Some remarks on Large Deviations

Transcription

1 Some remarks on Large Deviations S.R.S. Varadhan Courant Institute, NYU Conference in memory of Larry Shepp April 25, 2014 Some remarks on Large Deviations p.1/??

2 We want to estimate certain probabilities. Some remarks on Large Deviations p.2/??

3 We want to estimate certain probabilities. Large Deviation Theory is a tool. Some remarks on Large Deviations p.2/??

4 We want to estimate certain probabilities. Large Deviation Theory is a tool. Need to be set up properly Some remarks on Large Deviations p.2/??

5 We want to estimate certain probabilities. Large Deviation Theory is a tool. Need to be set up properly Look at three examples Some remarks on Large Deviations p.2/??

6 LDP Some remarks on Large Deviations p.3/??

7 LDP logp n [A] ninf x A I(x) Some remarks on Large Deviations p.3/??

8 LDP logp n [A] ninf x A I(x) Estimates are local. Some remarks on Large Deviations p.3/??

9 LDP logp n [A] ninf x A I(x) Estimates are local. Upper bound is valid for compact sets Some remarks on Large Deviations p.3/??

10 LDP logp n [A] ninf x A I(x) Estimates are local. Upper bound is valid for compact sets Lower bound is valid for open sets Some remarks on Large Deviations p.3/??

11 LDP logp n [A] ninf x A I(x) Estimates are local. Upper bound is valid for compact sets Lower bound is valid for open sets Upper bound for closed sets? Some remarks on Large Deviations p.3/??

12 LDP logp n [A] ninf x A I(x) Estimates are local. Upper bound is valid for compact sets Lower bound is valid for open sets Upper bound for closed sets? X is compact. Some remarks on Large Deviations p.3/??

13 LDP logp n [A] ninf x A I(x) Estimates are local. Upper bound is valid for compact sets Lower bound is valid for open sets Upper bound for closed sets? X is compact. If not, we need some estimates Some remarks on Large Deviations p.3/??

14 LDP logp n [A] ninf x A I(x) Estimates are local. Upper bound is valid for compact sets Lower bound is valid for open sets Upper bound for closed sets? X is compact. If not, we need some estimates Compactification or some control Some remarks on Large Deviations p.3/??

15 1 n logep n [exp[nf(x)] sup x[f(x) I(x)] Some remarks on Large Deviations p.4/??

16 1 n logep n [exp[nf(x)] sup x[f(x) I(x)] F(y) I(y) < F(x 0 ) I(x 0 ) fory x 0 Some remarks on Large Deviations p.4/??

17 1 n logep n [exp[nf(x)] sup x[f(x) I(x)] F(y) I(y) < F(x 0 ) I(x 0 ) fory x 0 1 Z n exp[nf(x)]dp n δ x0 Some remarks on Large Deviations p.4/??

18 1 n logep n [exp[nf(x)] sup x[f(x) I(x)] F(y) I(y) < F(x 0 ) I(x 0 ) fory x 0 1 Z n exp[nf(x)]dp n δ x0 LSC implies upper bound. Some remarks on Large Deviations p.4/??

19 1 n logep n [exp[nf(x)] sup x[f(x) I(x)] F(y) I(y) < F(x 0 ) I(x 0 ) fory x 0 1 Z n exp[nf(x)]dp n δ x0 LSC implies upper bound. G : X Y Some remarks on Large Deviations p.4/??

20 1 n logep n [exp[nf(x)] sup x[f(x) I(x)] F(y) I(y) < F(x 0 ) I(x 0 ) fory x 0 1 Z n exp[nf(x)]dp n δ x0 LSC implies upper bound. G : X Y Q n = P n G 1 Some remarks on Large Deviations p.4/??

21 1 n logep n [exp[nf(x)] sup x[f(x) I(x)] F(y) I(y) < F(x 0 ) I(x 0 ) fory x 0 1 Z n exp[nf(x)]dp n δ x0 LSC implies upper bound. G : X Y Q n = P n G 1 J(y) = inf x G 1 (y)i(x) Some remarks on Large Deviations p.4/??

22 1967 Larry Shepp Some remarks on Large Deviations p.5/??

23 1967 Larry Shepp x(t) is Gaussian process on [0,1] with mean 0. Some remarks on Large Deviations p.5/??

24 1967 Larry Shepp x(t) is Gaussian process on [0,1] with mean 0. ρ(s, t) is its covariance. Smooth. Some remarks on Large Deviations p.5/??

25 1967 Larry Shepp x(t) is Gaussian process on [0,1] with mean 0. ρ(s, t) is its covariance. Smooth. P[(λG) c ] exp[ c(g)λ 2 +o(λ 2 )] Some remarks on Large Deviations p.5/??

26 1967 Larry Shepp x(t) is Gaussian process on [0,1] with mean 0. ρ(s, t) is its covariance. Smooth. P[(λG) c ] exp[ c(g)λ 2 +o(λ 2 )] c(g) = inf f G ci(f) [ 1 I(f) =sup f(t)g(t)dt g ] ρ(s, t)g(s)g(t)dsdt Some remarks on Large Deviations p.5/??

27 IfG = {f : sup 0 t 1 f(t) 1} Some remarks on Large Deviations p.6/??

28 IfG = {f : sup 0 t 1 f(t) 1} c(g) = [2sup 0 t 1 ρ(t,t)] 1 Some remarks on Large Deviations p.6/??

29 IfG = {f : sup 0 t 1 f(t) 1} c(g) = [2sup 0 t 1 ρ(t,t)] 1 Tail is coming from the one with the largest variance. Some remarks on Large Deviations p.6/??

30 IfG = {f : sup 0 t 1 f(t) 1} c(g) = [2sup 0 t 1 ρ(t,t)] 1 Tail is coming from the one with the largest variance. Is this always true? Some remarks on Large Deviations p.6/??

31 IfG = {f : sup 0 t 1 f(t) 1} c(g) = [2sup 0 t 1 ρ(t,t)] 1 Tail is coming from the one with the largest variance. Is this always true? Does every almost surely bounded Gaussian process have a Gaussian tail? Some remarks on Large Deviations p.6/??

32 IfG = {f : sup 0 t 1 f(t) 1} c(g) = [2sup 0 t 1 ρ(t,t)] 1 Tail is coming from the one with the largest variance. Is this always true? Does every almost surely bounded Gaussian process have a Gaussian tail? Do the constants always match? Some remarks on Large Deviations p.6/??

33 1970 Landau and Shepp proved a Gaussian bound. Some remarks on Large Deviations p.7/??

34 1970 Landau and Shepp proved a Gaussian bound. Sankhya. Some remarks on Large Deviations p.7/??

35 1970 Landau and Shepp proved a Gaussian bound. Sankhya. It is enough to prove an exponential tail. Some remarks on Large Deviations p.7/??

36 1970 Landau and Shepp proved a Gaussian bound. Sankhya. It is enough to prove an exponential tail. P[ X C n] = P[ X 1 +X 2 + +X n n C n] P[ X 1 + X n Cn] exp[ cn] Some remarks on Large Deviations p.7/??

37 1970 Landau and Shepp proved a Gaussian bound. Sankhya. It is enough to prove an exponential tail. P[ X C n] = P[ X 1 +X 2 + +X n n C n] P[ X 1 + X n Cn] exp[ cn] Provided C > E[ X]] and E[e θ X ] < Some remarks on Large Deviations p.7/??

38 Skhorohod published a proof of an exponential bound in a Banach Space. Some remarks on Large Deviations p.8/??

39 Skhorohod published a proof of an exponential bound in a Banach Space. X = X(1). X(t) is a continuos process with independent increments. Some remarks on Large Deviations p.8/??

40 Skhorohod published a proof of an exponential bound in a Banach Space. X = X(1). X(t) is a continuos process with independent increments. P[ X(1) n] P[ sup X(t) n] 0 t 1 P[τ 1 +τ τ n 1] e[e[e (τ 1+ +τ n ) ]] = e[e[e τ ] n ] Some remarks on Large Deviations p.8/??

41 Fernique 1970 Some remarks on Large Deviations p.9/??

42 Fernique 1970 X,Y are two independent copies. Some remarks on Large Deviations p.9/??

43 Fernique 1970 X,Y are two independent copies. So are X+Y 2 and X Y 2 Some remarks on Large Deviations p.9/??

44 Fernique 1970 X,Y are two independent copies. So are X+Y 2 and X Y 2 F(t) = P[ X t] Some remarks on Large Deviations p.9/??

45 Fernique 1970 X,Y are two independent copies. So are X+Y 2 and X Y 2 F(t) = P[ X t] F(t)[1 F(s)] [F( t s 2 )] 2 Some remarks on Large Deviations p.9/??

46 Fernique 1970 X,Y are two independent copies. So are X+Y 2 and X Y 2 F(t) = P[ X t] F(t)[1 F(s)] [F( t s 2 )] 2 Uses this to to show the Gaussian estimate with some constant for X. i.elogf(t) ct 2. Some remarks on Large Deviations p.9/??

47 Fernique 1970 X,Y are two independent copies. So are X+Y 2 and X Y 2 F(t) = P[ X t] F(t)[1 F(s)] [F( t s 2 )] 2 Uses this to to show the Gaussian estimate with some constant for X. i.elogf(t) ct 2. Improves it to get the right constant. Some remarks on Large Deviations p.9/??

48 This would follow from a general LDP for sums of IID s in Banach Space. Some remarks on Large Deviations p.10/??

49 This would follow from a general LDP for sums of IID s in Banach Space. This was done in 1977 Some remarks on Large Deviations p.10/??

50 This would follow from a general LDP for sums of IID s in Banach Space. This was done in 1977 E[e θ X ] < for all θ > 0. Some remarks on Large Deviations p.10/??

51 This would follow from a general LDP for sums of IID s in Banach Space. This was done in 1977 E[e θ X ] < for all θ > 0. For a Gaussian this follows frome[e θ X ] < for someθ > 0. Some remarks on Large Deviations p.10/??

52 Example. Sourav Chatterjee Ifr = nx by Stirling s formula ( ) n = exp[ n[xlogx+(1 x)log(1 x)]+o(n)] r Some remarks on Large Deviations p.11/??

53 Example. Sourav Chatterjee Ifr = nx by Stirling s formula ( ) n = exp[ n[xlogx+(1 x)log(1 x)]+o(n)] r For coin tossing with a biased coin I(x) = xlog x p +(1 x)log 1 x 1 p Some remarks on Large Deviations p.11/??

54 Counting the number of graphs with specified subgraph counts. Some remarks on Large Deviations p.12/??

55 Counting the number of graphs with specified subgraph counts. N vertices. The number of possible subgraphs Γ withk vertices in a complete graph of size N is c(n,γ) c(γ)n k. Some remarks on Large Deviations p.12/??

56 Counting the number of graphs with specified subgraph counts. N vertices. The number of possible subgraphs Γ withk vertices in a complete graph of size N is c(n,γ) c(γ)n k. In a given graph G this may be smaller and the ratio is some fraction r(n,g,γ) 1 Some remarks on Large Deviations p.12/??

57 Count the number of graphsg having N vertices with specified valuesr(n,g,γ i ) = r i for a finite number of Γ s. Some remarks on Large Deviations p.13/??

58 Count the number of graphsg having N vertices with specified valuesr(n,g,γ i ) = r i for a finite number of Γ s. Their number is exp[n 2 J(Γ 1,r 1 ;...;Γ k,r k )+o(n 2 )] Some remarks on Large Deviations p.13/??

59 Count the number of graphsg having N vertices with specified valuesr(n,g,γ i ) = r i for a finite number of Γ s. Their number is exp[n 2 J(Γ 1,r 1 ;...;Γ k,r k )+o(n 2 )] Out of a total of2 (N 2) possible graphs with N vertices. Some remarks on Large Deviations p.13/??

60 Count the number of graphsg having N vertices with specified valuesr(n,g,γ i ) = r i for a finite number of Γ s. Their number is exp[n 2 J(Γ 1,r 1 ;...;Γ k,r k )+o(n 2 )] Out of a total of2 (N 2) possible graphs with N vertices. 0 J 1 2 log2 Some remarks on Large Deviations p.13/??

61 Count the number of graphsg having N vertices with specified valuesr(n,g,γ i ) = r i for a finite number of Γ s. Their number is exp[n 2 J(Γ 1,r 1 ;...;Γ k,r k )+o(n 2 )] Out of a total of2 (N 2) possible graphs with N vertices. 0 J 1 2 log2 expression forj Some remarks on Large Deviations p.13/??

62 0 x,y 1;f(x,y) = f(y,x); 0 f 1 Some remarks on Large Deviations p.14/??

63 0 x,y 1;f(x,y) = f(y,x); 0 f 1 r(γ,f) = [0,1] V(Γ) (i,j) E(Γ) f(x i,x j ) i V(Γ) dx i Some remarks on Large Deviations p.14/??

64 0 x,y 1;f(x,y) = f(y,x); 0 f 1 r(γ,f) = [0,1] V(Γ) (i,j) E(Γ) f(x i,x j ) i V(Γ) dx i H(f) = 1 2 [f logf +(1 f)log(1 f)]dxdy Some remarks on Large Deviations p.14/??

65 0 x,y 1;f(x,y) = f(y,x); 0 f 1 r(γ,f) = [0,1] V(Γ) (i,j) E(Γ) f(x i,x j ) i V(Γ) dx i H(f) = 1 2 [f logf +(1 f)log(1 f)]dxdy J = sup f:r(γ i,f)=r i 1 i k H(f) Some remarks on Large Deviations p.14/??

66 Let f(x,y) be a continuous function. Some remarks on Large Deviations p.15/??

67 Let f(x,y) be a continuous function. Consider a "random" graph with N vertices labeled {1,2,...,N}. (i,j) is an edge with probability f( i N, j N ). Some remarks on Large Deviations p.15/??

68 Let f(x,y) be a continuous function. Consider a "random" graph with N vertices labeled {1,2,...,N}. (i,j) is an edge with probability f( i N, j N ). The "expected number" of subgraphsγcan be easily calculated. Some remarks on Large Deviations p.15/??

69 Consider a map φ ofγonto {1,2,...,N}. Some remarks on Large Deviations p.16/??

70 Consider a map φ ofγonto {1,2,...,N}. There aren(n 1) (N k +1) of them Some remarks on Large Deviations p.16/??

71 Consider a map φ ofγonto {1,2,...,N}. There aren(n 1) (N k +1) of them The chance that one of them maps edges in Γ to edges in our random graph is Π (v,v ) E(Γ)f( φ(v) N, φ(v ) N ) Some remarks on Large Deviations p.16/??

72 Ratio of the expected number of subgraphs of typeγ to the number in a complete graph, for largen is r(γ,f) = Π (i,j) E(Γ) f(x i,x j )Π i V(Γ) dx i [0,1] V(Γ) Some remarks on Large Deviations p.17/??

73 Ratio of the expected number of subgraphs of typeγ to the number in a complete graph, for largen is r(γ,f) = Π (i,j) E(Γ) f(x i,x j )Π i V(Γ) dx i [0,1] V(Γ) Law of large numbers is valid. Some remarks on Large Deviations p.17/??

74 Ratio of the expected number of subgraphs of typeγ to the number in a complete graph, for largen is r(γ,f) = Π (i,j) E(Γ) f(x i,x j )Π i V(Γ) dx i [0,1] V(Γ) Law of large numbers is valid. w(g) = Π (i,j) E(G) f( i N, j N )Π (i,j)/ E(G)[1 f( i N, j N )] Some remarks on Large Deviations p.17/??

75 G G N,ǫ,r1,r 2,...,r k w(g) 1 Some remarks on Large Deviations p.18/??

76 The typical probabilityw(g) under the distribution determined by f has the propertylogw(g) = (i,j) E(G) logf( i N, j N )+ (i,j)/ E(G) log[1 f( i N, j N )] Some remarks on Large Deviations p.19/??

77 The typical probabilityw(g) under the distribution determined by f has the propertylogw(g) = (i,j) E(G) N2 2 H(f) logf( i N, j N )+ (i,j)/ E(G) log[1 f( i N, j N )] Some remarks on Large Deviations p.19/??

78 The typical probabilityw(g) under the distribution determined by f has the propertylogw(g) = (i,j) E(G) N2 2 H(f) logf( i N, j N )+ (i,j)/ E(G) You must have at least exp[ N2 2 log[1 f( i N, j N )] H(f)] graphs. Some remarks on Large Deviations p.19/??

79 X x 1,1 x 1,2 x 1,n x 2,1 x 2,2 x 2,n... x n,1 x n,2 x n,n Some remarks on Large Deviations p.20/??

80 k K n x 1,1 x 1,2 x 1,n x 2,1 x 2,2 x 2,n x n,1 x n,2 x n,n Some remarks on Large Deviations p.21/??

81 K = {k : k(x,y),[0,1] 2 [0,1]} Some remarks on Large Deviations p.22/??

82 K = {k : k(x,y),[0,1] 2 [0,1]} K N range ofn N matrices. Some remarks on Large Deviations p.22/??

83 K = {k : k(x,y),[0,1] 2 [0,1]} K N range ofn N matrices. P(k N ) = exp[ N2 2 log2] Some remarks on Large Deviations p.22/??

84 K = {k : k(x,y),[0,1] 2 [0,1]} K N range ofn N matrices. P(k N ) = exp[ N2 2 log2] logp[k N f] I(f) = N2 2 f log(2f)+(1 f)log(2(1 f))dxdy = N 2 [H(f) 1 2 log2] Some remarks on Large Deviations p.22/??

85 Topology? Weak? Some remarks on Large Deviations p.23/??

86 Topology? Weak? k r(γ,f) is not continuous. Some remarks on Large Deviations p.23/??

87 Topology? Weak? k r(γ,f) is not continuous. If LDP holds in a topology in which it is continuous, then J(Γ 1,r 1 ;...;Γ k,r k )) = 1 2 log2 inf k:r(γ i,k)=r i 1 i k = sup k:r(γ i,k)=r i 1 i k H(k) I(k) Some remarks on Large Deviations p.23/??

88 Topology? Weak? k r(γ,f) is not continuous. If LDP holds in a topology in which it is continuous, then J(Γ 1,r 1 ;...;Γ k,r k )) = 1 2 log2 inf k:r(γ i,k)=r i 1 i k = sup k:r(γ i,k)=r i 1 i k H(k) I(k) Strong topology likel p will be OK. Some remarks on Large Deviations p.23/??

89 Topology? Weak? k r(γ,f) is not continuous. If LDP holds in a topology in which it is continuous, then J(Γ 1,r 1 ;...;Γ k,r k )) = 1 2 log2 inf k:r(γ i,k)=r i 1 i k = sup k:r(γ i,k)=r i 1 i k H(k) I(k) Strong topology likel p will be OK. No chance. Fluctuations. Some remarks on Large Deviations p.23/??

90 Enter "cut" topology Some remarks on Large Deviations p.24/??

91 Enter "cut" topology cut metric isd (k 1,k 2 ) = [k 1 (x,y) k 2 (x,y)]φ(x)ψ(y)dxdy sup φ, ψ 1 Some remarks on Large Deviations p.24/??

92 Enter "cut" topology cut metric isd (k 1,k 2 ) = [k 1 (x,y) k 2 (x,y)]φ(x)ψ(y)dxdy sup φ, ψ 1 sup A,B A B [k 1 (x,y) k 2 (x,y)]dxdy Some remarks on Large Deviations p.24/??

93 Ifd (k n,k) 0 and sup n,x,y k n (x,y) C Some remarks on Large Deviations p.25/??

94 Ifd (k n,k) 0 and sup n,x,y k n (x,y) C r(γ,k n ) r(γ,k). Some remarks on Large Deviations p.25/??

95 Ifd (k n,k) 0 and sup n,x,y k n (x,y) C r(γ,k n ) r(γ,k). Limits of large graphs. Some remarks on Large Deviations p.25/??

96 Ifd (k n,k) 0 and sup n,x,y k n (x,y) C r(γ,k n ) r(γ,k). Limits of large graphs. Count the number of occurrences ofγin the graph. Some remarks on Large Deviations p.25/??

97 Ifd (k n,k) 0 and sup n,x,y k n (x,y) C r(γ,k n ) r(γ,k). Limits of large graphs. Count the number of occurrences ofγin the graph. Divide by the number in a complete graph. Some remarks on Large Deviations p.25/??

98 Assume the limit ψ(γ) of the ratio exists for every Γ. Some remarks on Large Deviations p.26/??

99 Assume the limitψ(γ) of the ratio exists for everyγ. What are possible limits? Graphons. Some remarks on Large Deviations p.26/??

100 Assume the limitψ(γ) of the ratio exists for everyγ. What are possible limits? Graphons. Representation. Some remarks on Large Deviations p.26/??

101 Assume the limitψ(γ) of the ratio exists for everyγ. What are possible limits? Graphons. Representation. There is s symmetric function f(x,y) on [0,1] [0,1] such that Some remarks on Large Deviations p.26/??

102 Assume the limitψ(γ) of the ratio exists for everyγ. What are possible limits? Graphons. Representation. There is s symmetric function f(x,y) on [0,1] [0,1] such that For any graphγwith verticesv(γ) and edgese(γ) r(γ,f) = Π (i,j) E(Γ) f(x i,x j )Π i V(Γ) dx i [0,1] V(Γ) Some remarks on Large Deviations p.26/??

103 r(γ,f) = r(γ,g) for all Γ if and only if f(x,y) = g(σx,σy) for someσ H. Some remarks on Large Deviations p.27/??

104 r(γ,f) = r(γ,g) for all Γ if and only if f(x,y) = g(σx,σy) for someσ H. Cut topology is the smallest topology on K/H that makes f r(γ, f) continuous for every Γ. Some remarks on Large Deviations p.27/??

105 r(γ,f) = r(γ,g) for all Γ if and only if f(x,y) = g(σx,σy) for someσ H. Cut topology is the smallest topology on K/H that makes f r(γ, f) continuous for every Γ. This topology works for LLN. 2 n 2 n << 2 n2 Some remarks on Large Deviations p.27/??

106 Upper Bound needs compactness, or exponential tightness. Some remarks on Large Deviations p.28/??

107 Upper Bound needs compactness, or exponential tightness. K is not compact under cut topology. Some remarks on Large Deviations p.28/??

108 Upper Bound needs compactness, or exponential tightness. K is not compact under cut topology. But K/H is by a theorem of Lovász-Szegedy Some remarks on Large Deviations p.28/??

109 Upper Bound needs compactness, or exponential tightness. K is not compact under cut topology. But K/H is by a theorem of Lovász-Szegedy It may be possible to prove the large deviation estimate in the topology induced by "cut" topology on K/H Some remarks on Large Deviations p.28/??

110 Need to estimate the probability of a neighborhood of the orbit. Some remarks on Large Deviations p.29/??

111 Need to estimate the probability of a neighborhood of the orbit. Szemerédi s regularity lemma Some remarks on Large Deviations p.29/??

112 Need to estimate the probability of a neighborhood of the orbit. Szemerédi s regularity lemma Replaces thehorbit by aπ n permutation orbit. Some remarks on Large Deviations p.29/??

113 Need to estimate the probability of a neighborhood of the orbit. Szemerédi s regularity lemma Replaces thehorbit by aπ n permutation orbit. logn! = o(n 2 ) Some remarks on Large Deviations p.29/??

114 Example. Chiranjib Mukherjee Brownian Motion Some remarks on Large Deviations p.30/??

115 Example. Chiranjib Mukherjee Brownian Motion L t = 1 T T 0 δ x(s)ds Some remarks on Large Deviations p.30/??

116 Example. Chiranjib Mukherjee Brownian Motion L t = 1 T T 0 δ x(s)ds λ(v) = lim T 1 = sup f 2 =1 = sup f 0 f 1 =1 T T loge[exp[ V(x(s))ds]] 0 [ V(x)[f(x)] 2 dx 1 2 [ V(x)f(x)dx 1 f 2 8 f f 2 dx ] dx Some remarks on Large Deviations p.30/??

117 P[L T f] = exp[ TI(f)+o(T)] Some remarks on Large Deviations p.31/??

118 P[L T f] = exp[ TI(f)+o(T)] I(f) = 1 8 f 2 dx f Some remarks on Large Deviations p.31/??

119 [ E exp[ 1 T T 0 T 0 ] V(β(s) β(t))dsdt] Some remarks on Large Deviations p.32/??

120 [ E exp[ 1 T T 0 T V(x) 0 as x 0 ] V(β(s) β(t))dsdt] Some remarks on Large Deviations p.32/??

121 [ E exp[ 1 T T 0 T V(x) 0 as x exp[ct +o(t)]?. 0 ] V(β(s) β(t))dsdt] Some remarks on Large Deviations p.32/??

122 [ E exp[ 1 T T 0 T V(x) 0 as x exp[ct +o(t)]?. 0 ] V(β(s) β(t))dsdt] c = sup f 0 f 1 =1 [ V(x y)f(x)f(y)dxdy 1 8 f 2 f ] dx Some remarks on Large Deviations p.32/??

123 [ E exp[ 1 T T 0 T V(x) 0 as x exp[ct +o(t)]?. 0 ] V(β(s) β(t))dsdt] c = sup f 0 f 1 =1 [ V(x y)f(x)f(y)dxdy 1 8 f 2 f ] dx Compactification ofm(r d )/R d Some remarks on Large Deviations p.32/??

124 If we only need to estimate [ T ] E exp[ V(β(s))ds] One point comactification of R d is enough. 0 Some remarks on Large Deviations p.33/??

125 If we only need to estimate [ T ] E exp[ V(β(s))ds] One point comactification of R d is enough. {f : f 0, f(x)dx 1}. 0 Some remarks on Large Deviations p.33/??

126 If we only need to estimate [ T ] E exp[ V(β(s))ds] One point comactification of R d is enough. {f : f 0, f(x)dx 1}. Vague topology is OK. 0 Some remarks on Large Deviations p.33/??

127 Translation invariant comapactification? Some remarks on Large Deviations p.34/??

128 Translation invariant comapactification? { µ α } Some remarks on Large Deviations p.34/??

129 Translation invariant comapactification? { µ α } α µ α(r d ) 1, dµ α = f α dx Some remarks on Large Deviations p.34/??

130 Translation invariant comapactification? { µ α } α µ α(r d ) 1, dµ α = f α dx I({f α }) = i 1 8 fα 2 f α dx = α I(f α) Some remarks on Large Deviations p.34/??

131 Translation invariant comapactification? { µ α } α µ α(r d ) 1, dµ α = f α dx I({f α }) = i 1 8 fα 2 f α dx = α I(f α) c = sup {f α } [ α V(x)(f α f α )(x)dx α ] I(f α ) Some remarks on Large Deviations p.34/??

132 Translation invariant comapactification? { µ α } α µ α(r d ) 1, dµ α = f α dx I({f α }) = i 1 8 c = sup {f α } fα 2 f α dx = α I(f α) [ V(x)(f α f α )(x)dx α α [ ] V(x)(f f)(x)dx I(f) c = sup f ] I(f α ) Some remarks on Large Deviations p.34/??

133 Needs Compactness Some remarks on Large Deviations p.35/??

134 Needs Compactness D( µ 1, µ 2 ) Some remarks on Large Deviations p.35/??

135 Needs Compactness D( µ 1, µ 2 ) 1 2 j F j (x 1,...,x kj )[Π k j r=1 µ 1(dx r ) Π k j r=1 µ 2(dx r ) Some remarks on Large Deviations p.35/??

136 Needs Compactness D( µ 1, µ 2 ) 1 2 j F j (x 1,...,x kj )[Π k j r=1 µ 1(dx r ) Π k j r=1 µ 2(dx r ) {F j } are translation invariant Some remarks on Large Deviations p.35/??

137 Needs Compactness D( µ 1, µ 2 ) 1 2 j F j (x 1,...,x kj )[Π k j r=1 µ 1(dx r ) Π k j r=1 µ 2(dx r ) {F j } are translation invariant F(x 1 +x,...,x k +x) = F(x 1,...,x k ) Some remarks on Large Deviations p.35/??

138 Needs Compactness D( µ 1, µ 2 ) 1 2 j F j (x 1,...,x kj )[Π k j r=1 µ 1(dx r ) Π k j r=1 µ 2(dx r ) {F j } are translation invariant F(x 1 +x,...,x k +x) = F(x 1,...,x k ) Complete with this metric. Some remarks on Large Deviations p.35/??

139 Needs Compactness D( µ 1, µ 2 ) 1 2 j F j (x 1,...,x kj )[Π k j r=1 µ 1(dx r ) Π k j r=1 µ 2(dx r ) {F j } are translation invariant F(x 1 +x,...,x k +x) = F(x 1,...,x k ) Complete with this metric. Completion is compact. Some remarks on Large Deviations p.35/??

140 What is in the completion? Some remarks on Large Deviations p.36/??

141 What is in the completion? { µ α } Some remarks on Large Deviations p.36/??

142 What is in the completion? { µ α } D({ µ α },{ µ β }) = 1 2 j [ α Πµ α (dx r ) β F j (x 1,...,x kj ) Πµ β (dx r ) Some remarks on Large Deviations p.36/??

143 What is in the completion? { µ α } D({ µ α },{ µ β }) = 1 2 j [ α Πµ α (dx r ) β F j (x 1,...,x kj ) Πµ β (dx r ) Need to show that ifd({ µ α },{ µ β }) = 0 then Some remarks on Large Deviations p.36/??

144 What is in the completion? { µ α } D({ µ α },{ µ β }) = 1 2 j [ α Πµ α (dx r ) β F j (x 1,...,x kj ) Πµ β (dx r ) Need to show that ifd({ µ α },{ µ β }) = 0 then { µ α } = { µ β } Some remarks on Large Deviations p.36/??

145 Identification of{ µ α } from Some remarks on Large Deviations p.37/??

146 Identification of{ µ α } from α F(x1,...,x k )Πµ α (dx r ) Some remarks on Large Deviations p.37/??

147 Identification of{ µ α } from α F(x1,...,x k )Πµ α (dx r ) ] m α[ F(x1,...,x k )Πµ α (dx r ) Some remarks on Large Deviations p.37/??

148 Identification of{ µ α } from α F(x1,...,x k )Πµ α (dx r ) α[ F(x1,...,x k )Πµ α (dx r ) F(x1,...,x k )Πµ α (dx r ) ] m Some remarks on Large Deviations p.37/??

149 Identification of{ µ α } from α F(x1,...,x k )Πµ α (dx r ) α[ F(x1,...,x k )Πµ α (dx r ) F(x1,...,x k )Πµ α (dx r ) ] m F = exp[ 1 t i x i ] provided i t i = 0 Some remarks on Large Deviations p.37/??

150 Identification of{ µ α } from α F(x1,...,x k )Πµ α (dx r ) α[ F(x1,...,x k )Πµ α (dx r ) F(x1,...,x k )Πµ α (dx r ) ] m F = exp[ 1 t i x i ] provided i t i = 0 Πφ(t i ) provided i t i = 0 Some remarks on Large Deviations p.37/??

151 φ(t) 2 Some remarks on Large Deviations p.38/??

152 φ(t) 2 φ(t) = φ(t) χ(t) Some remarks on Large Deviations p.38/??

153 φ(t) 2 φ(t) = φ(t) χ(t) Π i χ(t i ) = 1 if i t i = 0 Some remarks on Large Deviations p.38/??

154 φ(t) 2 φ(t) = φ(t) χ(t) Π i χ(t i ) = 1 if i t i = 0 χ(t+s) = χ(t)χ(s), χ(nt) = [χ(t)] n Some remarks on Large Deviations p.38/??

155 φ(t) 2 φ(t) = φ(t) χ(t) Π i χ(t i ) = 1 if i t i = 0 χ(t+s) = χ(t)χ(s), χ(nt) = [χ(t)] n χ(t) = e ita Some remarks on Large Deviations p.38/??

156 φ(t) 2 φ(t) = φ(t) χ(t) Π i χ(t i ) = 1 if i t i = 0 χ(t+s) = χ(t)χ(s), χ(nt) = [χ(t)] n χ(t) = e ita a is not determined. Some remarks on Large Deviations p.38/??

157 THANK YOU Some remarks on Large Deviations p.39/??