linear response theory). Among notable special cases of the FDT are the Einstein relation between particle diffusivity and its mobility [8] (or the recent accounts in [11,19,20]), and the Johnson–Nyquist formula [22] for the thermal noise in a resistor.
When deriving the FDT in equilibrium statistical physics one typically starts from a measureμ0(·)which is often a Gibbs measure, characterized by a HamiltonianH (·), and a dynamicsX(s) for which this measure is invariant.
For δ >0 small, one perturbs the dynamics so it becomes invariant for the Gibbs measure corresponding to the HamiltonianH (·)+δf (·). The linear response measures the effect of applying such perturbation at timeson the rate of change, asδ↓0 in the value of a test functiong(·)at timet > s(often takingf =gto be the state variable of the dynamics in question). This response function (ofsandt) is then compared to the rate of change insof the covariance betweenf (X(s))andg(X(t))at equilibrium, in the non-perturbed dynamics, whereby the FDT states that the ratio between these two functions is merelyβ, the inverse of the system’s temperature.
Whereas the FDT is well established and understood in physics, at least in or near thermal equilibrium, see [16, 17], our goal here is to provide its rigorous derivation from a mathematical perspective, as a result about perturbations of Markovian semi-groups. This is easy to do in special (simple) cases, most notably, when dealing with a Markov process on a finite state space. Aiming here instead for a unified derivation, we formalize in Definition2.7the concept of having a linearresponse functionin the general framework of a family of continuous time, homogeneous Markov processes Xf(·)that are invariant for the measures μf(·)=efμ0(·)(see Assumption2.5for the precise setting).
In our definition, the response functionRXf,g(s, t)depends on the initial position of the process at time 0, thereby allowing us to study the effect of the initial measure on the FDT relation. Though this function is uniquely defined per familyXf(·), it depends not only on the given Markov processX(·)but also on the chosen perturbationXf(·)of it (compare for example Theorems4.1and4.2). In Theorem2.10we characterize the set of all possible response func- tions for a given Markov processX(·)and show that they all satisfy the FDT relation (2.13). It states that if the initial measureμ0(·)is invariant for the underlying Markov processX(·), then the dissipation, that is the derivative insof the covariance betweenf (X(s))andg(X(t))fors < tequals theμ0-average of the infinitesimal responseRXf,g(s, t) to any Markovian fluctuationXδf(·)that alters the invariant measure ofX(·)in the direction of test functionf (·)at times, as registered at timet via test function g(·). We show in Corollary2.13that this FDT relation holds in the limits→ ∞andt−sfixed, whenever the initial measure is such that the law ofX(s)converges (in the appropriate sense) to an invariant measureμ0(·)and in Proposition2.14we specify the set of all possible response functions in case ofμf-symmetric (i.e., reversible) Markovian perturbations. Note that the FDT relation has to do with invariance ofμ0(·)but does not require the Markov processX(·)to be reversible with respect toμ0(·). To further demonstrate how widely this theory can be used, we construct in Section3two generic families of Markov perturbations satisfying Assumption2.5, which apply foranyMarkov process (subject only to mild restrictions on the domain of certain gener- ators). Namely, thetime changeof Propositions3.1and3.4and thegeneralized Langevin dynamicsof Proposition3.2 (in the symmetric case) and Proposition3.8(in the general, non-symmetric case). The bulk of the mathematical work in this paper is in proving that these two generic families admit a response function per Definition2.7. This is done in Theorems4.1and4.2of Section4which also provide an explicit formula for the response function in each case.
Moving to examples of specific Markov processes, in Proposition 5.1we use the simple sufficient condition of Proposition2.9for the existence of a response function, to show that essentially all choices of the response function that are possible per Theorem 2.10are indeed attainable in the context of pure jump processes on a discrete state space. We also demonstrate there how to create a host of perturbations via cycle decomposition (e.g., perturbation of a Metropolis dynamics, or of a Glauber dynamics). In Section6we illustrate our generic Langevin perturbation for dif- fusion processes on connected, compact, smooth, finite-dimensional manifolds without boundary, showing in Proposi- tion6.1that it is generated in this case by the addition of a smooth drift, which for symmetric diffusions is of gradient form. Finally, in Section 7we demonstrate the flexibility of our framework by considering such a perturbation for infinite-dimensional diffusion processes associated with stochastic spin systems in the setting of Gibbs distributions.
All our derivations and results apply even when the Markov processX(·)has more than one invariant measure (as is often the case in statistical physics when the system’s temperature is sufficiently low), and most of them apply also for non-reversible dynamics (i.e., having non-symmetric semi-groups).
We note in passing that the appearance ofβin the FDT in physics is merely due to the definition of Gibbs measure as proportional to e−βH, withH the corresponding Hamiltonian, and not countingβ as part of the perturbationδf. It makes more sense for the mathematical version of the FDT to not mentionβ(so it in effect corresponds to doing statistical physics atβ=1). Also, though from a mathematical point of view the response function cannot be defined solely in terms of the given Markov processX(·), this is never an issue in physics, whereby viewing the (Markov) dynamics as a classical approximation of a quantum system, the perturbation of its Hamiltonian in directionf (·) uniquely defines also the perturbed quantum system dynamics, hence its classical approximationXf(·)(e.g., see the physics based derivation in [14] of the FDT in quantum statistical mechanics).
2. General theory: FDT at or near equilibrium
A continuous time, homogeneous, strong Markov processX(t )with values in a complete, separable metric spaceS is defined on a fixed probability space(Ω,F,P). We assume thatX(t )has right continuous sample path and Markov semi-groupPth(x)=Ex(h(X(t)))∈Cb(S)for anyh∈Cb(S). LetD(Pt)denote the domain ofPtwith respect to the supremum norm. That is, the closed vector spaceD(Pt)= {h∈Cb(S):Pth−h∞→0 ast→0}on whichPt is strongly continuous and such thatPt:D(Pt)→D(Pt)and denote byD(L)the domain of the generatorL:D(L)→ D(Pt)of the semi-groupPt (i.e.,D(L):= {h∈D(Pt):t−1(Pth−h)→Lhin(D(Pt), · ∞)fort↓0}), which is a dense subset of(D(Pt), · ∞). Recall that ifg∈D(L)thent→Ptg:R+→D(L)is differentiable and∂tPtg= LPtg=PtLg.
For example, whenS is compact one often hasD(Pt)=Cb(S)whereas for S=Rd one typically hasD(Pt)= {h+c: c∈R, h∈C0(S)}for the spaceC0(S)of continuous functions that vanish at infinity.
The following hypothesis applies throughout.
Assumption 2.1. We assume thatD(Pt)is an algebra under point-wise multiplication and a dense subset ofL2(μ) for any probability measureμ.We consider test functions in a linear vector spaceG⊆D(L)such thatGis an algebra with1∈Gand thatGis dense in(D(Pt), · ∞).SettingG:= {Ptg:g∈G, t≥0},we further assume thatφ(g)∈G, Lg∈Gandghˆ∈D(L)for allg∈G,hˆ∈Gand anyφ∈C∞(R).
As usual we say that a finite measureμonSis invariant forPt ifPtgμ:=
Ptgdμ=
gdμfor allg∈BS and t≥0. SinceG⊆D(L)is dense inL2(μ), this is equivalent to
Lgˆdμ=0 for allg∈G.
A key role is to be played by the symmetric bi-linear Γ(f, g)=L(f g)−fL(g)−gL(f ),
whose domain isD(Γ):= {(f, g): f ∈D(L), g∈D(L), f g∈D(L)} ⊆D(Pt)×D(Pt). One may instead defineΓ as the “carré du champ” operator for the Dirichlet form associated with aμ0-symmetric semi-groupPt, thus possibly having a much larger domain (cf. [2], Proposition I.4.1.3). However, in either caseG×Gis a dense subset ofD(Γ), which is all we use in this paper.
Letν=ν0 denote the initial measure ofX(0)andνt:=ν◦Pt the corresponding measure ofX(t ). Fixing 0≤ s < t <∞andf ∈L2(νs),g∈L2(νt), we denote the covariance of the random variablesf (X(s))andg(X(t))by Kf,g(s, t). By the Markov property we have
Kf,g(s, t)=
Ps(fPt−sg)
ν− PsfνPtgν. (2.1)
The next lemma provides useful formulas for the derivative of the covariance with respect tos.
Lemma 2.2. For anyf, g∈Gand all0≤s < t <∞,
∂sKf,g(s, t)=
PsL(fPt−sg)
ν−
Ps(fPt−sLg)
ν− PsLfνPtgν (2.2)
=
SPsΓ(f,Pt−sg)dν+KLf,g(s, t). (2.3)
Supposeνis invariant forPt.Then,
∂sKf,g(s, t)= −fLPt−sgν. (2.4)
If in additionPt isν-symmetric(i.e.,fPtgν= gPtfνfor allf, g∈BS),then
∂sKf,g(s, t)=1 2
Γ(f,Pt−sg)
ν. (2.5)
Proof. Fixf, gands, tas in the statement of the lemma, andδ >0. LetDdenote the right-hand side of (2.2),h=Lg, δ=Pδ−Iandδ=δ−1(Pδ−I)−L. It is not hard to verify that
δ−1
Kf,g(s+δ, t )−Kf,g(s, t)
−D= PsδfPt−sgν− Ps+δfPt−s−δδgν
+ Ps+δfPt−s−δδhν− PsδfPt−shν+ PsδfνPtgν. Recall that{Pt}is contractive for the supremum norm whileδh∞→0 andδh∞→0 asδ↓0 for each fixed h∈D(L). Sincef is bounded withf,g,fPt−sgandfPt−shinD(L), it follows that
limδ↓0δ−1
Kf,g(s+δ, t )−Kf,g(s, t)
=D.
Similar computation applies for the case ofδ <0, thus establishing (2.2). SincePt−sLg=LPt−sg forg∈G, the equality (2.2) is a direct consequence of (2.1) and the definition ofΓ(·,·). To derive (2.4) note that
D=
L(fPt−sg)
νs − fLPt−sgνs− LfνsPtgν.
If ν is invariant for Ps then νs =ν and further Lhνs =0 for any h∈D(L). Consequently, in this case D=
−fLPt−sgν yielding (2.4). If in additionPt isν-symmetric, then obviouslyfLhν= hLfνfor allf, h∈D(L),
so (2.5) holds by the definition ofΓ(·,·).
Definition 2.3. The G-convergence ass→ ∞of probability measuresνs onS to a probability measureμon S, denotedνs→Gμ,means thathνs → hμass→ ∞,for any fixedh∈ {fg,ˆ L(fg):ˆ f∈G,gˆ∈G}.
For example, the weak convergence ofνs toμimplies thatνs→Gμas well.
Corollary 2.4. Ifνs=ν◦Ps→Gμass→ ∞,then(2.2)implies that for allf, g∈G,and any fixedτ,
slim→∞∂sKf,g(s, s+τ )=
L(fPτg)
μ− fPτLgμ− LfμPτgμ. If in additionPt isμ-symmetric then by(2.5),
slim→∞∂sKf,g(s, s+τ )=1 2
Γ(f,Pτg)
μ.
Given an invariant probability measureμ0forPt we consider throughout the following setting (where the Banach spaceBis typicallyL2(μ0)or(Cb(S), · ∞)).
Assumption 2.5. The norm topology on a Banach space (B, · )ofμ0-integrable functions is finer than the one induced byL1(μ0)andh ≤ h∞for allh∈D(Pt)⊆B.For eachf ∈G there exists a continuous time,homo- geneous Markov processXf(t)with a contractive semi-groupPft on(B, · )such thatPft h−h →0ast→0 for anyh∈D(Pt)and the finite,positive measureμf onS such thatdμf/dμ0=ef is invariant forPft .Further, X0(t)=X(t ),the subspaceGis contained in the domain of the generatorLf ofPft ,andLfg∈Gfor allg∈G.
Remark 2.6. It is easy to verify that all results and proofs of this section remain valid in caseμf =eΦ(f )μ0as long asΦ:G→BSis such that for any fixedf ∈Gthe functionsψδ:=δ−1(eΦ(δf )−δf −1)converge to zero inL1(μ0) whenδ↓0.
The FDT is about the relation between derivatives of the covariance at equilibrium and theresponseof the system to small perturbation out of equilibrium. We turn now to the rigorous definition of the latter (see also Proposition2.9 for an easy sufficient condition for existence of such response, in caseLis bounded on(B, · )).
Definition 2.7. Assume2.5and that for anyf ∈Gthere exists a linear operator Af:D(Af)→Bwhose domain D(Af)containsG,and such thats→AfPsgis strongly continuous in(B, · )for eachg∈G.If,moreover,for any T ≥0,gˆ∈G,allt∈ [0, T]andδ >0,
δ−1
Pδft −Pt
gˆ− t
0
PsAfPt−sgˆds
≤ηδt, (2.6)
andηδ=ηδ(f,g, T )ˆ →0asδ↓0,then we call
RXf,g(s, t)=PsAfPt−sg∈B (2.7)
(which for anyt≥0is strongly integrable on[0, t]),the response function(at timetand directiong)for the Markovian perturbationsXf(·)onG(applied at times).
Note thatRXf,g(s, t)is uniquely defined per given Markovian perturbationsXf(·). Indeed, suppose that for some f ∈G the inequality (2.6) holds for the same semi-groupsPδft and both linear operatorsAf andAf. Then, taking δ↓0 we see that the linear operatorA=Af −Af is such that for allt >0 andgˆ∈G,
t−1 t
0
PsAgˆds+t−1 t
0
Pt−sA(Ps−I)gˆds=0. (2.8)
WithPs strongly continuous, upon takingt↓0 the left-most term converges toAg, whereasˆ t−1t
0Pt−sA(Ps − I)gˆds→0 by the contractivity ofPt−s on(B, · )and the assumed strong continuity of APsg. Consequently,ˆ Af=Af on the setG, and so using either operator in (2.7) leads to the same response function.
As we demonstrate next,RXf,g(u, a+t+b)of (2.7) is merely the effect in “direction”gand at timea+t+b, of a small perturbation of the dynamics in “direction”f during the time intervalu∈ [a, a+t], in agreement with the less formal definition of response function one often finds in the literature.
Corollary 2.8. For anyf, g∈G,eachT ≥a, b, t≥0and allδ >0,a response function of the form(2.7)must satisfy the inequality
δ−1Pa
Pδft −Pt
Pbg− a+t
a
RXf,g(u, a+t+b)du
≤ηδt (2.9)
forηδ=ηδ(b, f, g, T )→0asδ↓0.
Proof. Using the expression (2.7) to write (2.9) more explicitly, one finds that fora=0 the latter is precisely (2.6) forv=b, hence it obviously holds. Moreover, in casea >0 we merely consider the norm ofPahδ, wherehδ is the element ofBthe norm of which we consider in (2.6). AsPais contractive on(B, · ), we have that (2.9) holds also
in this case.
We provide now an explicit sufficient condition for the existence of a response function of the form (2.7) whenLf, LandAf are bounded operators (as is the case in the setting of Section5).
Proposition 2.9. Assume2.5holds.If for eachf∈G,the operatorsAf,L,Lδf,δ >0are bounded on(B, · )and the corresponding operator norms are such that
limδ↓0
δ−1
Lδf−L
−Af =0, (2.10)
then(2.6)holds andAfPsgis strongly continuous,for eachg∈G.
Proof. SinceAfPs+vg−AfPvg ≤ AfPsgˆ− ˆgforgˆ=Pvg, the strong continuity ofs→AfPsgis a direct consequence of the strong continuity ofPs on its domain. Fixingf, g∈G andv≥0 it remains only to show that t−1ρtδ →0 asδ↓0, uniformly overt∈(0, T], where
ρtδ:=δ−1
Pδft −Pt
gˆ− t
0
Pt−uAfPugˆdu.
To this end, letrt:= −t
0Pt−uAfPugˆduand note that
∂tρtδ=δ−1
LδfPδft −LPt
gˆ−AfPtgˆ+Lrt,
which imply, after some algebraic manipulations, that
∂tρtδ=Lδfρtδ+ζtδ (2.11)
for
ζtδ:=
L−Lδf rt+
δ−1
Lδf −L
−Af
Ptg.ˆ (2.12)
It is not hard to show that the solutionρtδof (2.11) with initial conditionρ0δ=0, is ρtδ=
t
0
Pδft−sζsδds.
With bothPδft andPt contractive on(B, · ), we thus have thatrt ≤tAfgand for anyt∈ [0, T], ρtδ ≤
t
0
ζsδ ds≤t
T L−Lδf Af + δ−1
Lδf −L
−Af g =:ηδ(f, g, T )t.
Finally, note that from (2.10) we have thatηδ→0 asδ↓0.
Our next theorem characterizes the type of response functions one may find. It also proves the FDT, showing that ifXf(·)has a response functionRXf,g(s, t), then the average of the response function according to an initial measureν0=μ0which is invariant for X(·), equals the time derivative of the covariance ofX(·)under the same initial measure.
Theorem 2.10 (Fluctuation Dissipation Theorem). Letf∈G.IfXf(·)has a response function,thenAf1=0and Arf =rAf for allr >0.Further,then(Af +fL)gˆμ0 =0 for allgˆ∈Gand consequently,if the initial measure ν0=μ0,then for anys < t,
∂sKf,g(s, t)=
RXf,g(s, t)
ν0. (2.13)
Remark 2.11. IfPt isμ0-symmetric andf, g∈L2(μ0)with(f, g)∈D(Γ),then by spectral decomposition we have the Green–Kubo formula
−fLgμ0=1 2
Γ(f, g)
μ0= ∞
0
(PsLf )(Lg)
μ0
ds
(cf. [15],Theorem4.3.8).Applying it forf andPt−sg,we get the alternative expression
∂sKf,g(s, t)= ∞
0
(PsLf )(LPt−sg)
μ0
ds
for the dissipation term.In contrast with(2.13),this identity does not involve a perturbation of the given Markovian dynamic.
Proof of Theorem2.10. Fixf∈G. Ifg=1thenPug=Pδfu g=1for allδ >0 andu≥0, so in this case takingδ↓0 in (2.6) we find that for anyt >0,
t−1 t
0
PsAf1ds=0.
Thus, takingt↓0 we have thatAf1=0.
Next, fixingr >0, note thatPδ(rf )t =P(δr)ft for allδ >0, hence by (2.6) we have forA=Arf −rAf, t
0
PsAPt−sgˆds ≤
ηδ(rf,g, T )ˆ +rηδr(f,g, T )ˆ t.
Takingδ↓0 this implies that (2.8) holds forA=Arf −rAf and allt >0. Hence, by the argument we provided immediately following (2.8), we deduce thatAgˆ=0for allgˆ∈G. That is, without loss of generality we may assume thatArf =rAf, as claimed.
Letψδ=δ−1(eδf−δf−1). Sinceμδf is invariant forPδft andμ0is invariant forPt, it follows that δ−1
Pδft −Pt
gˆ
μ0 = ψδ
I−Pδft ˆ g
μ0− f
Pδft −Pt
gˆ
μ0−
f (Pt−I)gˆ
μ0
:=F1(δ)+F2(δ)+F3.
With (2.6) implying that(Pδft −Pt)gˆ →0 asδ↓0 (hence so does|(Pδft −Pt)gˆ|μ0), andf∞<∞we deduce thatF2(δ)→0. Further,|ψδ|μ0 →0 whenδ↓0 (sincef is bounded), while(I−Pδft )gˆ∞≤2 ˆg∞is uniformly bounded inδ, resulting withF1(δ)→0 as well. We thus deduce by considering the limitδ↓0 in (2.6), and applying Fubini’s theorem, that for anyt >0 andgˆ∈G,
f t−1(Pt−I)gˆ
μ0+t−1 t
0
Pt−sAfPsgˆμ0ds=0.
Further, withμ0invariant forPtand having assumed strong continuity ofs→AfPsg, upon takingˆ t↓0 we find that
fLgˆμ0+ Afgˆμ0=0, (2.14)
as claimed.
Next, recall (2.4) and (2.7) that forν0=μ0which is invariant forPsand for any finites < t,
∂sKf,g(s, t)−
RXf,g(s, t)
μ0= −fLPt−sgμ0− AfPt−sgμ0=0,
where the right-most identity is precisely (2.14).
Combining Corollary2.4and Theorem2.10 we deduce the existence of the FDT regime in out-of-equilibrium dynamics wheneverX(s)converges in law (in the appropriate sense), with the limiting measureμ0being invariant forX(·). That is, the FDT relation (2.13) then asymptotically holds fort−s=τfixed, in the limits→ ∞. Specifically, similar to Definition2.3we define the notion ofGf-convergence as follows.
Definition 2.12. The Gf-convergence of probability measuresνs onS to a probability measure μon S,denoted νs→Gf μ,means thathνs → hμass→ ∞,for any fixedh∈ {fg,ˆ L(fg),ˆ Afg:ˆ gˆ∈G} (implicitly assuming that Afgˆ∈L1(νs)for allslarge enough).
Corollary 2.13. Letν0denote the initial measure ofX(0)for anS-valued,continuous time,homogeneous,strong Markov processX(t )such thatXf(·)has a response function(in the sense of Definition2.7).Suppose further that νs=ν0◦Ps→Gfμ0ass→ ∞.Then,for anyg∈Gand fixedτ ≥0,
slim→∞∂sKf,g(s, s+τ )= −fLPτgμ0 = lim
s→∞
RXf,g(s, s+τ )
ν0. (2.15)
Proof. The left-hand side of the identity follows from the formula for the limit of∂sKf,g(s, s+τ )given in Corol- lary2.4and the fact thatμ0is invariant forPt soLhμ0=0 for allh∈D(L). Recall (2.7) thatRXf,g(s, s+τ )ν0= AfPτgνs which converges to AfPτgμ0 ass→ ∞. From Theorem2.10we have that (Af +fL)Pτgμ0 =0,
which thus yields the right side of the stated identity.
We conclude this section with a statement characterizing response functions for symmetric perturbations.
Proposition 2.14. Assume2.5holds for the Hilbert spaceB=L2(μ0)and that for eachf ∈G the semi-groupPft ofXf(t)is μf-symmetric.Then,any response function of the form(2.7)is based onAf =Bf −fLfor a linear operatorBf which isμ0-symmetric onG.
Remark 2.15. The stated μ0-symmetry of Bf =Af +fL is necessary in this setting of symmetric perturba- tions. However,typically more is required from Bf in order to assure the existence of a response function of the form(2.7).
Proof of Proposition2.14. Note that for anyδ >0 and allg,ˆ hˆ∈G, Fδ(ˆh,g)ˆ :=δ−1hˆ
Pδft −Pt
gˆ
μ0+hf (Pˆ t−I)gˆ
μ0
= hψˆ δ
I−Pδft ˆ g
μ0−hfˆ
Pδft −Pt
gˆ
μ0+δ−1hˆ
Pδft −I ˆ g
μδf −δ−1h(Pˆ t−I)gˆ
μ0
=:F1(δ,h,ˆ g)ˆ +F2(δ,h,ˆ g)ˆ +F3(δ,h,ˆ g)ˆ +F4(δ,h,ˆ g),ˆ
where ψδ =δ−1(eδf −δf −1). Recall that (I−Pδft )gˆ∞≤2 ˆg∞, hˆ is bounded and |ψδ|μ0 →0, hence F1(δ,h,ˆ g)ˆ →0 asδ↓0. Further,|(Pδft −Pt)gˆ|μ0 →0 hence alsoF2(δ,h,ˆ g)ˆ →0 asδ↓0. By theμδf-symmetry ofPδft and theμ0-symmetry ofPt, it follows that
F3(δ,h,ˆ g)ˆ =F3(δ,g,ˆ h),ˆ F4(δ,h,ˆ g)ˆ =F4(δ,g,ˆ h)ˆ for anyδ >0. Consequently, asδ↓0,
Fδ(ˆh,g)ˆ −Fδ(g,ˆ h)ˆ →0
which by (2.6) and theμ0-symmetry ofPt−uamounts to
Et(h,ˆ g)ˆ =Et(g,ˆ h),ˆ (2.16)
where
Et(h,ˆ g)ˆ :=
t
0
(Pt−sh)(Aˆ fPsg)ˆ
μ0ds+hf (Pˆ t−I)gˆ
μ0.
Sinces→AfPsgˆis strongly continuous (as part of Definition2.7of the response function), it follows that ast↓0, t−1
t
0 ˆhAfPsgˆμ0ds+t−1hf (Pˆ t−I)gˆ
μ0→ ˆhAfgˆμ0+ ˆhfLgˆμ0 = ˆhBfgˆμ0. Further, for allu >0,
(Pu−I)ˆh= u
0
PvLhˆdv,
hence(Pu−I)ˆh∞≤uLhˆ∞, while the strong continuity ofAfPsgˆimplies that sups≤tAfPsgˆ → Afgˆ<∞ ast↓0. Taken together, these imply that
limt↓0t−1 t
0
(Pt−s−I)hˆ
(AfPsg)ˆ
μ0ds=0, and consequently we have also that
t−1Et(h,ˆ g)ˆ → ˆhBfgˆμ0.
We thus conclude, based on (2.16), that ˆhBfgˆμ0= ˆgBfhˆμ0 for anyg,ˆ hˆ∈G, as claimed.
3. Generic Markov perturbations
In this section we construct several generic Markov perturbations for which Assumption2.5holds. Our presentation is somewhat technical because we aim at addressing a rather general framework. The reader may thus benefit from considering first the concrete examples of Sections5and6.
Time change
Our first construction corresponds to changing the clock as follows. For each fixedf ∈Gands≥0 let tf(s, ω):=
s
0
ef (X(u))du.
Note thattf:R+×Ω→R+is a measurable stochastic process the sample path of which are everywhere differentiable with dsdtf =ef (X(s)) bounded and bounded away from zero. Its inverse,τf(t):=inf{s≥0: tf(s)≥t}is thus also a measurable stochastic process, the sample path of which are everywhere differentiable with dtdτf =e−f (X(τf(t))) uniformly (intandω) bounded and bounded away from zero.
Proposition 3.1. Assumption2.5holds for(B, · )=(Cb(S), · ∞)and the Markov processXf0(t)=X(τf(t)).
Further,the generatorLf0 of the semi-group(Pf0)t ofXf0(t)is such thatD(L)=D(Lf0)andLf0g=e−fLgfor any g∈D(L).
Proof. Obviously,{τf(t)≤s} = {tf(s)≥t}is inFs=σ (X(u): 0≤u≤s)by the right continuity and boundedness of u→exp(f (X(u))). Hence, for each fixed t, the random variable τf(t)is a stopping time with respect to the canonical filtrationFs of{X(·)}. Similarly, the stochastic process
τf(t)= t
0
e−f (X0f(v))dv,
is adapted to the canonical filtrationFtf =σ (X0f(u): 0≤u≤t )of{Xf0(·)}. The strict monotonicity and continuity oft→τf(t)imply thatFtf=Fτf(t)and further it is not hard to check thatτf(t, ω)have the regeneration property
τf(t, ω)=τf(s, ω)+τf
t−s, θτf(s,ω)ω
for anyt > s >0 (whereθuω(·)=ω(u+ ·)denotes the usual shift operator). Thus, for anyh∈BS,t > s >0 and x∈S, by the strong Markov property ofX(·)atτf(s),
Ex
h
Xf0(t)Fsf
=Ex
h X
τf(t)Fτf(s)
=EX(τf(s))
hX
τf(t−s)
=EXf
0(s)
hX0f(t−s)
, (3.1)
whereX( ·)andX0f(·)denote independent copies ofX(·)andXf0(·), respectively. The Markov property ofX0f(·)then follows from (3.1) by standard arguments. SinceXf0(·)assumes its values in a complete, separable metric space, the contractive semi-groupPft h=Ex(h(Xf0(t)))is well defined on(Cb(S), · ∞). Further, by the change of variable v=τf(u)its resolvent is given by
Rfλh=Ex
∞
0
e−λuh Xf0(u)
du
=Ex
∞
0
e−λtf(v)ef (X(v))h X(v)
dv
. Note that λtf(s)=λcs−s
0ξ(X(u))du for positive, finite c=exp(f∞) and the continuous function ξ(x)= λ(c−ef (x))≥0 withξ∞< λc. The linear operatorRλcξ:D(Pt)→D(L)such that
(Rλcξ )g=Ex
∞
0
e−λcsξ X(s)
g X(s)
ds
is thus strictly contractive, and since the series Rfλh=
k≥0
(Rλcξ )kRλc
efh
converges uniformly, it follows thatRfλ: D(Pt)→D(Pt)and consequentlyPft :Cb(S)→Cb(S).
Fixing g∈ D(L) recall that g(X(·)) and (Lg)(X(·)) are bounded right-continuous functions, with M(t ):=
g(X(t))−g(X(0))−t
0(Lg)(X(v))dva right-continuous martingale with respect to the filtrationFt. WithM(0)=0 andτf(t)bounded above uniformly inω, by Doob’s optional sampling theoremExM(τf(t))=0 for anyx∈S. By the change of variablev=τf(u)and Fubini, this amounts to
Pft g(x)=g(x)+ t
0
Ex
e−fLg
Xf(u)
du=g(x)+ t
0
Pfue−fLg
(x)du. (3.2)
Letg=e−fLg∈D(Pt). Since Pfu is contractive onD(Pt), it follows from (3.2) thatPft g−g∞≤tg∞. In particular,Pft g−g∞→0 ast↓0, for allg∈G. WithGdense in the closed vector space(D(Pt), · ∞), we have that Pft is strongly continuous there. The uniformly bounded and strongly continuousPfug:[0, t] →Cb(S)is also strongly integrable, so (3.2) implies that for anyg∈D(L),
limt↓0
t−1
Pft g−g
−g
∞=lim
t↓0
t−1 t
0
Pfugdu−g ∞=0.
We have thus seen thatD(L)⊆D(Lf0)withLf0g=e−fLgfor allg∈D(L).
We next prove that ifg∈D(Lf0), then necessarilyg∈D(L). To this end, observe that the sample path ofXf(t)in- herits the right continuity of those ofX(t ), and fixingg∈D(Lf0)(which is thus continuous and bounded), we have the
