Strong Markov property of determinatal processes

Transcription

1 Strong Markov property of determinatal processes Hideki Tanemura Chiba university (Chiba, Japan) (August 2, 2013) Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 1 / 27

2 Introduction The configuration space of unlabelled particles: { M = ξ( ) = } δ xj ( ) : {j I : x j K} <, for any K compact j I M is a Polish space with the vague topology. The configuration space of noncolliding systems: { } M 0 = ξ M : ξ({x}) = 1, for any x supp ξ { } = {x j } : {j : x j K} <, for any K compact. A configuration space X is relative compact, if sup ξ(k) <, ξ X for any K R compact Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 2 / 27

3 Introduction The moment generating function of multitime distribution of the M-valued process Ξ(t) is defined as { M Ψ (t 1,...,t M ) (f 1,..., f M ) = Ψ t (f) = E exp m=1 for 0 t 1 < t 2 < < t M, f = (f 1, f 2,..., f M ) C 0 (R) M. Ψ t (f) = N m 0, 1 m M { M Q M m=1 RN m m=1 R 1 N m! dx(m) N m ρ ξ ( t 1, x (1) N 1 ;... ; t M, x (M) N M ), f m (x)ξ(t m, dx)} ] (1) N m ( χ m i=1 x (m) i ) } with χ m ( ) = e f m( ) 1 and the multitime correlation functions ρ ξ (... ). Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 3 / 27

4 Introduction A processs Ξ(t) is said to be determinantal if the moment generating function (1) is given by the Fredholm determinant ] Ψ t f] = δ st δ x (y) + K(s, x; t, y)χ t (y), (2) Det (s,t) {t 1,t 2,...,t M } 2, (x,y) R 2 In other words, the multitime correlation functions are represented as ] ( ) ρ t 1, x (1) N 1 ;... ; t M, x (M) N M = det K(t m, x (m) 1 j N j ; t n, x (n) k ). m,1 k N n 1 m,n M The function K is called the correlation kernel of the process Ξ(t). Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 4 / 27

5 Examples of determinatal processes (1) Non-colliding Brownian motion (The Dyson model): X j (t) = x j + B j (t) + k:1 k N k j t 0 ds X j (s) X k (s), 1 j N where B j (t), j = 1, 2,..., N are independent one dimensional BMs. (2) Non-colliding Bessel process: X j (t) = x j + B j (t) + 2ν t { k:1 k N k j 0 t 0 ds X j (s) 1 X j (s) X k (s) + 1 X j (s) + X k (s) } ds, 1 j N Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 5 / 27

6 Related results (O) Spohn: 1987, Prḧofer-Spohn: JSP02, Johanson: CMP02] Infinite many particle systems Ξ t obtaine by the Bulk or the Soft-edge scaling limits of the processes. equilibrium systems (1) Katori-T: JSP09, CMP10, JSP11, ECP13]: Infinite many particle systems in non-equilibrium (2) Katori-T: MPRF11] Markov property of the reversible Markov processes. (3) Oasda: PTRF12, SPA13, AOP13, Osada-T: in preparation, Osada-Honda: preprint] Constructing diffusion processes Ξ DF t by Dirichlet form technique and deriving ISDEs related to them. (4) Osada-T: in preparation] The uniqueness of solutions of the ISDE. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 6 / 27

7 Infinite dimensional SDEs for Ξ DF t = j N δ X j (t) Bulk scaling limit : Osada:PTRF12] β = 1, 2 and 4 dx j (t) = db j (t) + β 2 k N k j dt, j N. (3) X j (t) X k (t) Hard edge scaling limit : Honda-Osada: preprint] ν > 1 dx j (t) = db j (t) + 2ν X j (t) dt + β { 2 k N k j 1 X j (t) X k (t) + 1 X j (t) + X k (t) } dt, j N. (4) Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 7 / 27

8 Infinite dimensional SDEs for Ξ DF t = j N δ X j (t) Soft edge scaling limit : Osada-T: in preparation] β = 1, 2 and 4 where ρ(x) = dx j (t) = db j (t) + β 2 lim L k N,k j X k (t) L x1(x<0) π. 1 X j (t) X k (t) ρ(y) y L y dy dt, j N, (5) Questions 1. The Dyson model constructed by Spohn(1987) solves ISDE (1)? 2. The infinite particle system constructed by Prähofer-Spohn(2002), Johannson(2002) solves ISDE (2)? Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 8 / 27

9 Main Theorem In this talk we consider the case that β = 2, and ν = 1 2, 1 2. Theorem Let Ξ t be the determinantal process obtained by (i) or (ii) or (iii): (i) the bulk scaling limit of noncolliding BM (ii) the soft-edge scaling limit of noncolliding BM (iii) the hard-edge scalig limit of one or three dimensional Bessel process. There exists the state spaces S associated with the process Ξ t such that the process (P ξ, Ξ t ), ξ S has the strong Markov property. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 9 / 27

10 Answer the questions Strong Markov property of Ξ t the quasi-regularity of the Dirichlet form associated with Ξ t Ξ t solve the same ISDE as the coincidence of the processes Ξ t and Ξ DF t (by the technique in (3)) Ξ DF t Ξ DF t (by (4)) Remark. The state space of is uniquely determined up to quasi everwhere. That is S is a version of the state space. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 10 / 27

11 State Space M κ 1 L 0 (ζ), ζ M, κ 1 > 0, L 0 N, is the set of configurations ξ M 0 satisfying N κ 2 ζ(0, L] ξ(0, L]) L κ 1, ζ( L, 0] ξ( L, 0]) L κ 1, L L 0. (6) L 0, κ 2 > 0, L 0 N, is the set of configurations ξ M 0 satisfying ( ) ξ x e x κ2, x + e x κ2 ] \ {x} = 0, x supp ξ, x > L 0. (7) Put M κ 1 (ζ) = L 0 N S(ζ) = M κ 1 L 0 (ζ), κ (0,1) N κ 2 = L 0 N (M κ (ζ) N κ ). N κ 2 L 0. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 11 / 27

12 State space We introduce the following configurations: ζ ρ = δ x, ζ Ai = δ x, ζ (ν) = x:sin(ρx)=0 x:ai(x)=0 x:j ν(x)=0 Let µ sin : The determinatal point process with the sine kernel. µ Ai : The determinatal point process with the Airy kernel. µ Jν : The determinatal point process with the Bessel kernel. Let κ 2 > κ 1 > 0, It is readily checked that µ Jν ({ξ 2 = j N µ sin (M κ 1 (ζ ρ ) N κ 2 ) = 1, κ 1 > 0, µ Ai (M κ 1 (ζ Ai ) N κ 2 ) = 1 κ 1 > 1/2, δ x 2 j : ξ = j N δ x. δ xj M κ 1 (ζ (ν ) N κ 2 }) = 1 κ 1 > 0. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 12 / 27

13 Harmonic transformation (Brownian motions with a drift) B(t) = (B 1 (t),..., B N (t)) : N-dim BM Let g be a F t -adapted symmetric measurable function on C(0, T ], R N ). (1) Noncolliding Brownian motions with constant drift E x g(y( ))] = E x g(b( )) h N(B(T )) h N (x) N j=1 ] e D(B j x j )(T ) D2 T 2, where h N (x) = 1 i<j N (x j x i ) = det 1 i,j N x j 1 i ]. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 13 / 27

14 Harmonic transformation (Bessel processes ν = ± 1 2 ) Suppose that g(x 1, x 2,..., x N ) = g( x 1, x 2,..., x N ). (2) Noncolliding 3-dimensional Bessel process (Type C) where E x g(y( ))] = E x h (0) N (x) = 1 i<j N g(b( ))) h(0) N (B(T )) h (0) N (x) N j=1 (xj 2 xi 2 ) = det 1 i,j N ] B j (T ), x j x 2(j 1) i (3) Noncolliding 1-dimensional Bessel process (Type D) E x g 1 (Y( ))] = E x g(b( )) h(0) N (B(T ))] h (0) N (x). ]. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 14 / 27

15 Entire functions Suppose that ξ = ξ(r) i=1 δ u i M 0 with ξ(r) N. ( Φ 0 (ξ, u, z) = 1 z u ), z C, u R r u r supp ξ {u} (1)(Noncolliding Brownian motions with constant drift) Φ D (ξ, u, z) = e D(z u) Φ 0 (ξ, u, z), u R. (2)(Noncolliding 3-dimensional Bessel process (Type C)) Φ (1/2) (ξ, u, z) = z u Φ 0(ξ 2, u 2, z 2 ), u > 0. (3)( Noncolliding 1-dimensional Bessel process (Type D)) Φ ( 1/2) (ξ, u, z) = Φ 0 (ξ 2, u 2, z 2 ), u 0. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 15 / 27

16 Complex Brownian motion Put V j (t), 1 j N : indep. BMs with V j (0) = u j R W j (t), 1 j N : indep. BMs with W j (0) = 0, Z j (t) = V j (t) + 1W j (t), 1 j N : Complex BMs Z(t) = (Z 1 (t), Z 2 (t),..., Z N (t)), V(t) = (V 1 (t), V 2 (t),..., V N (t)), Noting that and for ξ = N j=1 δ x j, x C N, E W h N (Z(T ))] = h N (V(N)), h N (z) h N (x) = det Φ 0(ξ, x j, z k ). 1 j,k N Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 16 / 27

17 Theorem (Complex Brownian motion representations) Let 0 < t < T <. For any F(t)-measurable function F : (1)(Noncolliding Brownian motions with constant drift) ξ(r) ( )] ))] E ξ F Ξ( ) = Eu F det Φ D (ξ, u i, Z j (T. i=1 δ Vi ( ) 1 i,j ξ(r) (2)(Noncolliding 3-dimensional Bessel process (Type C)) ξ(r) ( )] ))] E ξ F Ξ( ) = Eu F det Φ (1/2) (ξ, u i, Z j (T. i=1 δ Vi ( ) 1 i,j ξ(r) (3)( Noncolliding 1-dimensional Bessel process (Type D)) ξ(r) ( )] ))] E ξ F Ξ( ) = Eu F det Φ ( 1/2) (ξ, u i, Z j (T. i=1 δ Vi ( ) 1 i,j ξ(r) Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 17 / 27

18 Entire functions for configurations of infinitely many points Lemma 1 If ξ M κ 1 (ζ ρ ) N κ 2 with 0 < κ 1 < κ 2 < 1, Φ 0 (ξ, a, z) = lim L Φ 0(ξ L, L], a, z), exists, and Φ 0 (ξ, a, z) C exp{c( a δ + z θ )}, a supp ξ, z C, for some C = C(κ 1, κ 2, L 0 ) > 0,c = c(κ 1, κ 2, L 0 ) > 0, δ (0, 1) and θ (1, 2). THEOREM 1 (Katori-T. CMP10, ECP13) Suppose that ξ S(ζ ρ ). Then (Ξ(t), P ξ L,L] ) (Ξ(t), P ξ ), L weakly on C(0, ) M 0 ). In particular, the process (Ξ(t), P ξ ) has a modification which is almost-surely continuous on 0, ) with Ξ(0) = ξ. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 18 / 27

19 CBM representation for infinte particle systems In case F is a polynomial, CBMR holds for ξ S(ζ ρ ). For example (1) For any measurable function f on R with compact support ] E ξ f (u)ξ t (du) = ξ(du)e u f (V 1 (t))φ 0 (ξ, u, Z 1 (T )] R R (2) For any measurable symmetric function f on R 2 with compact support ] E ξ φ(u 1, u 2 )Ξ t Ξ t (du 1 du 2 ) u 1 <u 2 ] = ξ(du)e (u1,u 2 ) f (V 1 (t), V 2 (t)) det Φ 0(ξ, u i, Z j (T )) u 1 <u 2 1 i,j 2 Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 19 / 27

20 State space Lemma 2 Let ζ = ζ ρ, ζ Ai or ζ (ν), ν = ± 1 2, 1/2 < κ 1 < κ 2 < 1. Then for any κ 1 > κ 1 and κ 2 > κ 2 ( ) P ξ Ξ t M κ 1 (ζ) N κ 2, t > 0 = 1. To prove this lemma we apply CBMR to functions F 1 and F 2 : ( ) F 1 δ Xj ( ) = sup f (X j (t)) f (X j (0)), j j t 0,T ] ( ) F 2 δ Xj ( ) = sup f ( X j (t) X k (t) ), j j,k t 0,T ] and the following identity: Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 20 / 27

21 Another key identity For a symmetric function F (v 1, v 2 ) ] E (u1,u 2 ) f (V 1, V 2 ) det Φ 0(ξ, u i, Z j ) 1 i,j 2 = E (u1,u 2 ) f (V 1, V 2 ) h 2(V 1 (T ), V 2 (T )) h 2 (u 1, u 2 ) ] Φ 0 (ξ {u 1, u 2 } c, u j, Z j (T )). (8) j=1,2 Then, (Y 1 (t), Y 2 (t)) can be treated as the non-colliding Brownian motion of 2-particles. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 21 / 27

22 Soft-edge case We introduce the entire function defined by { z } Φ 1 (ξ, 0, z) = exp {0} c x ξ(dx) Φ 0 (ξ, 0, z). Then Ai(z) = exp{d 1 z}φ 1 (ζ Ai, 0, z). The soft-edge scaling limit case is associated with the entire function for ξ with ξ(r) = N defined by { z } Φ Ai (ξ, 0, z) = exp d 1 z + {0} c x (ξ ζn Ai)(dx) Φ 1 (ζ Ai, 0, z), Φ Ai (ξ, u, z) = Φ Ai (τ u ξ, 0, z u). where ζ N Ai = N j=1 δ a j, with the decreasing sequence {a j } of zeros of Ai. Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 22 / 27

23 Hard-edge case J ν (z) = (z/2)ν Γ(ν + 1) Φ 0(ζ 2 J ν, z 2 ) The hard-edge scaling limit case is associated with the entire functions defined by Φ (1/2) (ξ, u, z) = z u Φ 0(ξ 2, u 2, z 2 ), Φ ( 1/2) (ξ, u, z) = Φ 0 (ξ 2, u 2, z 2 ). Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 23 / 27

24 Markov property Lemma 3 Let (ζ, Φ) = (ζ ρ, Φ 0 ), (ζ Ai, Φ Ai ) or (ζ (ν), Φ (ν) ), ν = ± 1 2, and let 1/2 < κ 1 < κ 2 < 1. (1) Suppose that ξ M κ 1 L 0 (ζ) N κ 2 L 0, with L 0 N. Then Φ(ξ L, L], u, z) Φ(ξ, u, z), L. (2) Suppose ξ n, ξ M κ 1 L 0 (ζ) N κ 2 L 0 (ζ), and u n supp ξ n, u supp ξ. If ξ n ξ, vaguely and u n u, n, then Φ(ξ n, u n, z) Φ(ξ, u, z), n Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 24 / 27

25 (P ξ L,L], Ξ t ) is a Markov process, that is, ] ] E ξ L,L] f 1 (Ξ s )f 2 (Ξ t ) = E ξ L,L] f 1 (Ξ s )E Ξs f 2 (Ξ t s )] for 0 < s < t and polynomials f 1, f 2 : ( f j (ξ) = Q j ϕ 1 (x)ξ(dx),..., R R ) ϕ k (x)ξ(dx) By taking L from Lemmas 1 and 3 ] ] E ξ f 1 (Ξ s )f 2 (Ξ t ) = E ξ f 1 (Ξ s )E Ξs f 2 (Ξ t s )] Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 25 / 27

26 Strong Markov property For a polynomial function f we put T t f (ξ) = E ξ f (Ξ t )], the semigroup associated with the Markov process (P ξ, Ξ t ). Suppose that ξ n, ξ M ε L 0 (ξ Z/ρ ) N κ L 0 for some L 0 N. From Lemmas 1 and 2, ξ n ξ, vaguely T t f (ξ n ) T t f (ξ). We equip S δ with the topology by the inductive limit. Then, for ξ n, ξ S δ We have the Feller property. ξ n ξ, T t f (ξ n ) T t f (ξ). Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 26 / 27

27 Thanks Thank you for your attention! Hideki Tanemura (Chiba univ.) () Markov process (August 2, 2013) 27 / 27