Point-Map Probabilities of a Point Process

Transcription

1 Point-Map Probabilities of a Point Process F. Baccelli, UT AUSTIN Joint work with M.O. Haji-Mirsadeghi TexAMP, Austin, November 23, 24

2 Structure of the talk Point processes Palm probabilities Mecke s invariant measure equation Point maps Dynamical systems Point map probabilities 1

3 COUNTING MEASURES Let φ be a finite or countably-infinite collection of points of IR d, without accumulation One can think ofφ either as a set of points: φ = {t n } IR d or as a counting measure: φ = n ε t n 2

4 POINT PROCESS N: space of all counting measures φ N : σ-field ofmgenerated byφ φ(b) IN,B Borel sets of IR d (Ω, F, IP): probability space A point process Φ is a(n,n)-valued random variable on (Ω,F,IP) 3

5 STATIONARY POINT PROCESS Let {θ t } t IR d be a measure preserving flow on (Ω,F,IP) A point process Φ is stationary if the translations ofφare a factor of the flowθ t : Φ θ t (B) = Φ(B +t) t, B Implies the existence of an intensity λ assumed finite below 4

6 Palm Probability of a Stationary Point Process ProbabilityIP 0 = IP 0 Φ on (N,N) such that Mecke: f(t,φ(θ t (ω)))φ(ω,dt)ip(dω) = f(t,z)λdtip 0 (dz), f 0 IR d Ω IR d N Matthes: IP 0 (A) = E 1 tn B1 Φ θt n A E, A N, B Borel 1 tn B The support of IP 0 is contained in N 0 : space of counting measures with a point at the origin. 5

7 Palm Probability of a Stationary Point Process (continued) Interpretation Conditional: distribution of the point process given that the origin is included in the point process Ergodic: empirical distribution of the sequence{φ θ tn } for all points t n of Φ in a large ball 6

8 Point-Shift on Point Processes Maps each point ofφto some point of Φ Point-Shifts in the literature: Point-Shift H. Thorisson 00 Allocation rule e.g. by A. Holroyd & Y. Peres 05 Initial motivations: Palm calculus Navigation on the points ofφ Cracks in materials 7

9 Example 1 of Point-Shifts on Point Processes Strip Routing PS P. A. Ferrari, C. Landim, H. Thorisson 05 8

10 Example 2 of Point-Shifts on Point Processes Directional PS F.B. & C. BORDENAVE 07 9

11 Example 3 of Point-Shifts on Point Processes Closest-Closest PS 10

12 Example 4 of Point Shifts on Point Processes Directional PS on the supercritical random geometric graph The PS a.s. leads to a trap even when departing from points in the infinite connected component

13 Factor Point-Shifts Point Shift f : supp(φ) supp(φ) : f(φ,t n ) = t m Factor Point-Shift: there exists a function g called the point map, defined on N 0 which associates to each φ N 0 a point of its support and s.t. f(t n ) = t n +g θ tn, n Notation: f g org f 12

14 Point Stationarity Theorem Mecke (1975) Iff is an a.s. bijective factor point shift, then its associated point map g preserves the Palm probability of all stationary point processes. i.e. (θ g ) IP Φ 0 = IP Φ 0, Φ Palm probabilities are the only probability measures on N 0 preserved by all a.s. bijective factor point shifts. 13

15 Point Stationarity (continued) Example: Closest-closest PS Example of a.s. bijective factor point shifts visiting all points of a Poisson point process in IR 2 in Ferrari, Landim, Thorisson 04 14

16 Mecke s Invariant Measure Equation Question Let f be a factor point shift. Let g denote its point map. What is the set of all probability measures on N 0 satisfying (θ g ) Q = Q? (1) The point stationarity theorem says that if f is bijective, the Palm measure of any stationary point process solves (1). 15

17 Question Iff is not bijective, can we construct a solution to (1) from the Palm probability of a stationary point process? Neither strip PM nor directional PM are bijective. Can we find solutions to (1) for these PMs? Invariant Measure Equation 16

18 Notation for Point Shifts Image of a point x Φ, f Φ (x) := f(φ,x). Image of the point process f Φ (Φ) := {f Φ (x),x Φ}. Notation: M 1 (N): set of probability measures on N M 1 (N 0 ): set of probability measures on N 0 17

19 Four Actions of IN Actions π = {π n } of (N,+) on the topological spacex: 1 X = N, equipped with the vague topology, and for all n N, π n (φ) = f n φφ 1 X = M 1 (N), equipped with the weak convergence of probability measures on N, and for all Q X,n IN, π n (Q) = (f φ φ) n Q M 1 (N) 18

20 Four Actions of IN (continued) 2 X = N 0, equipped with vague topology, and φ N 0 and n N, π n (φ) = θ g n (φ)(φ) N 0 2 X = M 1 (N 0 ), equipped with the same topology as M 1 (N), and for all Q M 1 (N 0 ) andn N, π n (Q) = (θ g n (φ)) Q M 1 (N 0 ) 19

21 Action 1 Possible behaviors: p-periodicity Evaporation: Action 1 converges a.s. to the null measure onφ Let I := {φ N 0 ; n IN, y φ s.t. fφ(y) n = 0} Lemma 1 For all factor point shifts f and all stationary point processes Φ, there is evaporation of Φ underf if and only if IP Φ 0 [I] = 0 20

22 Action 1 (continued) Example of p-periodicity Directional PS on the super-critical RGG: p = 1 Closest-Closest PS: p = 2 Examples of Evaporation Ferrari, Landim, Thorisson 04: If Φ is a stationary Poisson point process in IR 2, then the strip PS evaporates Φ The same holds true for the directional PS on a stationary Poisson point process in IR 2 21

23 Actions 1 and 2 Actions 1 is only of interest in the non evaporation case Action 2 is not a topological dynamical system in dim. > 1: Lemma 2 For d 2, there is no continuous point map on the whole N 0 other than the identity. 22

24 Action 2 First g-palm probability ofφ: IP g,1 0 = (θ g ) IP 0 Interpretation: distribution of Φ given that the origin is inf(φ), considering multiplicities of the points of the image process 23

25 Action 2 (continued) n-th g-palm probability of Φ: distribution of Φ given that 0 is in f n (Φ) taking multiplicities into account. IP g,n 0 = (θ g ) IP g,n 1 0 Interpretation: distribution of Φ seen from a typical point of f n (Φ). 24

26 Definition of Point Map Probabilities Let g be a point map Φ be a stationary point process with Palm distributionp 0 Definition Every element of theω-limit set ofp 0 under the action of{(θ g n) } n N will be called ag probability ofp 0 If the limit of the sequence {(θ g n) P 0 } n=1 = {P g,n 0 } n=1 exists, it is called theg probability of P 0 and denoted byp g 0 25

27 Neither Existence nor Uniqueness are granted Example with no converging subsequence: B r (x): Ball of radius r and center x m(x) := Φ(B 1 (x)) f(x) := argmin y x ;m(y) > 2m(x) IP g,n 0 [Φ(B 1 (0)) > 2n] = 1 Examples with convergent subsequences with different limits 26

28 Periodic Case Theorem 1 If f := f g is 1-periodic on Φ, then the g probability P g 0 of P 0 exists and is given by P g 0 = Φ P f Φ Φ 0 Furthermore P g 0 is absolutely continuous with respect top 0, with ( dp g 0 f φ φ ) ({0}) =. dp 0 φ({0}) In addition,p g 0 = (θ g) P g 0 Similar statements hold in thep-periodic case 27

29 Evaporation Case Theorem 2 Assume Φ evaporates under the action of f g, the g probabilityp g 0 of P 0 exists it satisfiesp g 0 = (θ g) P g 0, then P g 0 is singular with respect to P 0 Relies on Lemma 1: under P g 0, I is of probability 1; under P 0, it is of probability 0. 28

30 Sample of the Point Process under its Palm Probability Grey level proportional to the age of the point w.r.t. the PM 29

31 Sample of the Point Process unde its Point Map Probability A point is black if it has pre-images of all orders and grey otherwise 30

32 Mecke s Invariant Measure Equation Consider the Cesàro sums When the limit of P g,n 0 := 1 n n 1 i=0 P g,i 0, n N. P g,n 0 exists (w.r.t. the topology of M 1 (N 0 )), let P g 0 := lim n 1 n n 1 i=0 P g,i 0. In general, P g 0 is not ag probability. 31

33 Mecke s Invariant Measure Equation (continued) Theorem 3 Assume there exists a subsequence{ P g,n i 0 } i=1 converging to a probability P g 0 (θ g ) is continuous at P g 0 Then P g 0 solves Mecke s Invariant Measure Equation Proposition 1 Ifθ g is P g 0 -almost surely continuous, then (θ g) is continuous at P g 0 Proposition 2 Ifg is P g 0 -almost surely continuous, then (θ g) is P g 0 -continuous 32

34 Construction of Point Map Probabilities Tightness Markov 33

35 Tightness Approach Tightness Lemma Kallenberg 1986: The sequence of point processes with distributions Q n is sequentially compact iff for all bounded Borel B IR d, lim limsup Q n (Φ(B) > r) = 0. r n The tightness lemma gives a necessary and sufficient condition for the existence of converging subsequences ofip f,n 0. Example: Any point-map on the Matérn hard core model 34

36 Markovian Approach Illustrated by the Strip PS Proposition 3 Iff = f g is the strip PS on the stationary Poisson point process onir 2, then the sequence IP g,n 0 is convergent θ g isip g 0 -a.s. continuous 35

37 Sketch of Proof B r + (0): right open half disk of radiusr Br (0): left closed half disk of radiusr It is sufficient to show the convergence inb r + (0) and Br (0) for all r. A + B r + (0) and A Br (0). 36

38 Sketch of Proof + Strong Markov property of the Poisson point process. 37

39 Directional PM The edge process (U n = T n T n+1 ) is not a Markov Chain. U2 U3 U4 U2 There exists a sequence of finite stopping times (τ k ) s.t. U1 U1 D2 T0 Φ k = (U τk,...,u τk+1 1) is aψ-irreducible, aperiodic Markov Chain which admits a small set and is geometrically ergodic. T4 T3 T2 T1 38

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41 Conclusions Transformations associated with point shifts lead to a variety of topological and metrical dynamical systems on stationary point processes The main new object of this presentation, point map probabilities, generalize strictly Palm probabilities 40

42 Future Research Ergodicity and mixing properties of the measure preserving transformations: partial results available on Poisson point processes. Better understanding of the space of solutions of Mecke s invariant measure equation Random dynamical systems on point processes: partial results on random walks on this type of random medium 41

43 Reference F.B. and M.O. Mirsadeghi Compactification of the Action of a Point-Map on the Palm Probability of a Point Process Arxiv, 24 42