Quenched invariance principle for random walks and random divergence - PowerPoint PPT Presentation

Quenched invariance principle for random walks and random divergence forms in random media on cones Takashi Kumagai (RIMS, Kyoto University, Japan) On-going joint work with Z.Q. Chen (Seattle) and D.A. Croydon (Warwick). http://www.kurims.kyoto-u.ac.jp/~kumagai/ 28 September 2012: Stochastic Analysis and Applications at Okayama

Each bond “open” with prob. p “closed” with prob. 1-p “Open”, “closed” is indep for each bond 1 Introduction Bond percolation on Z d ( d ≥ 2) ∃ p c ∈ (0 , 1) s.t. p > p c ⇒ ∃ 1 ∞ -cluster G ( ω ) (random media!), p < p c ⇒ no ∞ -cluster

SRW on supercritical percolation cluster on Z d Known results • [Quenched invariance principle (QIP)] (Sidoravicius-Sznitman ’04, Berger-Biskup ’07, Mathieu-Piatnitski. ’07) n � 1 Y ! n 2 t ! B � t P ⇤ -a.s. ! for some � > 0 p ! t ( x, y ) := P x ( Y t = y ) /µ y . • [Gaussian heat kernel bounds] (Barlow ’04) d ( x, y ) 2 d ( x, y ) 2 c 1 t ( x, y )  c 3 )  p ! t d/ 2 exp( � c 2 t d/ 2 exp( � c 4 ) , t t P ⇤ -a.s. ! for t � d ( x, y ) _ 9 U x , x, y 2 C . Rem 1. ”Annealed” invariance principle: known since 80’s Rem 2. Generalization of the QIP to random conductance model is known.

Each bond “open” with prob. p “closed” with prob. 1-p “Open”, “closed” is indep for each bond Ex 1 RW on supercritical percolation cluster for L ⇢ Z d ( d � 2) (Our problem) L := { ( x 1 , · · · , x d ) 2 Z d : x j 1 , · · · , x j l � 0 } for some 1  j 1 < · · · < j l  d , l  d . 9 p c 2 (0 , 1) s.t. 9 1 1 -cluster C for p > p c , no 1 -cluster for p < p c .

P ⇤ ( · ) := P p ( ·| 0 2 C ), Y ! : SRW on C ( ! ). C ( ! ): 1 -cluster, [ n 2 t ] ! B � t , P ⇤ � a.e. ! (for some � > 0)? n � 1 Y ! (Q1) (How about RW on percolation on boxes?)

0 Ex 2 Random divergence form on a cone C : Lipschitz domain in R d � 1 D := { ( t, tx 1 , · · · , tx d � 1 ) 2 R d : t > 0 , ( x 1 , · · · , x d � 1 ) 2 C } : cone c 1 I  A ! ( x )  c 2 I for all x 2 D , P -a.e. ! . ( Ω , P ): Prob. space, ! 2 Ω , A ! ( x ) , 2 R d s.t. ˜ 9 ˜ A ! ( x ) = A ! ( x ) , x 2 D , ˜ A ! ( x ) = ˜ A ⌧ x ! (0), { ⌧ x } x 2 R d : ergo. shift. R D r f ( x ) A ! ( x ) r f ( x ) dx Y ! : corresponding di ff usion. E ( f, f ) = ) " Y ! (Q2) " � 2 t ! B � t , P � a.e. ! (for some � > 0)?

(Known results for the whole space) Random divergence form on R d c 1 I  A ! ( x )  c 2 I for all x 2 R d , P -a.e. ! , ( Ω , P ): Prob. space, ! 2 Ω , A ! ( x ) = A ⌧ x ! (0), { ⌧ x : x 2 R d } : ergo. shift. R R d r f ( x ) A ! ( x ) r f ( x ) dx Y ! : corresponding di ff usion. E ( f, f ) = ) • [Quenched invariance principle] (...., Osada ’83, Kozlov ’85) " Y ! " � 2 t ! B � t , P -a.s. ! for some � > 0 . • [Gaussian heat kernel bounds] d ( x, y ) 2 d ( x, y ) 2 c 1 t ( x, y )  c 3 )  p ! t d/ 2 exp( � c 2 t d/ 2 exp( � c 4 ) , (1) t t P -a.s. ! for t > 0, x, y 2 R d .

Problem in extending the results to cones All the results use corrector method, which requires translation invariance of the original space. Main results : Yes! (Q1) (box case as well) and (Q2) hold. Ideas • Full use of heat kernel estimates. • Use information of QIP on the whole space and Dirichlet form methods.

2 Framework and results D ⇢ R d : Lipschitz domain Z E ( f, f ) = C | r f ( x ) | 2 dx, 8 f 2 W 1 , 2 ( D ) , 2 D W 1 , 2 ( D ) = { f 2 L 2 ( D, m ) : r f 2 L 2 ( D, m ) } , m : Lebesgue meas. X : reflected BM corresponding to ( E , W 1 , 2 ( D )) X D : process killed on exiting D (i.e. X D corresponding to ( E , W 1 , 2 0 ( D ))). { D n } n � 1 ⇢ D : D n supports a meas. m n s.t. m n ! m weakly in D .

weak Theorem 2.1 { X n t } t � 0 : sym. Hunt proc. on L 2 ( D n ; m n ) , m n ! m on D . Assume that 8 { n j } subseq., 9 { n j k } sub-subseq. and 9 ( e X, e P x , x 2 D ) : m -sym. conserv. conti. Feller proc. on D starting at x s.t. n jk x j ) e (i) 8 x j ! x , P P x weakly in D ([0 , 1 ) , D ) , = X D where e X D is subprocess of e d (ii) e X D X killed upon leaving D , (iii) ( ˜ E , ˜ F ) : D-form of e X on L 2 ( D ; m ) satisfies C ⇢ ˜ ˜ F and E ( f, f )  K E ( f, f ) 8 f 2 C , where C : core for ( E , W 1 , 2 ( D )) and K � 1 . weak ) ( X n , P n x n ) ! ( X, P x ) in D ([0 , 1 ) , D ) as n ! 1 .

How to verify (i)-(iii)? (i) Use heat kernel esitmates etc. (ii) From QIP of the whole space (iii) By LLN-type arguments 2.1 About (i) Assume 0 2 D n , 8 n � 1, 9 � n 2 [0 , 1] with lim n !1 � n = 0 s.t. | x � y | � � n 8 x 6 = y 2 D n . Assumption 2.2 (I) 9 c 1 , c 2 , c 3 , ↵ , � , � > 0 , N 0 2 N s.t. the following hold for all n � N 0 , x 0 2 B (0 , c 1 n 1 / 2 ) \ D n , and all � 1 / 2  r  1 . n (a) E x [ ⌧ B ( x 0 ,r ) \ D n ( X n )]  c 2 r � , 8 x 2 B ( x 0 , r/ 2) \ D n , where ⌧ A := { t � 0 : X t / 2 A } . Ellip. Harnack: 8 h n : bdd. in D n and harm. (w.r.t. X n ) in B ( x 0 , r ) , then (b) | h n ( x ) � h n ( y ) |  c 3 ( | x � y | ) � k h n k 1 for all x, y 2 B ( x 0 , r/ 2) . (2) r

(II) 8 { x n 2 D n : n � 1 } and 8 x 2 D s.t. x j ! x 2 D , { P x n n } n is tight in D ( R + , D ) . R 1 d 0 e � u { 1 ^ (sup 0  t  u | X t � X t � | ) } du (III) J ( X ) := ! 0 . Proposition 2.3 Under Assumption 2.2, the following holds: 8 { n j } subseq., 9 { n j k } sub-subseq. and m -sym. di ff usion ( e X, e P x , x 2 D ) on D x j weak P x in D ([0 , 1 ) , D ) . ! e s.t. 8 x j ! x , P n jk Rem. Roughly, Gaussian-type heat kernel est. are enough to verify Assumption 2.2. (Obtain equi-H¨ older cont. for resolvents and use Ascoli-Arzela etc.: Z 1 n f ( y ) |  Cd ( x, y ) � 0 k f k 1 , where U � n f ( x ) = E x e � � t f ( X n | U � n f ( x ) � U � t ) dt. 0 For the case of random media, use Borel-Cantelli as well.)

3 Answer to (Q2) in Example 2 • Condition (ii): Whole space QIP by Osada, Kozlov ) (ii) holds. • Condition (i): By uniform ellipticity, Z Z | r f ( x ) | 2 dx, 8 f 2 W 1 , 2 ( D ) . r f ( x ) A ! ( x ) r f ( x ) dx ⇣ E ( f, f ) = (3) D D ) E " : D-form corresp. to " Y ! " � 2 · also satisfies (3). (Note: " D = D .) ) Gaussian-type HK est. (1) still holds uniformly for E " . (Due to the stability: (1) , (Vol. doubling)+ (Poincar´ e ineq.) i.e.) µ ( B ( x 0 , 2 R ) \ D )  C 1 µ ( B ( x 0 , R ) \ D ) , Z Z ( f ( x ) � f B ) 2 µ ( dx )  C 2 R 2 | r f ( x ) | 2 dx, 8 f 2 W 1 , 2 ( D ) , B ( x 0 ,R ) \ D B ( x 0 , 2 R ) \ D R 8 x 0 2 D, R > 0 where f B = B f ( x ) µ ( dx ). ) Assumption 2.2 holds. • Condition (iii): Any subsequ. limit ˜ E still satisfies (3) ) (iii) holds.

4 Answer to (Q1) in Example 1 • Condition (ii): Whole space QIP by Berger-Biskup, Mathieu-Piatnitski ) (ii) holds. • Condition (iii): LLN-type arguments as follows: Lemma 4.1 { ⌘ i } i : i.i.d. with E | ⌘ 1 | < 1 . P n P n k |  M 8 k, n , a := 9 lim n !1 1 k , 9 lim n !1 1 { a n k } n k =1 : a n k 2 R , | a n k =1 a n k =1 | a n k | . n n P n ) lim n !1 1 k =1 a n k ⌘ k = aE [ ⌘ 1 ] almost surely. n x = ( ] of bonds in C con. to x ), D n = n � 1 L , D = { ( x 1 , .., x d ) 2 R d : x j 1 , .., x j l � 0 } . Let µ ! Proposition 4.2 Let E ( n ) be D-form corresp. to n � 1 Y ! [ n 2 · ] . Z ˜ n !1 E ( n ) ( f, f ) = 2 d � l pd | r f ( x ) | 2 dx, 8 f 2 C 2 E ( f, f )  lim c ( D ) , (4) Z D X f ( x ) µ ! n d = 2 d � l pd nx lim f ( x ) dx, 8 f 2 C c ( D ) . (5) n !1 D x 2 D n

1 1 n j ˜ Proof of 1st ineq. of (4) : E ( f, f ) = sup t ( f � P t f, f ) = sup lim inf t ( f � P t f, f ) n j !1 t> 0 t> 0 1 n j n j !1 E ( n j ) ( f, f ) .  lim inf n j !1 sup t ( f � P t f, f ) = lim inf t> 0 For the 2nd ineq., suppose Supp f ⇢ B (0 , M ) \ D . Then X X E ( n ) ( f, f ) = n 2 � d ( f ( x ) � f ( y )) 2 µ nx,ny = 1 n 2 ( f ( x/n ) � f ( y/n )) 2 µ x,y , n d 2 x,y 2 D n ,x ⇠ y ( x,y ) 2 H n,f where H n,f := { ( x, y ) : x, y 2 L \ B (0 , nM ) , x ⇠ y } . Note ] M n,f ⇠ 2 d � l d ( nM ) d . ( x,y ) := n 2 ( f ( x/n ) � f ( y/n )) 2 2 [0 , 9 M 0 ] and ⌘ ( x,y ) := µ x,y . By IP of SRW, Let a n Z X n !1 (2 d � l d ( Mn ) d ) � 1 a n ( x,y ) = M � d | r f ( x ) | 2 dx. lim D ( x,y ) 2 H n,f

So by Lemma 4.1, Z X n !1 n � d a n ( x,y ) ⌘ ( x,y ) = 2 d � l dp | r f ( x ) | 2 dx. lim D ( x,y ) 2 H n,f X X f ( x ) µ nx n d = n � d Proof of (5): f ( x/n ) µ x,y , x 2 D n ( x,y ) 2 H n,f Z X n !1 n � d f ( x/n ) ± = 2 d � l d lim f ( x ) ± dx. D ( x,y ) 2 H n,f So by Lemma 4.1, we obtain (5). ⇤

• Condition (i): Strategy (Percolation est.) ) (HK estimates) ) Assumption 2.2 Lemma 4.3 (Percolation est.) 9 c 1 , c 2 , c 3 > 0 s.t. 8 x, y 2 L , P ( x, y 2 C and d ( x, y )  c 1 | x � y | )  c 2 e � c 3 | x � y | , � � x, y 2 C and d ( x, y ) � c � 1  c 2 e � c 3 | x � y | , 1 | x � y | P where | · � · | is the Euclidean dist. and d ( · , · ) is the graph dist. Rem. Z d case by Antal-Pisztora and we exteded to the case of L . NB: This is the only place where we need the restriction to half/square spaces.