Infinite-dimensional stochastic differential equations related to - PowerPoint PPT Presentation

Infinite-dimensional stochastic differential equations related to Airy random fields 2012/9/25/Tue Okayama • Random matrices and log gasses • Soft edge scaling limit and Airy random fields • General theory for ISDEs: quasi-Gibbs property & log derivative • Examples: Sine RPF, Bessel RPF, Airy RPF, Ginibre RPF, All canonical Gibbs measures

Gaussian ensembles & semi-circle law • The dist of eigen values of the G(O/U/S)E Random Matrices are given by ( β = 1 , 2 , 4) N β ( d x N ) = 1 | x i − x j | β e − β ∑ N i =1 | x i | 2 d x N , m N ∏ (1) 4 Z i<j • The distribution of N N − 1 under m N ∑ δ x i β i =1 converge the semi-circle law ς ( x ) dx = 1 √ 4 − x 2 dx (2) 2 π 2

Sine rpf (Dyson’s model)–Bulk scaling limit N β ( d x N ) = 1 ς ( x ) dx = 1 √ | x i − x j | β e − β ∑ N i =1 | x i | 2 d x N , m N 4 − x 2 dx ∏ 4 Z 2 π i<j √ • Take x i = s i / N in (1) and set N N sin ,β ( d s N ) = 1 e − β | s k | 2 / 4 N d s N µ N | s i − s j | β ∑ ∏ (3) Z i<j k =1 30p • The associated N particle system is given by the SDE: N 1 t + β dt − β dX i t = dB i 4 N X i ∑ t dt (4) t − X j 2 X i j ̸ = i t • So the ass ∞ particle system is given by ∞ t + β 1 dX i t = dB i ∑ dt t − X j 2 X i t j ̸ = i 3

Airy rpf – Soft edge scaling limit Airy rpf: µ Ai ,β ( S = R , β = 1 , 2 , 4) √ N + s i N − 1 / 6 in Take the scaling x i �→ 2 N β ( d x N ) = 1 | x i − x j | β e − β ∑ N i =1 | x i | 2 d x N m N ∏ 4 Z i<j and set N √ Ai ,β ( d s N ) = 1 | s i − s j | β e − β ∑ N N + N − 1 / 6 s i | 2 d s N . i =1 | 2 µ N ∏ 4 Z i<j Then µ Ai ,β is the TDL of µ N Ai ,β : N →∞ µ N lim Ai ,β = µ Ai ,β 4

Airy rpf – Soft edge scaling limit • β = 2 ⇒ µ Ai ,β is the det rpf gen by ( K Ai , dx ): K Ai ( x, y ) = Ai( x )Ai ′ ( y ) − Ai ′ ( x )Ai( y ) x − y Here Ai( · ) the Airy function such that Ai( z ) = 1 ∫ dk e i ( zk + k 3 / 3) , z ∈ C . (5) 2 π R If β = 1 , 4, the correlation func of µ Ai ,β are given by similar formula of quaternion determinant. 5

Airy rpf – Soft edge scaling limit • From √ Ai ,β ( d s N ) = 1 | s i − s j | β e − β ∑ N N + N − 1 / 6 s i | 2 d s N i =1 | 2 µ N ∏ 4 Z i<j we deduce the SDE of the N particle system: N t + β 1 dt − β 1 2 { N 1 / 3 + dX i t = dB i 2 N 1 / 3 X i ∑ t } dt t − X j X i 2 j =1 ,j ̸ = i t • Problem: What is the limit SDE? N 1 − N 1 / 3 } ∑ Does lim converge ? N →∞ { t − X j X i t j =1 ,j ̸ = i How to solve the limit SDE? 6

Airy rpf – Soft edge scaling limit Thm 1 (with Tanemura) . Let β = 1 , 2 , 4 . Then: • the limit ISDE is t + β 1 ϱ ( x ) ∫ dX i t = dB i ∑ 2 lim r →∞ { ( ) − − x dx } dt t − X j X i | x | <r j ̸ = i, | X j t t | <r √− x ϱ ( x ) = 1 ( −∞ , 0] ( x ) π • The above SDE has a unique, strong solution. • So far the sto dyn related to Airy RPF was constructed only for β = 2 by Spohn, Johansson, and others by the method of space-time cor funs. This sto dyn is same as the above. • ( X i t ) is a diffusion with state space R N . • X t = ∑ t is reversible w.r.t. µ airy ,β . i δ X i 7

General theorems for Infinite-dim SDE: set up Let S = R d , C , [0 , ∞ ). S : Configuration space over S ∑ S = { s = δ s i ; s i ∈ S, s ( | s | < r ) < ∞ ( ∀ r ∈ N ) } i µ : RPF on S . i.e. prob meas. on S . Prob: (1) To construct a natural stochastic dynamics X t = ( X i t ) i ∈ N (labeled dynamics) related to µ , i.e. ∑ X t = (unlabeled dynamics) δ X i t i ∈ N is reversible w.r.t. µ . (2) To find the ∞ -dim. SDE that X t satisfies. 8

General theorems for Infinite-dim SDE: set up • ρ n is called the n -correlation function of µ w.r.t. Radon m. m if n m s ( A i )! ∫ ∫ ρ n ( x n ) ∏ ∏ m ( dx i ) = ( s ( A i ) − k i )! dµ A k 1 ···× A km 1 × S m i =1 i =1 for any disjoint A i ∈ B ( S ), k i ∈ N s.t. k 1 + . . . + k m = n . • µ is called the determinantal RPF generated by ( K, m ) if its n -correlation fun. ρ n is given by ρ n ( x n ) = det[ K ( x i , x j )] 1 ≤ i,j ≤ n • Ginibre RPF S = C . µ gin is generated by ( K gin , 2 , g ) g ( dx ) = π − 1 e −| x | 2 dx K gin , 2 ( x, y ) = e x ¯ y 9

Gibbs measure • Ψ: Ruelle’s class interaction potential, Q r = {| x | ≤ r } , π r ( s ) = s ( · ∩ Q r ), π c r ( s ) = s ( · ∩ Q c r ) µ m r,ξ ( · ) = µ ( π r ∈ ·| s ( Q r ) = m, π c r ( s ) = π c r ( ξ )) • µ is called (Φ , Ψ)-Gibbs m. if it satisfies DLR eq: m 1 e −H r ( s ) −W r,ξ ( s ) dµ m e − Φ( s k ) ds k ∏ r,ξ = z r,ξ k =1 ∑ ∑ H r = Ψ( s i − s j ) , W r,ξ = Ψ( s i − ξ j ) s i ∈ Q r ,ξ j ∈ Q c s i ,s j ∈ Q r ,i<j r • Let Ψ( x ) = − 2 log | x | . Then, W r,ξ diverge, so DLR does not make sense 10

(Φ , Ψ)-Quasi Gibbs measures r = ∏ m Let ν m k =1 1 Q r ( s k ) e − Φ( s k ) ds k (Φ , Ψ)-Gibbs m. 1 dµ m e −H r −W r,ξ dν m r,ξ = (DLR eq) r z m r,ξ (Φ , Ψ)-quasi Gibbs m. ∃ c m r,ξ − 1 e −H r dν m c m r ≤ µ m r,ξ ≤ c m r,ξ e −H r dν m r r,ξ r,ξ diverge. But e −W r,ξ /z m • If µ is Airy RPF, W r,ξ and z m r,ξ conv. − 1 ≤ e −W r,ξ /z m c m r,ξ ≤ c m r,ξ r,ξ • Quasi-Gibbs is very mild restriction. If µ is (Φ , Ψ)- quasi-Gibbs m, then µ is also (Φ + f, Ψ)-quasi Gibbs m for any loc bdd m’able f . 11

Main theorems: Unlabeled level construction Let D be the canonical square field on S : s = ∑ i δ s i , s = ( s i ). D [ f, g ]( s ) = 1 ∑ ∇ s i ˜ f ( s ) · ∇ s i ˜ g ( s ) 2 i Let D be the set of local smooth fun with E µ 1 ( f, f ) < ∞ . ∫ E µ ( f, g ) = D [ f, g ] dµ S Thm 2. [O . 96[CMP] , 10[JMSJ] , 12?[AOP] (1) If µ is quasi-Gibbs with upper semi-cont potentials (Φ , Ψ) , then ( E µ , D , L 2 ( S , µ )) is closable. (2) If ( E µ , D , L 2 ( S , µ )) is closable & all correlation fun are loc bounded, then a diffusion X t associated with the closure ( E µ , D µ ) exists. If µ is Poisson rpf with Lebesgue intensity, then X t = ∑ i δ B i t . 12

Log derivative of µ – SDE representation of the stochastic dynamics • Let µ x be the (reduced) Palm m. of µ conditioned at x µ x ( · ) = µ ( · − δ x | s ( x ) ≥ 1) • Let µ 1 be the 1-Campbell measure on R d × S : ∫ µ 1 ( A × B ) = ρ 1 ( x ) µ x ( B ) dx A • d µ ∈ L 1 ( R d × S , µ 1 ) is called the log derivative of µ if ∫ ∫ ∇ x fdµ 1 = − f d µ dµ 1 ∀ f ∈ C ∞ 0 ( R d ) ⊗ D R d R d × × S S Here ∇ x is the nabla on R d , D is the space of bounded, local smooth functions on S . • Very informally d µ = ∇ x log µ 1 13

Main theorems: Infinite-dim SDE (A1) µ is a quasi-Gibbs measure. (closability) (A2) ∑ ∞ r ∈ L 2 ( S k k =1 kµ ( S k r ) < ∞ , σ k r , dx ) (quasi-regular) Here S r = {| x | < r } , S k r = { s ( S r ) = k } , σ k r is k -density fun on S r . (A3) The log derivative d µ ∈ L 1 loc ( µ 1 ) exists (SDE rep) (A4) { X i t } do not collide each other (non-collision) (A5) each tagged particle X i t never explode (non-explosion) Let u : S N → S such that u (( s i )) = ∑ i δ s i . 14

Main theorems: labeled diffusions Thm 3. (O . 12(PTRF)) (A1) – (A5) ⇒ ∃ S 0 ⊂ S such that µ ( S 0 ) = 1 , (6) and that, for ∀ s ∈ u − 1 ( S 0 ) , ∃ u − 1 ( S 0 ) -valued pr. ( X i t ) i ∈ N and ∃ S N -valued Brownian m. ( B i t ) i ∈ N satisfying t + 1 dX i t = dB i 2 d µ ( X i ( X i ∑ ) dt, 0 ) i ∈ N = s (7) t , δ X j t j ̸ = i 15

Main theorems: labeled diffusions t + 1 dX i t = dB i 2 d µ ( X i ( X i ∑ t , δ X j ) dt, 0 ) i ∈ N = s t j ̸ = i Thm 4 (O. (JMSJ 10)) . The family of processes { ( X i t ) i ∈ N } is a diffusion with state space u − 1 ( S 0 ) ⊂ S N . Remark 1 . (1) (A1)–(A5) can be checked for Ginibre RPF ( β = 2), Sine RPFs, Airy RPFs and Bessel RPFs ( β = 1 , 2 , 4). (2) We can calculate the log derivatives of these measures. (3) We have general theorems for quasi-Gibbs property and the log derivatives (O. PTRF12, to appear in AOP, preprint). The state- ments are too messy to be omitted here. 16

unique, strong solution H 1 = { ( x, s ) ∈ S × S ; d µ ( x, s ) is locally Lips cont. } Here “locally” means we regard d µ ( x, s ) as symmetric fun on S r with fixed particles outside S c r for ∀ r except a ca- pacity zero set (non-single points, say). Let H = { δ x + s ; ( x, s ) ∈ H 1 } Assume (A6) Cap µ ( H c ) = 0. Thm 5 (with Tanemura) . Assume (A1) – (A6) . Then the SDE has a unique, strong solution for initial starting points ( s i ) ∈ S N such that ∑ i δ s i ∈ H q.e.. Remark: (1) It is quite likely that all determinantal rpfs in continuous spaces satisfy (A1)–(A6). (2) It is likely that the conclusion of Thm 5 holds for all initial points s = ( s i ) such that ∑ i δ s i ∈ H . (in progress) 17