Discrete Parabolic Anderson Model with Heavy Tailed Potential

Transcript

1 Discrete Parabolic Anderson Model with Heavy Tailed Potential F. Caravenna 1, P. Carmona 2 and N. Pétrélis 2 1 Università degli Studi di Milano-Bicocca 2 Laboratoire Jean Leray, Université de Nantes 1March2011

2 1 The Model 2 Goals and former results 3 Single site localization of the endpoint 4 Path properties

3 1 The Model

4 1.1) Configurations. Let d N \{0} and for x Z d set x = x x d. Let S = {S k } k 0 be a (lazy) nearest-neighbor RW on Z d, i.e., S 0 =0, (S k+1 S k ) k 0 is i.i.d., {x Z d : P (S 1 = x) > 0} = {0} {y Z d : y =1}

5 1.2) The medium. Pick α>dand let (ξ(x)) x Z d be an i.i.d. field of Pareto-distributed random variables, i.e., P(ξ(0) t) = 1 t α t 1.

6 1.2) The medium. Pick α>dand let (ξ(x)) x Z d be an i.i.d. field of Pareto-distributed random variables, i.e., P(ξ(0) t) = 1 t α t ) Hamiltonian. Given ξ R Zd, each RW trajectory S is associated with the Hamiltonian N H ξ N (S) = ξ(s i )= l N (S, x) ξ(x). x Z d i=1

7 1.4) Perturbed measure. Given ξ R Zd, we let P ξ N be the perturbed law in size N, i.e., dp ξ exp H ξ N dp (S) = N (S) Z ξ, N and denote by p ξ N the law of S N, i.e., p ξ N (x) =P ξ N (S N = x) x Z d.

8 2 Goals and former results

9 2.1) Challenges. Pick N N, ξ R Zd and notice that B N = {x Z d : p ξ N (x) > 0} = {x Zd : x N}.

10 Determine the smallest A ξ N B N such that P-a.s. in ξ, lim N x A ξ N p ξ N (x) =1,

11 Determine the smallest A ξ N B N such that P-a.s. in ξ, lim N x A ξ N p ξ N (x) =1, Find a narrowed W ξ N {S : S N A ξ N } which still satisfies that P-a.s. in ξ, lim P ξ N N (S Wξ N )=1.

12 2.2) Two sites localization in continuous time. With a continuous time random walk on Z d Theorem (Konig, Lacoin, Morters and Sidorova (2008)) Let α>d.forallt>0 and ξ R Zd there exist z (1) such that t,ξ,z(2) t,ξ Zd lim t pξ t (z(1) t,ξ )+pξ t (z(2) t,ξ )=1 P-a.s. in ξ. Super-balistic localization : with q = d/(α d) > 0. z (1) t,ξ, z(2) t,ξ (t/ log t)1+q,

13 3 Single site localization of the endpoint

14 3.1) Modified field. Pick x B N and S : S N = x. The contribution of ξ(x) to H ξ N (S) is with l N (S, x) ξ(x) (N +1 x ) ξ(x) :=(N + 1) ψ N (x) ψ N (x) := 1 x N+1 ξ(x).

15 3.2) Order statistics of the modified field. Set such that z (1) N = argmax{ψ N(x): x B N }, z (k) N = argmax ψ N (x): x B N \{z (1) N,...,z(k 1) N }, ψ N (z (1) N ) >ψ N(z (2) N ) > >ψ N(z ( B N ) N ), is the order statistics of the field {ψ N (x)} x BN.

16 3.3) Localization. For all α>d, ξ R Zd and N 1 set w N,ξ := : argmax p ξ N (x) : x B N. Theorem (One-site localization) It comes that P-a.s. in ξ and Moreover, P lim N pξ N (w N,ξ) =1, lim P w N,ξ = z (1) N N,ξ =1. w N,ξ = z (2) N,ξ for infinitely many N =1.

17 3.4) Behavior of (z (1) N, z(2) N ). ξ(y ) >ξ(x) but 1 Y N 1 +1 ξ(y ) < 1 X N 1 +1 ξ(x) such that ψ N1 (Y ) <ψ N1 (X) (z (1) N 1, z (2) N 1 )=(X, Y )

18 N 2 =inf{n N 1 : ψ n (X) <ψ n (Y )} (z (1) N 1, z (2) N 1 )=(X, Y ) and (z (1) N 2 1, z(2) N 2 1 )=(X, Y )

19 (z (1) N 2 1, z(2) N 2 1 )=(X, Y ) and (z(1) N 2, z (2) N 2 )=(Y,X) ξ(z) >ξ(y ) >ξ(x) but ψ N2 (Z) <ψ N2 (X) <ψ N2 (Y )

20 (z (1) N 2, z (2) N 2 )=(Y,X) and (z (1) N 3, z (2) N 3 )=(Y,Z) ψ N3 (X) <ψ N3 (Z) <ψ N3 (Y )

21 3.5) Heuristic. Set for all S x N (S) =argmax{ξ(x): l N (S, x) > 0}.

22 3.5) Heuristic. Set for all S x N (S) =argmax{ξ(x): l N (S, x) > 0}. We can prove that P-a.s. in ξ lim N P ξ N (x N(S) / {z (1) N,z(2) N })=0.

23 3.5) Heuristic. Set for all S x N (S) =argmax{ξ(x): l N (S, x) > 0}. We can prove that P-a.s. in ξ Let S be such that lim P ξ N N (x N(S) / {z (1) N,z(2) N })=0. l N (S,z (1) N )=N +1 z(1) N,

24 3.5) Heuristic. Set for all S x N (S) =argmax{ξ(x): l N (S, x) > 0}. We can prove that P-a.s. in ξ Let S be such that lim P ξ N N (x N(S) / {z (1) N,z(2) N })=0. l N (S,z (1) N )=N +1 z(1) N, and pick S such that x N (S) / {z (1) N,z(2) N }.

25 Remark : P-a.s. in ξ and for N large enough, H ξ N (S ) (N + 1) ψ N (z (1) N ) u N = N 1+d/α (log log N) 1/α.

26 Set v N = N d/α (log N) 1/α H ξ N (S) H ξ N (S) = o( u N N ) and then x: ξ(x) v N l N (x) ξ(x)+ x: ξ(x) v N l N (x) ξ(x)+nv N. x: ξ(x) v N l N (x) ξ(x)

27 Set v N = N d/α (log N) 1/α H ξ N (S) H ξ N (S) = o( u N N ) and then x: ξ(x) v N l N (x) ξ(x)+ x: ξ(x) v N l N (x) ξ(x)+nv N. By definition of x N (S) it comes that x: ξ(x) v N l N (x) ξ(x) H ξ N (S) l N({x: ξ(x) v N }) ξ(x N (S)) + o(u N ).

28 Set v N = N d/α (log N) 1/α H ξ N (S) H ξ N (S) = o( u N N ) and then x: ξ(x) v N l N (x) ξ(x)+ x: ξ(x) v N l N (x) ξ(x)+nv N. By definition of x N (S) it comes that x: ξ(x) v N l N (x) ξ(x) H ξ N (S) l N({x: ξ(x) v N }) ξ(x N (S)) + o(u N ). Since S visits at least x N (S) distinct sites it comes l N ({x: ξ(x) v N }) N x N (S) + {x B N : ξ(x) v N }.

29 Remark : P-a.s. in ξ and for N large enough, {x B N : ξ(x) v N } (log N) 2.

30 Remark : P-a.s. in ξ and for N large enough, {x B N : ξ(x) v N } (log N) 2. Thus, for N large H ξ N (S) (N +1 x N(S) )ξ(x N (S)) + (log N) 2 ξ(x N (S)) + o(u N ).

31 Remark : P-a.s. in ξ and for N large enough, {x B N : ξ(x) v N } (log N) 2. Thus, for N large H ξ N (S) (N +1 x N(S) )ξ(x N (S)) + (log N) 2 ξ(x N (S)) + o(u N ). Remark : P-a.s. in ξ and for N large enough, max{ξ(x): x B N } N d/α log N.

32 Remark : P-a.s. in ξ and for N large enough, {x B N : ξ(x) v N } (log N) 2. Thus, for N large H ξ N (S) (N +1 x N(S) )ξ(x N (S)) + (log N) 2 ξ(x N (S)) + o(u N ). Remark : P-a.s. in ξ and for N large enough, max{ξ(x): x B N } N d/α log N. Therefore H ξ N (S) (N + 1)ψ N(x N (S)) + o(u N ),

33 Remark : P-a.s. in ξ and for N large enough, {x B N : ξ(x) v N } (log N) 2. Thus, for N large H ξ N (S) (N +1 x N(S) )ξ(x N (S)) + (log N) 2 ξ(x N (S)) + o(u N ). Remark : P-a.s. in ξ and for N large enough, max{ξ(x): x B N } N d/α log N. Therefore H ξ N (S) (N + 1)ψ N(x N (S)) + o(u N ), and x N (S) / {z (1) N,z(2) N }, then H ξ N (S) (N + 1)ψ N(z (3) N )+o(u N).

34 Remark : P-a.s. in ξ, ψ N (z (3) N )=o(ψ N(z (1) N )).

35 Remark : P-a.s. in ξ, ψ N (z (3) N )=o(ψ N(z (1) N )). Therefore, for every S : x N (S) / {z (1) N,z(2) N } H ξ N (S) =o H ξ N (S ).

36 4 Path properties

37 For all S, letτ N,ξ := inf{n 0: S n = w N,ξ }. Set C N,ξ := S : S is injec. on [0,τ N,ξ ], τ N,ξ w N,ξ + o(n), We then have the following result. Theorem It comes that P-a.s. in ξ, ξ(s) <ξ(w N,ξ ) on [0,τ N,ξ ], lim P N,ξ(C N,ξ )=1. N S = w N,ξ on [τ N,ξ,N],