Zurich Lectures in Advanced Mathematics Edited by Erwin Bolthausen (Managing Editor), Freddy Delbaen, Thomas Kappeler (Managing Editor), Christoph Schwab, Michael Struwe, Gisbert Wüstholz Mathematics in Zurich has a long and distinguished tradition, in which the writing of lecture notes volumes and research monographs plays a prominent part The Zurich Lectures in Advanced Mathematics series aims to make some of these publications better known to a wider audience The series has three main constituents: lecture notes on advanced topics given by internationally renowned experts, graduate text books designed for the joint graduate program in Mathematics of the ETH and the University of Zurich, as well as contributions from researchers in residence at the mathematics research institute, FIM-ETH Moderately priced, concise and lively in style, the volumes of this series will appeal to researchers and students alike, who seek an informed introduction to important areas of current research Previously published in this series: Yakov B Pesin, Lectures on partial hyperbolicity and stable ergodicity Sun-Yung Alice Chang, Non-linear Elliptic Equations in Conformal Geometry Sergei B Kuksin, Randomly forced nonlinear PDEs and statistical hydrodynamics in space dimensions Pavel Etingof, Calogero-Moser systems and representation theory Guus Balkema and Paul Embrechts, High Risk Scenarios and Extremes – A geometric approach Demetrios Christodoulou, Mathematical Problems of General Relativity I Camillo De Lellis, Rectifiable Sets, Densities and Tangent Measures Paul Seidel, Fukaya Categories and Picard–Lefschetz Theory Alexander H.W Schmitt, Geometric Invariant Theory and Decorated Principal Bundles Michael Farber, Invitation to Topological Robotics Alexander Barvinok, Integer Points in Polyhedra Christian Lubich, From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis Shmuel Onn, Nonlinear Discrete Optimization – An Algorithmic Theory Kenji Nakanishi and Wilhelm Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations Erwin Faou, Geometric Numerical Integration and Schrödinger Equations Published with the support of the Huber-Kudlich-Stiftung, Zürich Alain-Sol Sznitman Topics in Occupation Times and Gaussian Free Fields Author: Alain-Sol Sznitman Departement Mathematik ETH Zürich Rämistrasse 101 8092 Zürich Switzerland 2010 Mathematics Subject Classification: 60K35, 60J27, 60G15, 82B41 Key words: occupation times, Gaussian free field, Markovian loop, random interlacements ISBN 978-3-03719-109-5 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks For any kind of use permission of the copyright owner must be obtained © 2012 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the author’s TE X files: I Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321 Preface The following notes grew out of the graduate course “Special topics in probability”, which I gave at ETH Zurich during the Spring term 2011 One of the objectives was to explore the links between occupation times, Gaussian free fields, Poisson gases of Markovian loops, and random interlacements The stimulating atmosphere during the live lectures was an encouragement to write a fleshed-out version of the handwritten notes, which were handed out during the course I am immensely grateful to Pierre-Franỗois Rodriguez, Artởm Sapozhnikov, Balázs Ráth, Alexander Drewitz, and David Belius, for their numerous comments on the successive versions of these notes Support by the European Research Council through grant ERC-2009-AdG 245728-RWPERCRI is thankfully acknowledged Contents Preface v Introduction 1 Generalities 1.1 The set-up 1.2 The Markov chain X: (with jump rate 1) 1.3 Some potential theory 1.4 Feynman–Kac formula 1.5 Local times x 1.6 The Markov chain X: (with variable jump rate) 5 10 23 25 26 Isomorphism theorems 2.1 The Gaussian free field 2.2 The measures Px;y 2.3 Isomorphism theorems 2.4 Generalized Ray–Knight theorems 31 31 35 41 45 The Markovian loop 3.1 Rooted loops and the measure r on rooted loops 3.2 Pointed loops and the measure p on pointed loops 3.3 Restriction property 3.4 Local times 3.5 Unrooted loops and the measure on unrooted loops 61 61 70 74 75 82 Poisson gas of Markovian loops 4.1 Poisson point measures on unrooted loops 4.2 Occupation field 4.3 Symanzik’s representation formula 4.4 Some identities 4.5 Some links between Markovian loops and random interlacements 85 85 87 91 95 100 References 111 Index 113 Introduction This set of notes explores some of the links between occupation times and Gaussian processes Notably they bring into play certain isomorphism theorems going back to Dynkin [4], [5] as well as certain Poisson point processes of Markovian loops, which originated in physics through the work of Symanzik [26] More recently such Poisson gases of Markovian loops have reappeared in the context of the “Brownian loop soup” of Lawler and Werner [16] and are related to the so-called “random interlacements”, see Sznitman [27] In particular they have been extensively investigated by Le Jan [17], [18] A convenient set-up to develop this circle of ideas consists in the consideration of a finite connected graph E endowed with positive weights and a non-degenerate killing measure One can then associate to these data a continuous-time Markov chain x Xt , t 0, on E, with variable jump rates, which dies after a finite time due to the killing measure, as well as the Green density g.x; y/, x; y E, (0.1) (which is positive and symmetric), xt the local times Lx D Z t x 1fXs D xg ds; t 0, x E (0.2) In fact g ; / is a positive definite function on E Gaussian process 'x , x E, such that E, and one can define a centered cov.'x ; 'y /.D EŒ'x 'y / D g.x; y/; for x; y E (0.3) This is the so-called Gaussian free field x1 It turns out that 'z , z E, and Lz , z E, have intricate relationships For instance Dynkin’s isomorphism theorem states in our context that for any x; y E, x1 Lz C 2 'z z2E under Px;y ˝ P G (0.4) has the “same law” as 2 'z /z2E under 'x 'y P G , (0.5) where Px;y stands for the (non-normalized) h-transform of our basic Markov chain, with the choice h / D g ; y/, starting from the point x, and P G for the law of the Gaussian field 'z , z E Introduction Eisenbaum’s isomorphism theorem, which appeared in [7], does not involve htransforms and states in our context that for any x E, s 6D 0, x1 Lz C 'z C s/2 z2E under Px ˝ P G (0.6) has the “same law” as 'z C s/2 Á z2E under C 'x Á G P s (0.7) The above isomorphism theorems are also closely linked to the topic of theorems of Ray–Knight type, see Eisenbaum [6], and Chapters and of Marcus–Rosen [19] Originally, see [13], [21], such theorems came as a description of the Markovian character in the space variable of Brownian local times evaluated at certain random times More recently, the Gaussian aspects and the relation with the isomorphism theorems have gained prominence, see [8], and [19] Interestingly, Dynkin’s isomorphism theorem has its roots in mathematical physics It grew out of the investigation by Dynkin in [4] of a probabilistic representation formula for the moments of certain random fields in terms of a Poissonian gas of loops interacting with Markovian paths, which appeared in Brydges–Fröhlich–Spencer [2], and was based on the work of Symanzik [26] The Poisson point gas of loops in question is a Poisson point process on the state space of loops on E modulo time-shift Its intensity measure is a multiple ˛ of the image of a certain measure rooted , under the canonical map for the equivalence relation identifying rooted loops that only differ by a time-shift This measure rooted is the -finite measure on rooted loops defined by Z P dt t Qx;x d / ; (0.8) rooted d / D t x2E t where Qx;x is the image of 1fX t D xg Px under Xs /0ÄsÄt , if X: stands for the Markov chain on E with jump rates equal to attached to the weights and killing measure we have chosen on E The random fields on E alluded to above, are motivated by models of Euclidean quantum field theory, see [11], and are for instance of the following kind: Z Z Q Q '2 Á '2 Á 1 hF '/i D F '/ e E.';'/ h x d'x e E.';'/ h x d'x 2 RE RE x2E x2E (0.9) with Z h.u/ D e vu d v/, u 0, with a probability distribution on RC ; 100 Poisson gas of Markovian loops Special case: loops going through a point We specialize the above formula (4.42) to find the probability that loops in the Poisson cloud going through a base point x all avoid some K not containing x: E x K Figure 4.3 Corollary 4.13 (˛ > 0) Consider x E, and K  E not containing x, then P˛ Œ all loops going through x not intersect K Ex ŒHK < 1; g.XHK ; x/ Á˛ D : g.x; x/ (4.51) Proof In the notation of (4.42) we pick K1 D fxg, K2 D K Setting U D EnK, (4.42) yields that the left-hand side of (4.51) equals gU x; x/ Á˛ (1.49) g.x; x/ D g.x; x/ Ex ŒHK < 1; g.XHK ; x/ Á˛ ; g.x; x/ and (4.51) follows 4.5 Some links between Markovian loops and random interlacements In this section we discuss various limiting procedures making sense of the notion of “loops going through infinity”, and see random interlacements appear as a limit object We begin with the case of Zd , d Random interlacements have been introduced in [27], and we refer to [27] for a more detailed discussion of the Poisson point process of random interlacements We will recover random interlacements on Zd , 4.5 Some links between Markovian loops and random interlacements 101 d 3, by the consideration of “loops going through infinity” More precisely, we consider d 3, and Un , n 1, a non-decreasing sequence of finite connected S U n D Zd , subsets of Zd , with (4.52) n as well as x Zd ; a “base point”: (4.53) For fixed n 1, we endow the connected subset Un , playing the role of E in (1.1), with the weights n cx;y D 2d 1fjx and with the killing measure P n Äx D 2d y2Zd nUn yj D 1g; for x; y Un ; 1fjx yj D 1g; for x Un ; very much in the spirit of what is done in Example 2) above (1.18) (except for the fact P n n we now replace by 2d ) Note that n D y2Un cx;y C Äx D 1, for all x Un x We write n for the space corresponding to (4.2), of pure point measures on the n set of unrooted loops contained in Un , and P˛ for the corresponding Poisson gas of Markovian loops at level ˛, see (4.6) Un x Figure 4.4 An unrooted loop contained in Un and going through x ; first the limit n ! 1, then the limit x ! We want to successively take the limit n ! 1, and then x ! The first limit corresponds to the construction of a Poisson gas of unrooted loops on Zd We will not really discuss this Poisson measure, which can be defined in a rather similar fashion to what we have done at the beginning of this chapter, but of course escapes the set-up of a finite state space E with weights and killing measure satisfying (1.1)–(1.5) For the second limit (i.e x ! 1), we will also adjust the level ˛ as a function of x The fashion in which we tune ˛ to x is dictated by the Green function of 102 Poisson gas of Markovian loops simple random walk on Zd : gZd x; y/ D Zd Ex hZ i 1fX t D ygdt ; for x; y Zd , (4.54) d Z where Px denotes the canonical law of continuous-time simple random walk with 0, the canonical process Taking jump rate on Zd starting at x, and X t , t advantage of translation invariance we introduce the function def g.x/ D gZd 0; x/; for x Zd (so gZd x; y/ D g.y The function g / is known to be positive, finite (recall d g x/ D g.x/, and has the asymptotic behavior g.x/ cd jxj where cd D ; as x ! 1; d D 2/ jB.0; 1/j d (4.54’) 3), symmetric, that is, d 2/ d x/): Á 1 d ; (4.55) where jxj stands for the Euclidean norm of x, and jB.0; 1/j for the volume of the unit ball of Rd (see for instance [15], p 31) We will choose ˛ according to the formula ˛Du g.0/ jx j2.d cd 2/ ; with u 0: (4.56) We introduce for ! n , the subset of Un of points visited by the unrooted loops in the support of the pure point measure !, which pass through the base point x : ˚ in the support of«!, Jn;x !/ D z Un I there is a (4.57) which goes through x and z , for ! n Note that Jn;x !/ D , when x … Un , and Jn;x !/ x , when at least one in the support of ! goes through x For the next result we will use the fact that (1.57) and (1.58) in the case of continuous-time simple random walk with jump rate on Zd take the following form: when K is a finite subset of Zd , P Zd gZd x; y/ eK y/; for x Zd Px ŒHK < 1 D (4.58) y2K z (with HK as in (1.45)); where the equilibrium measure Z z eK y/ D Py ŒHK D 1 1K y/; y Zd z (with HK as in (1.45)); d 103 4.5 Some links between Markovian loops and random interlacements is the unique measure supported on K such that the equality in (4.58) holds for all x K Its mass capZd K/ is the capacity of K The next theorem relates the so-called “random interlacement at level u” to the n set Jn;x when n ! 1, and then x ! 1, under the measure P˛ , with ˛ as in (4.56) In this set of notes we will not introduce the full formalism of the Poisson point process of random interlacements but only content ourselves with the description of the random interlacement at level u, see Remark 4.15 below Theorem 4.14 (d and K  Zd finite, one has 3) For u lim P n lim x !1 ˛Du g.0/ jx j2.d n!1 2/ ŒJn;x \ K D D e u capZd K/ : (4.59) c d Proof By (4.51) we have, as soon as x Un and x … K, n n P˛ ŒJn;x \ K D D P˛ Œall loops going through x not meet K  d D Z Ex ŒHK < TUn ; gUn XHK ; x / gUn x ; x / Ã˛ ; with gUn ; / the Green function of simple random walk on Zd killed when exiting Un Clearly, by monotone convergence, gUn x; y/ " gZd x; y/; for x; y Zd , when n ! 1: So we see that when x … K, then  lim n n P˛ ŒJn;x \K D D d Z Ex ŒHK < 1; gZd XHK ; x / gZd x ; x / Ã˛ (the formula holds also when x K) Now gZd x ; x / D g.0/, and, as x ! 1, we have by (4.55) (4.55) Z Px ŒHK < 1 cd jx j (4.58) d Z Ex ŒHK < 1; gZd XHK ; x / cd jx j (4.55) d d 2/ d 2/ / capZd K/; and, in particular, with ˛ as in (4.56), lim ˛ x !1 g.0/ d Z Ex ŒHK < 1; gZd XHK ; x / D u capZd K/: Coming back to (4.60) we readily obtain (4.59) (4.60) 104 Poisson gas of Markovian loops Remark 4.15 One can define a translation invariant random subset of Zd denoted by « u , the so-called random interlacement at level u, see [27], with distribution characterized by the identity: PŒ« u \ K D D e u capZd K/ ; for all K  Zd finite (4.61) Coming back to (4.59), note that for any disjoint finite subsets K; K of Zd one has by an inclusion-exclusion argument: n P˛ ŒJn;x \ K D and Jn;x à K i hY Y c c D En 1Jn;x x/ 1Jn;x x/ ˛ x2K x2K D P AÂK n 1/jAj P˛ ŒJn;x \ K [ A/ D ; where we expanded the last product in the second line to find the last line In the same fashion, we see that for disjoint finite subsets K; K of Zd we have PŒ« u \ K D and « u à K P P D 1/jAj PŒ« u \ K [ A/ D D 1/jAj e AÂK u capZd K[A/ : AÂK As a result, Theorem 4.14 can be seen to imply that under the measure Pn ˛Du g.0/ jx j2.d 2/ ; c d the law of Jn;x converges in an appropriate sense (i.e convergence of all finite dimensional marginal distributions) to the law of « u , as n ! 1, and then x ! We continue with the discussion of links between random interlacements and “loops going through infinity” in the Poisson cloud of Markovian loops 3, where we will We begin with a variation on (4.59) in the context of Zd , d give a different meaning to the informal notion of “loops going through infinity” We consider a sequence Un , n 1, as in (4.52) of finite connected subsets of Zd , d 3, which increases (in the wide sense) to Zd The role of the base point x , cf (4.53), is now replaced by the complement of the Euclidean ball: def BR D fx Zd I jxj Ä Rg; with R > By analogy with (4.57), we introduce for ! (4.62) n, in the support of !, Kn;R !/ D fz Un I there is a c which goes through BR and zg (4.63) 105 4.5 Some links between Markovian loops and random interlacements Zd R K Un BR first the limit n ! 1, then the limit R ! an unrooted loop contained in Un c and touching BR Figure 4.5 We now choose ˛ according to ˛Du Rd , with u cd 0, and cd as in (4.55) (4.64) The corresponding statement to (4.59) is now the following By the argument of Remark 4.15, it can be interpreted as a convergence of the law of Kn;R to the law of « u , as n ! and then R ! Theorem 4.16 (d lim R!1 and K  Zd finite, one has 3) For u lim P n n!1 ˛Du Rd cd ŒKn;R \ K D D e u capZd K/ : (4.65) Proof We assume that R is large enough so that K  BR and n sufficiently large so that BR Un In the notation of (4.42), we chose K1 D K and K2 D Un nBR , so that K1 \ K2 D Then (4.42) yields that  n P˛ ŒKn;R detK \K D D detK GBR K GU n K By (1.49), we write detK K GBR D det.An R/ Bn /; Ã˛ : 106 Poisson gas of Markovian loops where An is the K K-matrix An x; y/ D gUn x; y/; for x; y K; R/ and Bn , the K K-matrix d R/ Z c Bn x; y/ D Ex ŒHBR < TUn ; gUn XHB c ; y/; for x; y K: R Likewise, by the above definitions, we find that detK K GUn D det.An /: When n ! 1, lim An D A; where A.x; y/ D gZd x; y/, for x; y K n (4.66) d R/ Z c lim Bn D B R/ ; where B R/ x; y/ D Ex ŒHBR < 1; gZd XHB c ; y/; (4.67) n R for x; y K The matrix A is known to be invertible (one can base this on a similar calculation as in the proof of (1.35), see also [25], P2, p 292) So we find that  lim P n ŒKn;R \ K n!1 ˛ det.A B R/ / D D det A Ã˛ D det.I A ˛ B R/ / : (4.68) d Z c For x K, Px -a.s., HBR < 1, and XHB c @BR , so that R Zd B R/ x; y/ D Ex ŒgZd XHB c ; y/ R (4.55) cd ; for x; y K, as R ! 1: Rd It follows that det.I A where 1K K B R/ / D denotes the K cd Rd  Tr.A 1K K/ C o à Rd ; as R ! 1; (4.69) K matrix with all coefficients equal to Coming back to (4.58), we see that A 1K K D C , where C is the K K-matrix P with coefficients C.x; y/ D eK x/, for x; y K Since x2K eK x/ D capZd K/, we have found that det.I A B R/ /D1 cd Rd  capZd K/ C o à Rd ; as R ! 1: (4.70) Inserting this formula into (4.68), with ˛ as in (4.64), immediately yields (4.65) 4.5 Some links between Markovian loops and random interlacements 107 Complement: random interlacements and Poisson gas of loops coming from infinity on a transient weighted graph So far we only discussed links between random interlacements and a Poisson cloud of “loops going though infinity”, in the case of Zd , d We now discuss another construction, which applies to the general set-up of an (infinite) transient weighted graph with no killing We consider a countable (in particular infinite) set endowed with non-negative weights cx;y , x; y (i.e satisfying (1.2) with in place of E), so that endowed with the set of edges consisting of fx; yg such that cx;y > 0, is connected, locally finite, (i.e each x has a finite number of neighbors), (4.71) and the simple random walk with jump rate on induced by these weights cx;y , x; y , is transient (4.72) This is what we mean by a transient weighted graph (i.e with no killing) We consider, as in (4.52), Un , n 1, a non-decreasing sequence of finite connected subsets S of increasing to (i.e Un D and Un  UnC1 ), (4.73) n as well as a point x not in (which will play the role of the point “at infinity for each Un ”) (4.74) We consider the finite graph with vertex set En D Un [ fx g, endowed with the c weights obtained by collapsing UN on x : n cx;y D cx;y ; when x; y Un , P n n cx ;y D cy;x D cx;y ; when y Un : (4.75) x2GnUn n In addition we choose on En the killing measure Äx , x En , concentrated on x , so that n Äx D n Äx n > 0; with lim n D 1; D 0; for x Un D En nfx g: (4.76) For the continuous-time walk on with jump rate 1, one can show that when K is P a finite subset of G, setting D y2 cx;y , for x , and g ; / for the Green x 108 Poisson gas of Markovian loops Un " G Un G Figure 4.6 function (i.e g x; y/ D y R1 Ex Œ Px ŒHK < 1 D 1fX t D ygdt, for x; y ), P g x; y/ eK y/; for x ; (4.77) y2K where eK is the equilibrium measure of K: z eK y/ D Py ŒHK D 1 1K y/ y; for y (4.78) (for instance one approximates the left-hand side of (4.77) by Px ŒHK < TUn , with n ! 1, and applies (1.57), (1.53) to the walk killed when exiting Un ) The total mass of eK is the capacity of K: P eK y/: cap K/ D (4.79) y2K n We write n for the space of unrooted loops on En and P˛ for the Poisson gas of Markovian loops at level ˛ > 0, on the above finite set En endowed with the weights c n in (4.75) and the killing measure Ä n in (4.76) We also introduce the random subset of Un : in the support of ! Jn !/ D fz Un I there is a which goes through x and zg: (4.80) 4.5 Some links between Markovian loops and random interlacements 109 We now specify ˛ via the formula, see (4.76), ˛Du n; with u 0: (4.81) The statement corresponding to (4.59) and (4.65), which in the present context links the Poisson gas of loops on En going through “the point x at infinity”, with the interlacement at level u on G is coming next We refer to Remark 1.4 of [27] and [29] for a more detailed description of the Poisson point process of random interlacements in this context Theorem 4.17 For u and K  G finite, one has n lim P˛Du n ŒJn \ K D D e u capG K/ n!1 : (4.82) Proof For large n, K  Un , and by (4.51) we can write  n Ex ŒHK < 1; gn XHK ; x / gn x ; x / n P˛ ŒJn \ K D D Ã˛ ; n where Px stands for the law of the walk on En with unit jump rate, starting at x En , attached to the weights and killing measure in (4.75), (4.76) and gn ; / for the corresponding Green function By (2.71), we know that gn x ; z/ D n ; for all z En ; (4.83) and, as a result, n P˛ ŒJn \ K D D (1.57) D (4.83) n Px ŒHK < 1 n ˛ ˛ (4.84) capn K/ ; where capn K/ stands for the capacity of K in En By (1.53) and the fact that Ä n vanishes on Un , we know that P n z Px ŒHK D 1 : capn K/ D x x2K n In addition, we know that Px -a.s., z z fHK D 1g D fHx < HK g \ ÂHx fHK D 1g/; (4.85) 110 Poisson gas of Markovian loops because Un is finite and the walk is only killed at x So applying the strong Markov property at time Hx we find that n z n z Px ŒHK D 1 D Px ŒHx < HK z D Px ŒTUn < HK n Px ŒHK D 1 n Px ŒHK < 1/; using the fact that the walk on G and on En “agree up to time TUn ” Note, in addition, that (1.57) P P (4.76) n (4.83) Px ŒHK < 1 Ä gn x ; y/ y D ! 0; y n (1.53) y2K and that y2K n!1 z z Px ŒTUn < HK # Px ŒHK D 1; as n ! 1: Coming back to (4.85), we have shown that lim capn K/ D n P x2K z Px ŒHK D 1 (4.78) x D (4.79) cap K/: If we now insert this identity in (4.84) and keep in mind that ˛ D u find (4.82) (4.86) n, we readily Remark 4.18 1) By a similar argument as described in Remark 4.15, the above n theorem can be seen to imply that under P˛Du n , the law of Jn converges to the law u of « in the sense of finite dimensional marginal distributions, as n goes to infinity 2) A variation on the approximation scheme, which we employed to approximate random interlacements on a transient weighted graph, can be used to prove an isomorphism theorem for random interlacements, see [28] One can define the random field Lx;u /x2 of occupation times of continuous-time random interlacements at level u (this random field is governed by a probability denoted by P) One can also define the canonical law P G on R of the Gaussian free field attached to the transient weighted graph under consideration: under P G the canonical field 'x /x2 is a centered Gaussian field with covariance E G Œ'x 'y D g x; y/, for x; y , with g ; / the Green function The isomorphism theorem from [28] states that Lx;u C 2 'x x2 under P ˝ P G 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4, 56, 110 [29] A Teixeira, Interlacement percolation on transient weighted graphs Electron J Probab 14 (2009), 1604–1627 109 Index capacity, 16, 18 variational problems, 18 conductance, 16 continuous-time loop, 65 stationarity property, 65 time-reversal invariance, 68 Dirichlet form, orthogonal decomposition, 19 trace form, 19, 21, 29 tower property, 23 discrete loop, 65 stationarity property, 65 time-reversal invariance, 68 energy, 6, 11 entrance time, 14 equilibrium measure, 16, 18 equilibrium potential, 16, 18 exit time, 14 Feynman diagrams, 92 Feynman–Kac formula, 23, 29 Gaussian free field, 31, 90 conditional expectations, 33 generalized Ray–Knight theorems first Ray–Knight theorem, 45 second Ray–Knight theorem, 52, 56, 110 Green function, killed, 14 jump rate 1, killing measure, local time, 25, 75 loops going through a point, 100 loops going through infinity, 101, 104, 107 x Markov chain X: , 26 Markov chain X: , Markovian loops, 90 measure Px;y , 35, 81 occupation field, 87 occupation field of Markovian loop, 87 occupation field of non-trivial loops, 89 occupation time, 90 pointed loops, 62 measure p on pointed loops, 70 Poisson gas of Markovian loops, 86 Poisson point measure, 86 potential operators, 11 random interlacements, 56, 95, 100, 104, 107 at level u, 103, 104, 109 Ray–Knight theorem, 45 restriction property, 74, 80 rooted loops, 61 measure r on rooted loops, 63 rooted Markovian loop, 61 hitting time, 14 Symanzik’s representation formula, 94 Isomorphism theorem time of last visit, 14 Dynkin isomorphism theorem, 41 Eisenbaum isomorphism theorem, 43 trace Dirichlet form, 19, 20, 29 for random interlacements, 110 trace process, 30 114 transient weighted graph, 107 transition density, killed transition density, 14 Index unrooted loops, 81 measure r on unrooted loops, 81 variable jump rate, 26 unit weight, 82 weights, ... some of the links between occupation times and Gaussian processes Notably they bring into play certain isomorphism theorems going back to Dynkin [4], [5] as well as certain Poisson point processes... Huber-Kudlich-Stiftung, Zürich Alain-Sol Sznitman Topics in Occupation Times and Gaussian Free Fields Author: Alain-Sol Sznitman Departement Mathematik ETH Zürich Rämistrasse 101 8092 Zürich Switzerland 2010 Mathematics... objectives was to explore the links between occupation times, Gaussian free fields, Poisson gases of Markovian loops, and random interlacements The stimulating atmosphere during the live lectures was