aa r X i v : . [ m a t h . P R ] J a n RANDOM INTERLACEMENTS VIA KUZNETSOV MEASURES
STEFFEN DEREICH AND LEIF D ¨ORING
Abstract.
The aim of this note is to give an alternative construction of inter-lacements - as introduced by Sznitman - which makes use of classical potentialtheory. In particular, we outline that the intensity measure of an interlacementis known in probabilistic potential theory under the name ”approximate Markovchain” or ”quasi-process”. We provide a simple construction of random inter-lacements through (unconditioned) two-sided Brownian motions (resp. two-sidedrandom walks) involving Mitro’s general construction of Kuznetsov measures anda Palm measures relation due to Fitzsimmons. In particular, we show that randominterlacement is a Poisson cloud (‘soup’) of two-sided random walks (or Brownianmotions) started in Lebesgue measure and restricted on being closest to the originat time T ∈ [0 ,
1] - modulus time-shift. Introduction
Since the seminal article of Sznitman [11] the subject of random interlacements hasbeen very active. Roughly speaking, a random interlacement is a random ‘soup’of trajectories indexed by the real time-axis such that a finite Poisson distributednumber of trajectories can be observed when looking into an arbitrary compactset. For transient processes, such as simple random walk (SRW) on Z d , d ≥
3, orBrownian motion (BM) on R d , d ≥
3, the construction is non-trivial and done via aprojective limit procedure. The original motivation to study random interlacementsstems from connections to earlier studied objects such as the random walks in torior cylinders and the Gaussian free field (see for instance [16], [12]) and a correlationstructure that allows a detailed study of percolative properties of the vacant set notcovered by the trajectories (see for instance [13], [14]).The aim of the present note is to look at the model of random interlacementsitself from a different angle rather than proving further properties. We explain whyinterlacements are very natural objects that existed in the literature long beforeunder the name quasi-process and resulted in general excursion theory. Those wereintroduced by Hunt [5] in the probabilistic study of Martin entrance boundaries atthat time under the name approximate Markov chain. Quasi-processes are variantsof a Markov process with random birth and death in such a way that the occupationtime measure is excessive for the Markov process.There is a reason why the interpretation of random interlacements as quasi-processesis interesting: Quasi-processes are linked one-to-one to so-called Kuznetsov mea-sures and under duality assumptions a simple two-sided constructions of Kuznetsovmeasures exists due to Mitro [7]. We are not aware of such a simple constructionfor quasi-processes but using that quasi-process can be obtained as Palm measuresof Kuznetsov measures Mitro’s construction can be employed to construct quasi-processes in a direct way. We apply this idea to random interlacements and derivea representation via two-sided SRW or BM.
From now on all arguments are given for Brownian interlacements, allarguments work analogously for random walk interlacements.
We should note that all we do works also in more generality without any changes:For instance the underlying motion can be replaced by a non-simple RW or anarbitrary L´evy process.After writing the article we learnt from Jay Rosen that in the recent article [10] healso used a relation between random interlacements and quasi-processes. Rosen in-troduced L´evy-driven random interlacements as a ‘soup’ of L´evy quasi-processes (i.e.Poisson point measure with quasi-process intensity) and used tools from probabilis-tic potential theory (in particular Revuz measures) for the analysis of interesectionlocal times.
Outline:
In Section 2 we recall the definition of random interlacements due to Sznit-man. An overview of some results and objects from probabilistic potential theory ispresented in Section 3. Finally, in Section 4 we explain how random interlacementscan be constructed from unconditioned two-sided BM.2.
Random Interlacements
In this section we give the definition of random interlacements following [12]. Asmentioned above, we stick to Brownian interlacements; the notation and resultspresented here are analogous in the discrete setting.Denote by W + the subspace of C ( R + , R d ), d ∈ { , , . . . } , of continuous R d -valuedtrajectories tending to infinity at infinity and by W the subspace of C ( R , R d ) con-sisting of continuous trajectories indexed by the real-line which diverge at bothends. The canonical process will be denoted ( ω t ) t ≥ (resp. ( ω t ) t ∈ R ) and the shift-operators by σ t , t ≥
0, (resp. σ t , t ∈ R ). We work with the canonical σ -algebras W + and W generated by the shift-operators. The first-hitting time of a closed set F shall be denoted by H F . The same symbol H F will be used for W + and W withoutcausing confusion.Let W ∗ denote the set W modulus identification by time-change, that is W ∗ = W/ ∼ with ω ∼ ω ′ if there is some t ∈ R such that ω ( s ) = ω ′ ( s + t ) for s ∈ R . Let π denote the projection from W to W ∗ and W ∗ the induced σ -algebra on W ∗ . Forany compact set B ⊂ R d define W B (resp. W ∗ B ) the set of trajectories hitting B (resp. equivalence classes with one (and then all) representatives hitting B ).For y ∈ R d we denote by P y the law of Brownian motion started in y and for acompact set B not containing y , P By ( · ) = P y ( · | H B = ∞ ). We extend the definitionof P By for y ∈ ∂B by taking weak limits P By = w-lim x → y,x ∈ B c P Bx provided that the limit exists. As is well known this is always the case if B is a ball.The law P By will be called the Brownian motion conditioned on avoiding B .For a compact set B let e B denote the Brownian equilibrium measure on B withcapacity cap( B ) = e B ( B ) (see for instance [8]). There are several descriptions of e B but all we need is a relation to the harmonic measure (from infinity):w-lim | y |→∞ P y ( W H B ∈ dy | H B < ∞ ) = e B ( dy )cap( B )(1) ANDOM INTERLACEMENTS VIA KUZNETSOV MEASURES 3 and the invariance property P e B ′ ( H B ∈ dx ) = e B ( dx ) , for B ⊂ B ′ , (2)which implies P e B ′ ( H B < ∞ ) = cap( B ).The crucial ingredients of random interlacements are the measures Q B on W definedas follows: Q B (cid:0) ( ω − t ) t ≥ ∈ A, ω ∈ dy, ( ω t ) t ≥ ∈ A ′ (cid:1) = P By ( A ) e B ( dy ) P y ( A ′ ) , ∀ A, A ′ ∈ W + , for a closed ball B ⊂ R d . That is, from a starting point y ∈ ∂B sampled accordingto e B a Brownian motion is started in positive time-direction and independently aconditioned Brownian motion is started in negative time-direction. For a generalcompact set K ⊂ B define Q K by restriction: Q K := σ H K ◦ (1 H K < ∞ Q B ) . (3)By definition the Q K are concentrated on paths hitting the compact set K for thefirst time at time 0. One then defines the measures Q ∗ K on ( W ∗ K , W ∗ K ) by projecting Q K onto W ∗ K : Q ∗ K := π ◦ Q K . Definition 1.
A measure ν on ( W ∗ , W ∗ ) such that for compact sets K ⊂ R d the restrictions to ( W ∗ K , W ∗ K ) are Q ∗ K (i.e. W ∗ K ν = Q ∗ K ) is called interlacementintensity. Brownian interlacement intensities have been constructed in [12] by an abstractprojective limit procedure. The proof is similar to the original argument from [11]for random walks.
Theorem 2 (Sznitman [12]) . The interlacement intensity exists and is unique.
We will give a more direct and perhaps easier construction via unconditioned two-sided Brownian motion.It was shown in Remark 2.3 of [12] that ν has a remarkable time-inversion duality,i.e. the measure ˆ ν obtained by reversing the time-direction is identical to ν . As weshall see in Corollary 11 below this is also classical in probabilistic potential theory.Finally, we should also mention the definition of the object of main interest: Definition 3.
A Poisson point process ξ on W ∗ with intensity measure αν is calledBrownian random interlacement at level α > . In Section 4 we shall use a standard relation for Poisson point processes to givea simple construction of random interlacements in terms of conditioned two-sidedBrownian motion.3.
Elements of Probabilistic Potential Theory
Suppose X = ( X t ) t ≥ is a Feller process on a locally compact separable metricspace E with transition semigroup ( P t ) t ≥ . We will assume that X is transientmeaning that there exists f > R ∞ P t f dt <
1. As usual the statespace E is compactified by a cemetery state ∂ .A central aim of probabilistic potential theory is the probabilistic analysis of ex-cessive measures, i.e. σ -finite measures η with ηP t ( dy ) := R η ( dx ) P t ( x, dy ) ≤ η ( dy )for t ≥
0. For the study of excessive measures two versions of extended Markov
STEFFEN DEREICH AND LEIF D ¨ORING processes have been introduced:
Kuznetsov measures and quasi-processes . Both con-cepts also turned out valuable in the general theory of stochastic processes, such asin excursion theory and in the theory of reversing Markov processes.Let us first introduce Kuznetsov measures and consider the space Ω of RCLL (rightcontinuous with left limits) trajectories ω : R → E ∪ { ∂ } that are E -valued on someopen interval ( α ( ω ) , β ( ω )), taking the value ∂ outside [ α ( ω ) , β ( ω )). Equip Ω withthe canonical σ -fields G t = σ ( ω s , s ≤ t ) and G = σ ( ω t , t ∈ R ). Then a measure on(Ω , G ) is called Kuznetsov measure for ( P, η ) if Q does not charge the trivial path ω ≡ ∂ and Q has finite dimensional marginals Q (cid:0) α ( w ) < t , w t ∈ dx , · · · , w t n ∈ dx n , t n < β ( w ) (cid:1) = η ( dx ) P t − t ( x , dx ) · · · P t n − t n − ( x n − , dx n ) . Under a Kuznetsov measure Q the canonical process is also called a Markov processwith random birth and death: α ( ω ) = inf { t : ω t ∈ E } is called the time of birth, β ( ω ) = sup { t : ω t ∈ E } the time of death and ζ = β − α the life-time. The life-timeis well-defined and the strong Markov property holds under Q . Theorem 4 (Kuznetsov, [6]) . If η is excessive for ( P t ) , then there is a uniqueKuznetsov measure for ( P, η ) . There are different proofs of the theorem, the original proof due to Kuznetsov [6]is based on Kolmogorov’s extension theorem; a simple construction under dualityassumptions is recalled below.To get an idea of the need for finite and infinite birth times let us mention aprobabilistic representation (due to Fitzsimmons, Maisonneuve [3]) of the Rieszdecomposition of an excessive measure η into the sum η = η I + η P , where η I is aninvariant measure (i.e. ηP t = η for all t ≥
0) and η P is a purely excessive measure(i.e. η P P t ( f ) ↓ t → ∞ for any f ≥ η P ( f ) < ∞ ). The probabilisticrepresentation of η I , η P is obtained by separating paths of Q according to infinitebirth time and finite birth time: With Q I ( · ) = Q ( · , α = −∞ ) and Q P ( · , −∞ < α )one has for t ∈ R η I ( · ) = Q I ( ω t ∈ · , t < β ) and η P ( · ) = Q P ( ω t ∈ · , α < t < β ) . We now turn to the second type of generalized Markov process that goes back toHunt [5] in discrete time under the original name approximate Markov chain andhas been extended by Weil [17] under the name quasi-process to continuous time.Consider again the space (Ω , G ) of trajectories equipped with the canonical σ -field.An R ∪ {∞} -valued random variable S is called stationary time, if S = S ◦ σ t + t and α ≤ S < β on { S < ∞} . Further, it is called intrinsic, if it is almost everywherefinite. For quasi-processes one does not work on the entire σ -field G but instead onthe σ -field A of σ t -invariant events [Note: this corresponds to taking the quotientspace in Section 2]. If η is excessive for ( P t ), then a measure P on (Ω , A ) is calleda quasi-process for ( P, η ) if η ( · ) = P Z β ( ω ) α ( ω ) · ( ω s ) ds ! ANDOM INTERLACEMENTS VIA KUZNETSOV MEASURES 5 and for any intrinsic stopping time S under P | S ∈ R the forward in time process( ω S + t ) t ≥ is strong Markov with transitions ( P t ).Both, Kuznetsov measure and quasi-process have advantages and disadvantages inthe study of excessive measures but surprisingly both concepts are equivalent viathe notion of Palm measures as noticed by Fitzsimmons: Theorem 5 (Fitzsimmons [2]) . If Q is a Kuznetsov measure for ( P, η ) and S is afinite stationary time then the Palm measure of P and S P ( A ) := Q ( A, S ∈ [0 , , A ∈ A , (4) is a ( P, η ) quasi-process and P is independent of the choice of S . Conversely, if P is a quasi-process for ( P, η ) , then Q ( f ) := P (cid:18)Z ∞−∞ f ◦ σ t dt (cid:19) , f is G -measurable , defines a Kuznetsov measure for ( P, η ) . In words: Given a Kuznetsov measure and a stationary time S , restricting to tra-jectories with S ∈ [0 ,
1] and forgetting about the time parametrization results ina quasi-process. Conversely, given a quasi-process defined on the invariant σ -field,then one obtains the corresponding Kuznetsov measure by independently attribut-ing to each path a reference time.A simple consequence is the existence and uniqueness of quasi-processes, deducedfrom the analogue statement for Kuznetsov measures: Corollary 6.
Given an excessive measure η for a Feller transition semigroup ( P t ) there is a unique quasi-process for ( P, η ) . It depends on the problem whether it is more useful to work with the (
P, η )Kuznetsov measure or quasi-process. A general advantage of Kuznetsov measures isemployed in the next section: Under a duality assumption there is always a simpleconstruction for Kuznetsov measures as (unconditioned) two-sided process, whereasthere is no simple construction for quasi-processes.4.
Random Interlacements through Mitro’s Construction
A particularly simple construction of Kuznetsov measures for Markov processes induality with respect to an invariant measure η is due to to Mitro. Suppose X is aFeller process on E with transition semigroup ( P t ) and X is in duality to a secondFeller process ˆ X with respect to an invariant measure η , that is, for all h, g boundedand measurable, E η [ h ( X t ) g ( X )] = ˆ E η (cid:2) h ( ˆ X ) g ( ˆ X t ) (cid:3) , ∀ t ≥ , or, equivalently, in the more usual notation (cid:10) P t h, g i η = (cid:10) h, ˆ P t g i η . Hunt’s achieve-ment was to relate duality and time-reversal in the setting of quasi-processes: Theorem 7 (Hunt [5]) . If P is the quasi-process for ( P, η ) , then the measure ˆ P obtained under time-reversal and switching to the RCLL version is the quasi-processfor ( ˆ P , η ) . Time-reversal for dual processes also holds for Kuznetsov measures and is reflectedin an important construction:
STEFFEN DEREICH AND LEIF D ¨ORING
Theorem 8 (Mitro [7]) . The Kuznetsov measure Q for ( P, η ) is the unique measurewith finite dimensional marginals Z E ˆ P x [ ˆ X s ∈ dy , · · · , ˆ X s l ∈ dy l ] η ( dx ) P x [ X t ∈ dx , · · · , X t k ∈ dx k ] at the times s l < · · · < s < ≤ t < · · · t k . In words, to build Q one samples the invariant measure η at time 0, and from theoutcome starts an independent copy of X to the right and an independent copy ofthe dual ˆ X to the left.Mitro’s theorem allows to give a simple argument for Hunt’s time-reversal theorem: Proof of Theorem 7. If X and ˆ X are in duality with respect to η , then Mitro’s con-struction explains how to construct Q and how to construct ˆ Q . In particular, thisshows that one being the image under time-reversal of the other. Also note that sta-tionary times for Q are stationary for ˆ Q as well, thus, Fitzsimmons’ representation(Theorem 5) implies the claim. (cid:3) Example 9. A d -dimensional Brownian motion is in duality with itself with respectto Lebesgue measure λ d on R d . Since Brownian motion is in duality to itself, Mitro’s theorem gives a simple con-struction of the Kuznetsov measure of a Brownian motion with Lebesgue invariantmeasure. Taking the Palm measure with respect to any stationary time yields a sim-ple representation of the corresponding quasi-process which, in fact, is Sznitman’sinterlacement intensity ν : Theorem 10. (a) Suppose X is a d -dimensional BM and λ d the Lebesgue measureon R d , then the Brownian interlacement intensity is the quasi-process P for ( P, βλ d ) for some β > .(b) The Brownian interlacement intensity ν is obtained by taking a two-sided BMstarted in Lebesgue measure, restricting an arbitrary intrinsic (i.e. finite and sta-tionary) time to [0 , and identifying paths that are translations of each other. Recall from the discussion of the Riesz decomposition above that α = −∞ for Q -almost all trajectories since λ d is invariant. The transience of Brownian motionon R d for d ≥ S = argmin t ∈ R {| ω t |} .We emphasize once more that the choice of the intrinsic time is irrelevant. Proof of Theorem 10.
Only (a) needs a proof, (b) is then a consequence of Mitro’sconstruction and Fitzsimmons’ theorem.It is enough to show that for some β >
P, βλ d )has the defining properties of the (unique) interlacement intensity. The multiple β will be chosen in due course.(i) In this first part of the proof we calculate the first-hitting distribution P ( ω H B ∈ · )of a compact set B . Let us first show that P ( H B < ∞ ) < ∞ . By monotonicity wecan restrict attention to B with strictly positive capacity. By definition P is the ANDOM INTERLACEMENTS VIA KUZNETSOV MEASURES 7 quasi-process for (
P, βλ d ), hence, the strong Markov property yields ∞ > βλ d ( B ) = P (cid:18)Z ∞−∞ B ( ω t ) dt (cid:19) = P (cid:18)Z ∞ H B B ( ω t ) dt, H B < ∞ (cid:19) = Z B P ( ω H B ∈ dx, H B < ∞ ) E x (cid:20)Z ∞ B ( X t ) dt (cid:21) ≥ P ( H B < ∞ ) inf x ∈ B E x (cid:20)Z ∞ B ( X t ) dt (cid:21) . The infimum is strictly positive, thus, P ( H B < ∞ ) < ∞ .From the strong Markov property we obtain, for any N ∈ N large enough with B ⊂ B N = { x : | x | ≤ N } , P ( ω H B ∈ dx, H B < ∞ )= P (cid:16) P ω HBN [ X H B ∈ dx | H B < ∞ ] P ω HBN [ H B < ∞ ] 1 H BN < ∞ (cid:17) . Using that by construction P a.a. sample paths have left-limits ∞ in norm andadditionally (1) we obtain P ( ω H B ∈ dx, H B < ∞ )= lim N →∞ P ( ω H B ∈ dx, H B < ∞ , H B N < ∞ )= lim N →∞ P (cid:16) P ω HBN [ X H B ∈ dx | H B < ∞ ] P ω HBN ( H B < ∞ )1 H BN < ∞ (cid:17) = e B ( dx )cap( B ) lim N →∞ P (cid:16) P ω HBN [ H B < ∞ ] 1 H BN < ∞ (cid:17) = e B ( dx )cap( B ) P ( H B < ∞ ) , where e B ( dx ) is the equilibrium measure on B . We now claim that P can be normal-ized so that P ( H B < ∞ ) = e B ( B ) for all compact B with strictly positive capacity.First, by the strong Markov property and (2), we obtain for B ⊂ B ′ , P ( H B < ∞ ) = P (cid:0) P ω HB ′ ( H B < ∞ )1 H B ′ < ∞ (cid:1) = Z P ( ω H B ′ ∈ dx, H B ′ < ∞ ) P x ( H B < ∞ )= Z e B ′ ( dx )cap( B ′ ) P ( H B ′ < ∞ ) P x ( H B < ∞ )= P ( H B ′ < ∞ ) 1cap( B ′ ) P e B ′ ( H B < ∞ )= P ( H B ′ < ∞ ) 1cap( B ′ ) cap( B )which proves that for general compact sets B ⊂ B ′ with strictly positive capacity P ( H B < ∞ ) = cap( B ) ⇐⇒ P ( H B ′ < ∞ ) = cap( B ′ ) . (5) STEFFEN DEREICH AND LEIF D ¨ORING
Normalizing P (i.e. choosing β appropriately) such that P ( H B < ∞ ) = cap( B ),equivalence (5) implies P ( H B < ∞ ) = e B ( B ) for all B compact. Hence, we haveproved P ( ω H B ∈ dx, H B < ∞ ) = e B ( dx ) , ∀ B compact , for a suitable choice of β >
0. Hence, the hitting distribution is the one needed forDefinition 1.(ii) Since H B is a stationary stopping time, the forward in time process after H B under P ( · , H B < ∞ ) is a BM as needed for Definition 1.(iii) Finally, we need to determine the backwards dynamics. Let B denote a closedball and denote by ˆ H B the last exit time from B . Using that Brownian motionstarted in Lebesgue measure is self-dual we conclude that under P the time-reversedprocess ( ω H B − t ) t ≥ up to time H B has the same ‘distribution’ as the forwards intime process ( ω ˆ H B + t ) t ≥ after time ˆ H B . Recall that ( ω H B + t ) t ≥ is a Brownian mo-tion started in the finite measure P ( ω H B ∈ · , H B < ∞ ). Hence, by the last exitdecomposition of Getoor [4], Theorem 2.12, we know that under P the pre-exitprocess and the post-exit processes of ( ω H B + t ) t ≥ are independent conditionally onthe point of last exit with the post-exit process being Brownian motion conditionedon not returning to B .(i)-(iii) show that for any closed ball B the defining property P ( · , H B < ∞ ) = Q ∗ B ( · )holds. Using the definition (3) of Q K for a compact set K ⊂ B , the interlacementproperty for P is seen as follows: P ( · , H K < ∞ ) = P ( · , H B < ∞ , H K < ∞ ) = Q ∗ B ( · , H K < ∞ ) = Q ∗ K ( · ) . (cid:3) Having shown that the interlacement intensity ν is a quasi-process for a self-dualMarkov process, a direct consequence of Hunt’s time-reversibility theorem is thetime-reversibility mentioned at the end of Section 2: Corollary 11. If ˆ ν is obtained from ν by time-reversal, then ˆ ν = ν . We can finally turn back to random interlacements and give an alternative con-struction:
Theorem 12.
Take a Poisson point process Ξ whose intensity measure is two-sidedBrownian motion started from Lebesgue measure λ d on R d . Restrict Ξ on the pathsbeing closest to the origin at time T ∈ [0 , and identify paths that are translationsof each other. This yields a random interlacement for some level β > .Proof. This is a consequence of Theorem 10 using the fact that restriction of theintensity turns into restriction of the Poisson point process. (cid:3)
Acknowledgement.
The authors would like to thank Pat Fitzsimmons and JayRosen for drawing our attention to reference [10].
ANDOM INTERLACEMENTS VIA KUZNETSOV MEASURES 9
References [1] C. Dellacherie, B. Maisonneuve, P.A. Meyer: ”Probabilit´es et Potential, Processes de Markov(fin) Compl´ements de calcul stochastique.” Hermann, (1992)[2] P.J. Fitzsimmons: ”On a connection between Kuznetsov processes and quasi-processes.” Sem.Stoch. Proc., pp. 123-134, (1987)[3] P.J. Fitzsimmons, B. Maisonneuve: ”Excessive measures and Markov processes with randombirth and death.” ZW, pp. 319-336, (1986)[4] R.K. Getoor: ”Splitting times and shift functionals.” ZW, pp. 69-81, (1979)[5] G.A. Hunt: ”Markoff chains and Martin boundaries.” Illinois Jour. Math, pp. 313-340, (1960)[6] S.E. Kuznetsov: ”Construction of Markov processes with random birth and death.” Th. Prob.Appl., pp. 571-574, (1974)[7] J.B. Mitro: ”Dual Markov processes: construction of a useful auxiliary process.” ZW, pp.139-156, (1979)[8] P. M¨orters, Y. Perez: ”Brownian Motion.” Cambridge University Press, (2010)[9] B. Rath and A. Sapozhnikov: ”Connectivity properties of random interlacement and inter-section a of random walks.” ALEA 9, (2012)[10] J. Rosen: ”Intersection local times for interlacements.” arXiv:1308.3469[11] A.-S. Sznitman: ”Vacant set of random interlacements and percolation.” Ann. Math., pp.2003-2087, (2010)[12] A.-S. Sznitman: ”On Scaling Limits and Brownian Interlacements.” Bull. Braz. Math. Soc.,New Series 44(4), pp. 555-592, (2013)[13] A.-S. Sznitman: ”Decoupling inequalities and interlacement percolation.” Invent. Math. 187(3), 645-706, (2012)[14] V. Sidoravicius, A.-S. Sznitman: ”Percolation for the Vacant Set of Random Interlacements.”Comm. Pure Appl. Math. 62 (6), pp. 831-858, (2009)[15] A. Teixeira: ”On the uniqueness of the infinite cluster of the vacant set of random interlace-ments.” Ann. Appl. Probab. 19 (1), 454-466, (2009)[16] D. Windisch: ”Random walk on a discrete torus and random interlacements.” Electron. Com-mun. Probab. 13, pp. 140-150, (2008)[17] M. Weil: ”Quasi-processus.” S´eminaire de Probabiit´es IV, pp. 216-239, (1970)[1] C. Dellacherie, B. Maisonneuve, P.A. Meyer: ”Probabilit´es et Potential, Processes de Markov(fin) Compl´ements de calcul stochastique.” Hermann, (1992)[2] P.J. Fitzsimmons: ”On a connection between Kuznetsov processes and quasi-processes.” Sem.Stoch. Proc., pp. 123-134, (1987)[3] P.J. Fitzsimmons, B. Maisonneuve: ”Excessive measures and Markov processes with randombirth and death.” ZW, pp. 319-336, (1986)[4] R.K. Getoor: ”Splitting times and shift functionals.” ZW, pp. 69-81, (1979)[5] G.A. Hunt: ”Markoff chains and Martin boundaries.” Illinois Jour. Math, pp. 313-340, (1960)[6] S.E. Kuznetsov: ”Construction of Markov processes with random birth and death.” Th. Prob.Appl., pp. 571-574, (1974)[7] J.B. Mitro: ”Dual Markov processes: construction of a useful auxiliary process.” ZW, pp.139-156, (1979)[8] P. M¨orters, Y. Perez: ”Brownian Motion.” Cambridge University Press, (2010)[9] B. Rath and A. Sapozhnikov: ”Connectivity properties of random interlacement and inter-section a of random walks.” ALEA 9, (2012)[10] J. Rosen: ”Intersection local times for interlacements.” arXiv:1308.3469[11] A.-S. Sznitman: ”Vacant set of random interlacements and percolation.” Ann. Math., pp.2003-2087, (2010)[12] A.-S. Sznitman: ”On Scaling Limits and Brownian Interlacements.” Bull. Braz. Math. Soc.,New Series 44(4), pp. 555-592, (2013)[13] A.-S. Sznitman: ”Decoupling inequalities and interlacement percolation.” Invent. Math. 187(3), 645-706, (2012)[14] V. Sidoravicius, A.-S. Sznitman: ”Percolation for the Vacant Set of Random Interlacements.”Comm. Pure Appl. Math. 62 (6), pp. 831-858, (2009)[15] A. Teixeira: ”On the uniqueness of the infinite cluster of the vacant set of random interlace-ments.” Ann. Appl. Probab. 19 (1), 454-466, (2009)[16] D. Windisch: ”Random walk on a discrete torus and random interlacements.” Electron. Com-mun. Probab. 13, pp. 140-150, (2008)[17] M. Weil: ”Quasi-processus.” S´eminaire de Probabiit´es IV, pp. 216-239, (1970)