A large deviation principle for a RWRC in a box

Transcription

1 A large deviation principle for a RWRC in a box 7th Cornell Probability Summer School Michele Salvi TU Berlin July 12, 2011 Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

2 Outline 1 Large Deviations for dummies What are Large Deviations? A proper denition 2 Random Walk among Random Conductances The model The main theorem Related elds Some heuristics Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

3 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

4 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Strong law of large numbers (SLLN): ( 1 P n n j=1 X j ) n 0 = 1; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

5 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Strong law of large numbers (SLLN): ( 1 P n n j=1 Central limit theorem (CLT): ( 1 P σ n n j=1 X j ) n 0 = 1; ) n X j A 1 e y2 2 dy; 2π A Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

6 Large Deviations for dummies What are Large Deviations? X 1, X 2,... i.i.d. R-valued random variables with E [X 1 ] = 0 and Var(X1) = σ 2 R. Strong law of large numbers (SLLN): ( 1 P n n j=1 Central limit theorem (CLT): ( 1 P σ n n j=1 X j ) n 0 = 1; ) n X j A 1 e y2 2 dy; 2π A Large Deviation Principle (LDP) (+ nite exponential moments): ( 1 P n n j=1 ) X j x e ni(x), x 0. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

7 Large Deviations for dummies What are Large Deviations? Large Deviation Theory deals with asymptotic computation of small probabilities on an exponential scale. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

8 Large Deviations for dummies What are Large Deviations? Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

9 Large Deviations for dummies What are Large Deviations? The function I(x) is convex, has compact level sets (= is lower semi-continuous), I(x) 0 and equality holds i x = µ = E [X 1 ]. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

10 Large Deviations for dummies A proper denition Denition Let X be a Polish space. A function I : X [0, ] is called rate function if I I has compact level sets (= lower semicontinuous) Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

11 Large Deviations for dummies A proper denition Denition Let X be a Polish space. A function I : X [0, ] is called rate function if I I has compact level sets (= lower semicontinuous) Denition Let γ n be a sequence in R +. A sequence of probability measures {µ n } on (X, B(X )) satises a large deviation principle with rate function I and speed γ n if Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

12 Large Deviations for dummies A proper denition Denition Let X be a Polish space. A function I : X [0, ] is called rate function if I I has compact level sets (= lower semicontinuous) Denition Let γ n be a sequence in R +. A sequence of probability measures {µ n } on (X, B(X )) satises a large deviation principle with rate function I and speed γ n if 1 For every open set O, lim inf n 2 For every closed set C, lim sup n 1 log µ n (O) inf I(x); γ n x O 1 log µ n (C) inf I(x). γ n x C Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

13 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

14 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

15 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; 3 µ n (X ) = 1 = inf x X I(x) = min I(x) = 0; x X Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

16 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; 3 µ n (X ) = 1 = inf x X I(x) = min I(x) = 0; x X 4 if!x s.t. I(x) = 0, the LDP implies SLLN ; Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

17 Large Deviations for dummies A proper denition Remarks 1 I lower semi-continuous = attains a minimum on every compact set; 2 for a "nice" set A µ n (A) e γn inf A I ; 3 µ n (X ) = 1 = inf x X I(x) = min I(x) = 0; x X 4 if!x s.t. I(x) = 0, the LDP implies SLLN ; 5 in general no relation between LDP and CLT. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

18 The model The model Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

19 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

20 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

21 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

22 The model The model Consider the lattice Z d and assign to any bond (x, x + e) a random weight ω x,e such that ω x,e = ω x+e, e (symmetry), {ω x,e } x Zd,e E are i.i.d., ω x,e 0 (positivity). Denition The Random Walk among Random Conductances (RWRC) is the continuous-time process generated by ω f (x) := ( ) ω x,e f (x + e) f (x). x Z d,e E Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

23 The model Let (X t ) t [0, ) be the RWRC. For x B Z d, B nite and connected set, dene the local time l t (x) := t 0 1l {Xs=x} ds. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

24 The model Let (X t ) t [0, ) be the RWRC. For x B Z d, B nite and connected set, dene the local time l t (x) := t 0 1l {Xs=x} ds. We want to study the annealed behaviour of l t : ( 1 ) P ω 0 t l t g 2 where g 2 M 1 (B) and is the expectation w.r.t. the conductances. The conductances can attain arbitrarily small values. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

25 The model Let (X t ) t [0, ) be the RWRC. For x B Z d, B nite and connected set, dene the local time l t (x) := t 0 1l {Xs=x} ds. We want to study the annealed behaviour of l t : ( 1 ) P ω 0 t l t g 2 where g 2 M 1 (B) and is the expectation w.r.t. the conductances. The conductances can attain arbitrarily small values. Three noises: the conductances; the waiting times; the embedded discrete-time RW. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

26 The main theorem Hypothesis: B Z d nite and connected; log Pr( ωx,e < ε ) ε η, for ε 0, η > 1. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

27 The main theorem Hypothesis: B Z d nite and connected; log Pr( ωx,e < ε ) ε η, for ε 0, η > 1. Theorem (joint work with Wolfgang König and Tilman Wol) The process of empirical measures ( 1l t t) t R + of the Random Walk among Random Conductances under the annealed law P ω( {supp(l 0 t) B}) satises a large deviation principle on M 1 (B) with speed γ t = t η η+1 and rate function J given by J(g 2 ) := C η g(z + e) g(z) 2η η+1 = C η g z,e for all g 2 M 1 (B), where C η := ( η ) η 1 1+η. 2η 1+η 2η 1+η Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

28 The main theorem This means P ω ( {1 0 t l t g 2} { supp(l t ) B }) 2η γt Cη g(z+e) g(z) η+1 e z,e. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

29 The main theorem This means P ω ( {1 0 t l t g 2} { supp(l t ) B }) 2η γt Cη g(z+e) g(z) η+1 e z,e. In particular: Corollary The annealed probability of non-exit from the box B for the Random Walk among Random Conductances for t 0 is log P ω 0 ( supp(l t ) B ) t η 1+η inf g 2 M 1 (B) C η z,e g(z + e) g(z) 2η η+1. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

30 Related elds Related elds: Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

31 Related elds Related elds: By a Fourier expansion: P ω 0 ( ) d #B supp(l t ) B = e tλω(b) k f k (0) f k, 1l e tλω 1 (B), k=1 where λ ω 1 (B) is the bottom of the spectrum of ω restricted to the box B. Relation with Random Schrödinger operators! Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

32 Related elds Related elds: By a Fourier expansion: P ω 0 ( ) d #B supp(l t ) B = e tλω(b) k f k (0) f k, 1l e tλω 1 (B), k=1 where λ ω 1 (B) is the bottom of the spectrum of ω restricted to the box B. Relation with Random Schrödinger operators! Parabolic Anderson model with random Laplace operator: { t u(x, t) = ω u(x, t) + ξ(x)u(x, t), t (0, ), x Z d u(x, 0) = δ 0 (x) x Z d. Feynman-Kac formula gives u(x, t) = E ω x is a RWRC. [ t ] e 0 ξ(xs)ds δ 0 (X t ), where X t Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

33 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

34 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); combine "classical" LDP's for the empirical measure of a (weighted) random walk by Donsker and Varadhan (1979) (speed t 1 r ) and for the conductances (speed t rη ); Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

35 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); combine "classical" LDP's for the empirical measure of a (weighted) random walk by Donsker and Varadhan (1979) (speed t 1 r ) and for the conductances (speed t rη ); "physicists' trick" (t 1 r = t rη ); Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

36 Some heuristics Sketch of the proof (heuristics): rescale the conductances (t r ω = ϕ); combine "classical" LDP's for the empirical measure of a (weighted) random walk by Donsker and Varadhan (1979) (speed t 1 r ) and for the conductances (speed t rη ); "physicists' trick" (t 1 r = t rη ); optimization over the rescaled shape of the conductances. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

37 Some heuristics Technical obstacles: Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

38 Some heuristics Technical obstacles: Lower bound for open sets. Problem: need to understand the asymptotics of inf P ϕ( 1 ) ϕ A t l t. There seems to be no monotonicity, but there is some kind of continuity of the map ϕ P ϕ 0 ( ). Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

39 Some heuristics Technical obstacles: Lower bound for open sets. Problem: need to understand the asymptotics of inf P ϕ( 1 ) ϕ A t l t. There seems to be no monotonicity, but there is some kind of continuity of the map ϕ P ϕ 0 ( ). Upper bound for closed sets. Problem: t r ω is not bounded. We need a compactication argument for the space of rescaled conductances. Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15

40 Some heuristics Thank you! Michele Salvi (TU Berlin) An LDP for a RWRC in a nite box July 12, / 15