Local fluctuations of critical Mandelbrot cascades. Konrad Kolesko

Local fluctuations of critical Mandelbrot cascades Konrad Kolesko joint with D. Buraczewski and P. Dyszewski Warwick, 18-22 May, 2015

Random measures µ µ 1 µ 2 For given random variables X 1, X 2 s.t. E [ e X 1 + e X 2] = 1 we are interested in random measures µ on [0, 1) satisfying self similar property: µ(b) = e X 1 µ 1 ( 2(B [0, 1/2)) ) + e X 2 µ 2 ( 2(B [1/2, 1) 1) ), where µ 1 µ 2 X 1, X 2 and Lµ = Lµ 1 = Lµ 2. Goal: Understand local properties of µ Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 2 / 19

Phase transition Define ψ(t) := log 2 E ( e tx 1 + e tx 2). By assumption ψ(0) = 1, ψ(1) = 0, ψ. 6 Supercritical Critical 5 4 3 2 Subcritical (x^(2) (x^(2) (x^(2) 1 0 0 1 2 3 4 5-1 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 3 / 19

Subcritical case ψ (1) = m < 0 Theorem (Liu) Suppose that ψ (1) = m < 0, then for any δ > 0, almost all realization of µ, µ-almost all x and sufficiently large n µ(b(x, 2 n )) 2 nm (1+δ) 2σ 2 n log log n µ(b(x, 2 n )) 2 nm+(1+δ) 2σ 2 n log log n, where σ 2 = ψ (1) ψ(1). Moreover µ(b(x, 2 n )) 2 nm (1 δ) 2σ 2 n log log n µ(b(x, 2 n )) 2 nm+(1 δ) 2σ 2 n log log n i.o. i.o. Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 4 / 19

Supercritical & critical case Barral, Rhodes, Vargas When ψ (1) > 0 then µ is purely atomic Barral, Kupiainen, Nikula, Saksman, Webb If ψ (1) = 0, then the random measure µ almost surely has no atoms. Moreover for any k and δ > 0, µ-a.e. x µ(b(x, 2 n )) e 6 log 2 n(log n+(1/3+δ) log log n) for sufficiently large n µ(b(x, 2 n )) e ( 2 log 2+δ) n log n i.o. µ(b(x, 2 n )) n k for sufficiently large n. Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 5 / 19

Main theorem Theorem (D. Buraczewski, P. Dyszewski, K.K.) Let ψ (1) = 0, k N and δ > 0. Then for almost all realizations of µ, µ-almost all x [0, 1) and sufficiently large n we have ( µ(b(x, 2 n )) exp (1 + δ) ) 2σ 2 n log log n ( µ(b(x, 2 n ) n )) exp k i=1 log (i)n (log (k+1) n) 2 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 6 / 19

Main theorem Theorem (D. Buraczewski, P. Dyszewski, K.K.) Let ψ (1) = 0, k N and δ > 0. Then for almost all realizations of µ, µ-almost all x [0, 1) and sufficiently large n we have ( µ(b(x, 2 n )) exp (1 + δ) ) 2σ 2 n log log n ( µ(b(x, 2 n ) n )) exp k i=1 log (i)n (log (k+1) n) 2 e K1 n(log n+(1/3+δ) log log n) n k e (K2+δ) n log n Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 6 / 19

Notations Dyadic intervals on [0,1) vertices of a binary tree T x [0, 1) θ T. B(v) = {θ T : v oθ}. For a random measure µ ω we are interested in a pointwise estimates of µ ω (B(θ n )) on the enlarged measure space P(dω, dθ) := P(dω)µ ω (dθ) i.e. we are looking for deterministic φ 1, φ 2 s.t. φ 1 (n) µ ω (B(θ n )) φ 2 (n), for P-almost all (ω, θ) and large n. P can be replaced by P(dω, dθ) = P(dω)µ ω (dθ), where µ ω is the normalized measure µ ω. Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 7 / 19

Branching random walk o P probability measure on the set of labeled trees X(v) the sum of random variables along the path between o and v (X(u) = X 2 + X 21 + X 212 ) {X(v)} v T Branching random walk (BRW) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 8 / 19

Branching random walk X 1 X 2 o P probability measure on the set of labeled trees X(v) the sum of random variables along the path between o and v (X(u) = X 2 + X 21 + X 212 ) {X(v)} v T Branching random walk (BRW) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 8 / 19

Branching random walk X 1 X 2 o X 11 X 12 P probability measure on the set of labeled trees X(v) the sum of random variables along the path between o and v (X(u) = X 2 + X 21 + X 212 ) {X(v)} v T Branching random walk (BRW) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 8 / 19

Branching random walk X 1 X 2 o X 11 X 12 X 21 X 22 P probability measure on the set of labeled trees X(v) the sum of random variables along the path between o and v (X(u) = X 2 + X 21 + X 212 ) {X(v)} v T Branching random walk (BRW) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 8 / 19

Branching random walk X 1 X 2 o X 11 X 12 X 21 X 22 X 212 u P probability measure on the set of labeled trees X(v) the sum of random variables along the path between o and v (X(u) = X 2 + X 21 + X 212 ) {X(v)} v T Branching random walk (BRW) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 8 / 19

Stopping line Since E v =1 e X(v) = 1 and E v =1 X(v)e X(v) = 0 the equation Ef(Y ) := E f(x(v))e X(v) v =1 defines distribution of a driftless r.v. Y Let h be a harmonic function on some set A (a solution of a Dirichlet problem), V n = Y 1 + + Y n and σ = min{k : s + V k / A}. Then the process h(s + V min(n,σ) ) is a martingale. Let τ = {w : s + X(w) / A for the first time}, v τ = min(v, τ). Then Wn s = h(s + X(v τ ))e X(vτ ) is a martingale. v =n Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 9 / 19

Martingales Some natural martingales h 1: h(x) = x: W n = v =n e X(v) D n = v =n X(v)e X(v) A = [0, ), h(x) x 0: W s n = v =n h(s + X(v))1 [s+x(v )>0, for v v]e X(v) µ(b(v)) := lim n w =n, w<v X(w)e X(w) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 10 / 19

Martingales Some natural martingales h 1: W n = v =n e X(v) 0 h(x) = x: D n = v =n X(v)e X(v) D > 0 A = [0, ), h(x) x 0: W s n = v =n h(s + X(v))1 [s+x(v )>0, for v v]e X(v) µ(b(v)) := lim n w =n, w<v X(w)e X(w) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 10 / 19

W s n Wn s = h(s + X(v))1 [s+x(v )>0, for v v]e X(v) v =n For any s D define W s n W s P-a.s. and L 1 P s := W s h(s) P P[supp W s ] 1 1/s Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 11 / 19

P s = BRW 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 12 / 19

P s BRW = 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 12 / 19

Spinal decomposition of P s BRW s 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW X 1 1 X 1 2 s 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW X 1 1 X 1 2 e X1 1 h(s + X 1 1 ) e X1 2 h(s + X 1 2 ) s 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 1 =s+x 1 2 S 0 =s 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW X 2 1 X 2 2 S 1 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW X 2 1 X 2 2 S 1 e X2 1 h(s1 + X 2 1) e X2 2 h(s1 + X 2 2) S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 1 S 2 =S 1 +X 1 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 1 S 2 X 3 1 X 3 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 1 S 2 X 3 1 X 3 2 e X3 1 h(s2 + X 3 1) e X3 2 h(s2 + X 3 2) S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 1 S 3 =S 2 +X3 1 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 1 S 3 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 4 S 1 S 3 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s BRW S 5 S 4 S 1 S 3 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s Random tree T s with a distinguished ray Θ T s BRW S 5 S 4 S 1 S 3 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s P(T s dω) = P s (dω) BRW S 5 S 4 S 1 S 3 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

Spinal decomposition of P s P s (dω, dθ) := P(T s dω, Θ dθ) BRW S 5 S 4 S 1 S 3 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 13 / 19

P s vs. P We have P s P. The converse is not true, but for any set A P s (A) = 1 P(A) 1 ɛ(s), for some ɛ(s) 0 as s. In particular, if for any s > 0 P s (A) = 1 then P(A) = 1. Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 14 / 19

Probabilistic interpretation of local fluctuations e (S 0 s) C 0 e (S 1 s) S 5 S 4 R 0 e (S 2 s) C 2 R 2 C 1 R 1 S 1 S 3 S 2 S 0 0 Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 15 / 19

Probabilistic interpretation of local fluctuations Under P s the sequence µ ω (B(θ n )) has the same law as e (Sk s) C k R k, k n where S k is a random walk starting form s conditioned to stay positive, R k independent random variables. Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 15 / 19

Useful information We are looking for LIL for k n e S k+s C k R k, Hambly, Kersting, Kyprianou ψ(t) t lim sup n dt< iff S n > nψ(n) eventually S n 2nσ = 1 2 log log n nψ(n) < Sn < (1 + δ) 2nσ 2 log log n P s -a.s. for n sufficiently large Kyprianou 1 Ee tr tl(t) near 0 = ER γ < for γ < 1 In particular, by Borel-Cantelli lemma, R n < n 2 for all but finitely many n. Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 16 / 19

Useful information We are looking for LIL for k n e S k Rk, Hambly, Kersting, Kyprianou ψ(t) t lim sup n dt< iff S n > nψ(n) eventually S n 2nσ = 1 2 log log n nψ(n) < Sn < (1 + δ) 2nσ 2 log log n P s -a.s. for n sufficiently large Kyprianou 1 Ee tr tl(t) near 0 = ER γ < for γ < 1 In particular, by Borel-Cantelli lemma, R n < n 2 for all but finitely many n. Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 16 / 19

Upper bound Take any ψ such that ψ(t)dt <. We have that t e S k R k k n is eventually bounded by e kψ(k) k 2. k n On the other hand, if ψ(t)dt t k n = then e S k R k e Sn R n i.o. e nψ(n) R n i.o. δe nψ(n) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 17 / 19

Upper bound Take any ψ such that ψ(t)dt <. We have that t e S k R k k n is eventually bounded by e kψ(k). k n On the other hand, if ψ(t)dt t k n = then e S k R k e Sn R n i.o. e nψ(n) R n i.o. δe nψ(n) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 17 / 19

Upper bound Take any ψ such that ψ(t)dt <. We have that t e S k R k is eventually bounded by k nψ(n) k n On the other hand, if ψ(t)dt t k n e k polynomial(k). = then e S k R k e Sn R n i.o. e nψ(n) R n i.o. δe nψ(n) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 17 / 19

Upper bound Take any ψ such that ψ(t)dt <. We have that t e S k R k k n is eventually bounded by k nψ(n) e k. On the other hand, if ψ(t)dt t k n = then e S k R k e Sn R n i.o. e nψ(n) R n i.o. δe nψ(n) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 17 / 19

Upper bound Take any ψ such that ψ(t)dt <. We have that t e S k R k is eventually bounded by k n e nψ(n). On the other hand, if ψ(t)dt t k n = then e S k R k e Sn R n i.o. e nψ(n) R n i.o. δe nψ(n) Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 17 / 19

Lower bound Take q > 1 and by N(n) := q log q n. Borel-Cantelli s lemma implies that for sufficiently large n, sup R k δ 0. q n <k q n+1 for some q. k n e Sk R k k n qn(n) k=n(n) e (1+δ) 2kσ 2 log log k R k e (1+δ) 2kσ 2 log log k R k δ 0 e (1+δ) 2qN(n)σ 2 log log(qn(n)) δ 0 e (1+δ) 2q 2 nσ 2 log log(q 2 n) e (1+2δ) 2nσ 2 log log n, Konrad Kolesko Local fluctuations of critical Mandelbrot cascades 18-22 May, 2015 18 / 19