Topics in Occupation Times and Gaussian Free Fields doc

[...]... related to the model of random interlacements [27], which loosely speaking corresponds to “loops going through in nity” It 4 Introduction appears as well in the recent developments concerning conformally invariant scaling limits, see Lawler–Werner [16], Sheffield–Werner [24] As for random interlacements, interestingly, in place of (0.11), they satisfy an isomorphism theorem in the spirit of the generalized... at least one y in K, since otherwise the chain starting from any x in K would a.s never reach  By (1.75) we thus see that Ä does not vanish everywhere on K In addition (1.7) holds by (1.82) We have thus proved (1.79) (1.77): Expanding the square in the first sum in the right-hand side of (1.77), we see using the symmetry of cx;y , (1.82), and the second line of (1.74), that the right-hand side 23 1.4... The Markov chain X: (with jump rate 1) We introduce in this section the continuous-time Markov chain on E [ fg (absorbed in the cemetery state ), with discrete skeleton described by Zn , n 0, and exponential holding times of parameter 1 We also bring into play some of the natural objects attached to this Markov chains The canonical space DE for this Markov chain consists of right-continuous functions... Markov chain Zn , n 0, with starting point a.s equal to x, and an independent sequence of positive variables Tn , n 1, the “jump times , increasing to in nity, with increments TnC1 Tn , n 0, i.i.d exponential with parameter 1 (with the convention T0 D 0) The continuous-time chain X t , t 0, will then be expressed as X t D Zn ; for Tn Ä t < TnC1 , n 0: Of course, once the discrete-time chain reaches... weights and killing measure) of the gas of loops with intensity 1 , and the Pxi ;yi , 1 Ä i Ä k are 2 defined just as below (0.4), (0.5) The Poisson point process of Markovian loops has many interesting properties We will for instance see that when ˛ D 1 (i.e the intensity measure equals 1 ), 2 2 2 Lx /x2E has the same distribution as 1 'x /x2E , where 2 'x /x2E stands for the Gaussian free field in (0.3)...3 Introduction and E.'; '/ the energy of the function ' corresponding to the weights and killing measure on E (the matrix E.1x ; 1y /, x; y 2 E is the inverse of the matrix g.x; y/, x; y 2 E in (0.3)) w3 w2 y1 x3 x2 w1 y2 x1 y3 Figure 0.1 The paths w1 ; : : : ; wk in E interact with the gas of loops through the random potentials The typical representation formula for the moments of the random field in. .. tending to 1 In particular it is an increasing bijection of RC , and using the formula for the derivative of the inverse one can write for the inverse function of L: , Z u Z u 0I L t ug D (1.95) x u D infft X v dv D Xv dv; 0 0 where we have introduced the time changed process (with values in E [ fg) x def Xu D X u ; for u 0 (1.96) x (the path of X: thus belongs to DE , cf above (1.17)) x We also introduce... shown in (1.42) that for all f; g W E ! R, E.f; g/ D h Lf; gi D hf; Lgi: Since L D I (1.44) P /, we also find, see (1.11) for notation, E.f; g/ D I P /f; g/ D f; I P /g/ : (1.440 ) 14 1 Generalities As a next step we introduce some important random times for the continuous-time Markov chain X t , t 0 Given K  E, we define HK D infft 0I X t 2 Kg; the entrance time in K; z HK D infft > 0I X t 2 K and there... setting Äx ; for x 2 E, and p; D 1; (1.14) px; D x so the corresponding discrete-time Markov chain on E [ fg is absorbed in the cemetery state  once it reaches  We denote by Zn ; n 0, the canonical discrete Markov chain on the space of discrete trajectories in E [ fg, which after finitely many steps reaches  and from then on remains at , (1.15) 1.2 The Markov chain X: (with jump rate 1) 7 and. .. Then X t , t 0, when starting in x 2 E, corresponds to the simple random walk in Zd with exponential holding times of parameter 1 killed at the first time it exits E Our next step is to introduce some natural objects attached to the Markov chain X: , such as the transition semi-group, and the Green function 1.2 The Markov chain X: (with jump rate 1) 9 Transition semi-group and transition density Unless . Sznitman Topics in Occupation Times and Gaussian Free Fields Author: Alain-Sol Sznitman Departement Mathematik ETH Zürich Rämistrasse 101 8092 Zürich Switzerland. between occupation times and Gaussian processes. Notably they bring into play certain isomorphism theorems going back to Dynkin [4], [5] as well as certain