Localization for Brownian motion in a heavy tailed Poissonian - PowerPoint PPT Presentation

Localization for Brownian motion in a heavy tailed Poissonian potential Ryoki Fukushima Tokyo Institute of Technology 10th Workshop on Stochastic Analysis on Large Scale Interacting Systems In celebration of Prof. Funaki’s 60’s birthday Kochi University, December 5-7, 2011 1 / 18

Motivation To understand the behavior of Brownian motion among randomly distributed (repulsive) obstacles. 2 / 18

Motivation To understand the behavior of Brownian motion among randomly distributed (repulsive) obstacles. − → Brownian motion conditioned to avoid the obstacles. 2 / 18

Motivation To understand the behavior of Brownian motion among randomly distributed (repulsive) obstacles. − → Brownian motion conditioned to avoid the obstacles. − → kill the Brownian motion by a random potential and condition to survive. 2 / 18

1. Setting ( ) : κ ∆-Brownian motion on R d • { B t } t ≥ 0 , P x ( ) ∑ : Poisson point process on R d • ω = δ ω i , P i with unit intensity 3 / 18

1. Setting ( ) : κ ∆-Brownian motion on R d • { B t } t ≥ 0 , P x ( ) ∑ : Poisson point process on R d • ω = δ ω i , P i with unit intensity Potential For a non-negative function v , ∑ V ω ( x ) := v ( x − ω i ) . i (Typically v ( x ) = 1 B (0 , 1) ( x ) or | x | − α ∧ 1 with α > d .) 3 / 18

Annealed path measure We are interested in the behavior of Brownian motion under the measure ∫ t Q t ( · ) = 1 { } exp − V ω ( B s ) d s P ⊗ P 0 ( · ) , Z t 0 ∫ t [ { }] Z t = E ⊗ E 0 exp − V ω ( B s ) d s . 0 The configuration is not fixed and hence Brownian motion and ω i ’s try to avoid each other. 4 / 18

• 0 • B t 5 / 18

• 0 • B t 5 / 18

Brief history (only annealed) ◮ Anderson (1958): Localization of electron in random potentials. ◮ Donsker-Varadhan (1975): Asymptotics of Z t in the case v ( x ) = o ( | x | − d − 2 ). ◮ Pastur (1977): Asymptotics of Z t in the case v ( x ) ∼ c | x | − α for α ∈ ( d , d + 2). ◮ Okura (1981): Asymptotics of Z t in the case v ( x ) ∼ c | x | − d − 2 . ◮ G¨ artner-Molchanov (1990, 1998): Localization of diffusion particle in general random potentials (weak sense). ◮ Sznitman (1991), Bolthausen (1994), Povel (1999): Strong localization of diffusion particle for compactly supported v . ◮ Asymptotics of Z t in various settings in the name “parabolic Anderson model”: G¨ artner, Molchanov, K¨ onig, Biskup, van der Hofstad, M¨ orters, Sidorova,... 6 / 18

2. Light tailed case Donsker and Varadhan (1975) When v ( x ) = o ( | x | − d − 2 ) as | x | → ∞ , ∫ t [ { }] E ⊗ E 0 exp − V ω ( B s ) ds 0 { d } d +2 (1 + o (1)) = exp − c ( d ) t ( 1 ) ( 1 ) d +2 R 0 ) d +2 R 0 )) = 0 = P 0 B [0 , t ] ⊂ B ( x , t ω ( B ( x , t , P as t → ∞ . 7 / 18

2. Light tailed case Donsker and Varadhan (1975) When v ( x ) = o ( | x | − d − 2 ) as | x | → ∞ , ∫ t [ { }] E ⊗ E 0 exp − V ω ( B s ) ds 0 { d } d +2 (1 + o (1)) = exp − c ( d ) t ( 1 ) ( 1 ) d +2 R 0 ) d +2 R 0 )) = 0 = P 0 B [0 , t ] ⊂ B ( x , t ω ( B ( x , t , P as t → ∞ . Remark This is related to spectral asymptotics of − κ ∆ + V ω (Lifshiz tail). 7 / 18

■ 1 B t R 0 t d +2 ❘ • • 0 8 / 18

One specific strategy gives dominant contribution to the partition function. ⇓ It occurs with high probability under the annealed path measure. 9 / 18

Sznitman (1991, d = 2) and Povel (1999, d ≥ 3) When v has a compact support, there exists 1 ( d +2 ( R 0 + o (1)) ) D t ( ω ) ∈ B 0 , t such that )) t →∞ ( 1 ( d +2 ( R 0 + o (1)) B [0 , t ] ⊂ B D t ( ω ) , t − − − → 1 . Q t Remark Bolthausen (1994) proved the corresponding result for two-dimensional random walk model. 10 / 18

3. Heavy tailed case Pastur (1977) When v ( x ) ∼ | x | − α as | x | → ∞ with α ∈ ( d , d + 2), ∫ t [ { }] { } d E ⊗ E 0 exp − V ω ( B s ) d s = exp − a 1 t α 0 as t → ∞ , where ( α − d ) a 1 := | B (0 , 1) | Γ . α 11 / 18

■ 1 α ) O ( t ✻ 1 ❘ α ) o ( t • 0 ❄ B t 12 / 18

F. (2011) When v ( x ) = | x | − α ∧ 1 with α ∈ ( d , d + 2), ∫ t [ { }] E ⊗ E 0 exp − V ω ( B s ) d s 0 { } d α + d − 2 α − ( a 2 + o (1)) t = exp − a 1 t 2 α as t → ∞ , where {∫ } κ |∇ φ ( x ) | 2 + C ( d , α ) | x | 2 φ ( x ) 2 dx a 2 := inf . ∥ φ ∥ 2 =1 13 / 18

F. (2011) When v ( x ) = | x | − α ∧ 1 with α ∈ ( d , d + 2), ∫ t [ { }] E ⊗ E 0 exp − V ω ( B s ) d s 0 { } d α + d − 2 α − ( a 2 + o (1)) t = exp − a 1 t 2 α as t → ∞ , where {∫ } κ |∇ φ ( x ) | 2 + C ( d , α ) | x | 2 φ ( x ) 2 dx a 2 := inf . ∥ φ ∥ 2 =1 Remark The proof relies on a general machinery developed by G¨ artner-K¨ onig (2000). 13 / 18

|∇ φ ( x ) | 2 d x ) ∫ We believe that the 2nd term (in particular expresses the “effort to confine the Brownian motion”. ( ) ≈ exp {− tr ( t ) − 2 } . sup | B s | < r ( t ) P 0 0 ≤ s ≤ t 14 / 18

|∇ φ ( x ) | 2 d x ) ∫ We believe that the 2nd term (in particular expresses the “effort to confine the Brownian motion”. ( ) ≈ exp {− tr ( t ) − 2 } . sup | B s | < r ( t ) P 0 0 ≤ s ≤ t tr ( t ) − 2 = t α + d − 2 α − d +2 ⇔ r ( t ) = t . 2 α 4 α 14 / 18

■ 1 α ) O ( t ✻ α − d +2 ❘ O ( t ) • 0 4 α ❄ B t 15 / 18

Main Theorem 2 + ϵ )) t →∞ ( ( α − d +2 1 4 α (log t ) Q t B [0 , t ] ⊂ B 0 , t − − − → 1 , 16 / 18

Main Theorem 2 + ϵ )) t →∞ ( ( α − d +2 1 4 α (log t ) Q t B [0 , t ] ⊂ B 0 , t − − − → 1 , { } in law t − α − d +2 4 α B t − − − → OU-process with α − d +2 s 2 α s ≥ 0 “random center”. 16 / 18

Thank you! & Happy birthday professor Funaki! 17 / 18

0 the below is the • section along this line V ( ω, x ) Light tailed case t α ( t ) ≪ t /β ( t ) 2 α ( t ) ✛ ✲ β ( t ) 18 / 18

0 the below is the • section along this line V ( ω, x ) Heavy tailed case t α ( t ) ≫ t /β ( t ) 2 α ( t ) ✛ ✲ β ( t ) 18 / 18