Brownian Motion on infinite graphs of finite total length

Transcription

1 Brownian Motion on infinite graphs of finite total length Technische Universität Graz

2 Our setup: l-top l-top

3 Our setup: l-top l-top let G = (V, E) be any graph

4 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e)

5 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e) this induces a metric: d(v, w) := inf{l(p) P is a v-w path}

6 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e) this induces a metric: d(v, w) := inf{l(p) P is a v-w path} let G l be the completion of the corresponding metric space

7 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e) this induces a metric: d(v, w) := inf{l(p) P is a v-w path} let G l be the completion of the corresponding metric space

8 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e) this induces a metric: d(v, w) := inf{l(p) P is a v-w path} let G l be the completion of the corresponding metric space

9 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e) this induces a metric: d(v, w) := inf{l(p) P is a v-w path} let G l be the completion of the corresponding metric space

10 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e) this induces a metric: d(v, w) := inf{l(p) P is a v-w path} let G l be the completion of the corresponding metric space

11 Our setup: l-top l-top let G = (V, E) be any graph give each edge a length l(e) this induces a metric: d(v, w) := inf{l(p) P is a v-w path} let G l be the completion of the corresponding metric space Theorem (G 06 (easy)) If e E(G) l(e) < then G l G.

12 Applications of G l Applications of G l aa(l-top)

13 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80)

14 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87)

15 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87) used by Benjamini and Schramm for Random Walks/harmonic functions/sphere Packings (Invent. math. 96, Preprint 09)

16 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87) used by Benjamini and Schramm for Random Walks/harmonic functions/sphere Packings (Invent. math. 96, Preprint 09) application in the study of the Cycle Space of an infinite graph (G & Sprüssel, Electr. J. Comb)

17 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87) used by Benjamini and Schramm for Random Walks/harmonic functions/sphere Packings (Invent. math. 96, Preprint 09) application in the study of the Cycle Space of an infinite graph (G & Sprüssel, Electr. J. Comb) application in Electrical Networks (G, JLMS 10)

18 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87) used by Benjamini and Schramm for Random Walks/harmonic functions/sphere Packings (Invent. math. 96, Preprint 09) application in the study of the Cycle Space of an infinite graph (G & Sprüssel, Electr. J. Comb) application in Electrical Networks (G, JLMS 10) Carlson studied the Dirichlet Problem at the boundary (Analysis on graphs and its applications, 08)

19 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87) used by Benjamini and Schramm for Random Walks/harmonic functions/sphere Packings (Invent. math. 96, Preprint 09) application in the study of the Cycle Space of an infinite graph (G & Sprüssel, Electr. J. Comb) application in Electrical Networks (G, JLMS 10) Carlson studied the Dirichlet Problem at the boundary (Analysis on graphs and its applications, 08) Colin de Verdiere et. al. use it to study self-adjointness of the Laplace and Schrödinger operators (Mathematical Physics, Analysis and Geometry, 10)

20 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87) used by Benjamini and Schramm for Random Walks/harmonic functions/sphere Packings (Invent. math. 96, Preprint 09) application in the study of the Cycle Space of an infinite graph (G & Sprüssel, Electr. J. Comb) application in Electrical Networks (G, JLMS 10) Carlson studied the Dirichlet Problem at the boundary (Analysis on graphs and its applications, 08) Colin de Verdiere et. al. use it to study self-adjointness of the Laplace and Schrödinger operators (Mathematical Physics, Analysis and Geometry, 10)

21 Applications of G l Applications of G l aa(l-top) used by Floyd to study Kleinian groups (Invent. math. 80) Gromov showed that his hyperbolic compactification is a special case of G l (Hyperbolic Groups... 87) used by Benjamini and Schramm for Random Walks/harmonic functions/sphere Packings (Invent. math. 96, Preprint 09) application in the study of the Cycle Space of an infinite graph (G & Sprüssel, Electr. J. Comb) application in Electrical Networks (G, JLMS 10) Carlson studied the Dirichlet Problem at the boundary (Analysis on graphs and its applications, 08) Colin de Verdiere et. al. use it to study self-adjointness of the Laplace and Schrödinger operators (Mathematical Physics, Analysis and Geometry, 10) All above authors discovered G l independently!

22 Our plan Problem Construct and study brownian motion on G l.

23 Our plan Problem Construct and study brownian motion on G l. Theorem (G 06 (easy)) If e E(G) l(e) < then G l G.

24 Our plan Problem Construct and study brownian motion on G l. Theorem (G 06 (easy)) If e E(G) l(e) < then G l G. Strategy: construct brownian motion on G l as a limit of brownian motions on finite subgraphs.

25 Our three topologies Level 1: The graph G l (with boundary)

26 Our three topologies Level 1: The graph G l (with boundary) Level 2: The space of sample paths C = C([0, T ] G l )

27 Our three topologies Level 1: The graph G l (with boundary) Level 2: The space of sample paths C = C([0, T ] G l ) with the metric d (b, c) := sup x G d l (b(x), c(x))

28 Our three topologies Level 1: The graph G l (with boundary) Level 2: The space of sample paths C = C([0, T ] G l ) with the metric d (b, c) := sup x G d l (b(x), c(x)) Level 3: The space M = M(C) of measures on C

29 Our three topologies Level 1: The graph G l (with boundary) Level 2: The space of sample paths C = C([0, T ] G l ) with the metric d (b, c) := sup x G d l (b(x), c(x)) Level 3: The space M = M(C) of measures on C with the weak topology,

30 Our three topologies Level 1: The graph G l (with boundary) Level 2: The space of sample paths C = C([0, T ] G l ) with the metric d (b, c) := sup x G d l (b(x), c(x)) Level 3: The space M = M(C) of measures on C with the weak topology, i.e. basic open sets of an element µ are of the form { ν M : fi dν f i dµ < ɛ i, i = 1,..., k }

31 Our three topologies Level 1: The graph G l (with boundary) Level 2: The space of sample paths C = C([0, T ] G l ) with the metric d (b, c) := sup x G d l (b(x), c(x)) Level 3: The space M = M(C) of measures on C with the weak topology, i.e. basic open sets of an element µ are of the form { ν M : fi dν f i dµ < ɛ i, i = 1,..., k } where the f i are bounded continuous real functions on C

32 Convergence in M Let G n be a sequence exhausting G.

33 Convergence in M Let G n be a sequence exhausting G. Let C, µ n be the brownian motion on G n.

34 Convergence in M Let G n be a sequence exhausting G. Let C, µ n be the brownian motion on G n. Theorem (classic) Let Γ M. Then Γ is compact iff for every ɛ there is a function ω ɛ (δ), with ω 0 as δ 0, such that µ({x : w x (δ) ω ɛ (δ) for all δ}) > 1 ɛ/2 for all µ Γ, where w x (δ) := sup t s <δ x(t) x(s) is the modulus of continuity of x.

35 Convergence in M Let G n be a sequence exhausting G. Let C, µ n be the brownian motion on G n. Theorem (classic) Let Γ M. Then Γ is compact iff for every ɛ there is a function ω ɛ (δ), with ω 0 as δ 0, such that µ({x : w x (δ) ω ɛ (δ) for all δ}) > 1 ɛ/2 for all µ Γ, where w x (δ) := sup t s <δ x(t) x(s) is the modulus of continuity of x. => {µ n } n has an accumulation point Remark: It is known that M(X) is compact iff X is compact; this would have allowed us to circumvent the above theorem if C were compact, but it isn t (although G l is).

36 brownian motion on G l Theorem (G & K. Kolesko 11+) For every G, l such that e E l(e) <, there is a brownian motion B l on G l with the following properties

37 brownian motion on G l Theorem (G & K. Kolesko 11+) For every G, l such that e E l(e) <, there is a brownian motion B l on G l with the following properties it behaves locally like standard BM on R

38 brownian motion on G l Theorem (G & K. Kolesko 11+) For every G, l such that e E l(e) <, there is a brownian motion B l on G l with the following properties it behaves locally like standard BM on R It is the limit of SRW s of finite subgraphs;

39 brownian motion on G l Theorem (G & K. Kolesko 11+) For every G, l such that e E l(e) <, there is a brownian motion B l on G l with the following properties it behaves locally like standard BM on R It is the limit of SRW s of finite subgraphs; It is unique;

40 brownian motion on G l Theorem (G & K. Kolesko 11+) For every G, l such that e E l(e) <, there is a brownian motion B l on G l with the following properties it behaves locally like standard BM on R It is the limit of SRW s of finite subgraphs; It is unique; It is recurrent (thus its sample paths are wild );

41 brownian motion on G l Theorem (G & K. Kolesko 11+) For every G, l such that e E l(e) <, there is a brownian motion B l on G l with the following properties it behaves locally like standard BM on R It is the limit of SRW s of finite subgraphs; It is unique; It is recurrent (thus its sample paths are wild ); Transition probabilities coincide with potentials of the corresponding non-elusive electrical current.

42 The discrete Network Problem The setup:

43 The discrete Network Problem The setup: A graph G = (V, E)

44 The discrete Network Problem The setup: A graph G = (V, E) a function r : E R + (the resistances)

45 The discrete Network Problem The setup: A graph G = (V, E) a function r : E R + (the resistances) a source and a sink p, q V

46 The discrete Network Problem The setup: A graph G = (V, E) a function r : E R + (the resistances) a source and a sink p, q V a constant I R (the intensity of the current)

47 The discrete Network Problem The setup: A graph G = (V, E) a function r : E R + (the resistances) a source and a sink p, q V a constant I R (the intensity of the current) Find a p-q flow in G with intensity I that satisfies Kirchhoff s cycle law: The problem:

48 The discrete Network Problem The setup: A graph G = (V, E) a function r : E R + (the resistances) a source and a sink p, q V a constant I R (the intensity of the current) Find a p-q flow in G with intensity I that satisfies Kirchhoff s cycle law: The problem: C e E(C) v( e) = 0

49 The discrete Network Problem The setup: A graph G = (V, E) a function r : E R + (the resistances) a source and a sink p, q V a constant I R (the intensity of the current) Find a p-q flow in G with intensity I that satisfies Kirchhoff s cycle law: The problem: C e E(C) v( e) = 0 where v( e) := f ( e)r(e) (Ohm s law)

50 Random Walks & Electrical networks p q x y

51 Random Walks & Electrical networks Every edge e has a weight c(e) p q x y

52 Random Walks & Electrical networks Every edge e has a weight c(e) Go from x to y with probability P x y := c(xy) c(x) where c(x) := xv E c(xv) p q x y

53 Random Walks & Electrical networks Every edge e has a weight c(e) Go from x to y with probability P x y := c(xy) c(x) where c(x) := xv E c(xv) p q x y P pq (x) := the probability that if you start in x you will hit p before q.

54 Random Walks & Electrical networks Every edge e has a weight c(e) Go from x to y with probability P x y := c(xy) c(x) where c(x) := xv E c(xv) p q x y P pq (x) := the probability that if you start in x you will hit p before q. Connect a source of voltage 1 to p, q

55 Random Walks & Electrical networks Every edge e has a weight c(e) Go from x to y with probability P x y := c(xy) c(x) where c(x) := xv E c(xv) p q x y P pq (x) := the probability that if you start in x you will hit p before q. Connect a source of voltage 1 to p, q P pq (x) = P(x)

56 Non-elusive flows p q

57 Non-elusive flows p q The solution is not necessarily unique!

58 Non-elusive flows p q The solution is not necessarily unique! p = q

59 Non-elusive flows p q The solution is not necessarily unique! p Non-elusive flow: The net flow along any such cut must be zero: = q

60 Uniqueness of non-elusive currents Theorem (G 08) In a network with e E r(e) < there is a unique non-elusive flow with finite energy that satisfies Kirchhoff s cycle law.

61 Uniqueness of non-elusive currents Theorem (G 08) In a network with e E r(e) < there is a unique non-elusive flow with finite energy that satisfies Kirchhoff s cycle law.

62 Uniqueness of non-elusive currents Theorem (G 08) In a network with e E r(e) < there is a unique non-elusive flow with finite energy that satisfies Kirchhoff s cycle law.

63 Uniqueness of non-elusive currents Theorem (G 08) In a network with e E r(e) < there is a unique non-elusive flow with finite energy that satisfies Kirchhoff s cycle law. Energy of f : 1 2 e E f 2 (e)r(e)

64 Uniqueness of non-elusive currents Theorem (G 08) In a network with e E r(e) < there is a unique non-elusive flow with finite energy that satisfies Kirchhoff s cycle law. Energy of f : 1 2 e E f 2 (e)r(e)

65 Random Walks & Electrical networks P pq (x) = P(x) p q x y P pq (x) := the probability that if you start in x you will hit p before q

66 Random Walks & Electrical networks P pq (x) = P(x) p q x y P pq (x) := the probability that if you start in x you will hit p before q x p q

67 Wild circles A wild circle

68 Wild circles A wild circle i.e. a homeomorphic image of S 1 in G (discovered by Diestel & Kühn)

69 Wild circles A wild circle i.e. a homeomorphic image of S 1 in G (discovered by Diestel & Kühn) Contains ℵ 0 double-rays aranged like the rational numbers

70 Wild circles A wild circle i.e. a homeomorphic image of S 1 in G (discovered by Diestel & Kühn) Contains ℵ 0 double-rays aranged like the rational numbers The gaps between the double-rays are filled by a Cantor set of ends

71 Wild circles A wild circle i.e. a homeomorphic image of S 1 in G

72 Wild circles A wild circle i.e. a homeomorphic image of S 1 in G More than 30 papers written on wild circles & paths relating to Cycle space (Homology) Hamilton circles Extremal graph theory