Relaxed Sector Condition and Random Walk in Divergence Free Random Drift Field

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Relaxed Sector Condition and Random Walk in Divergence Free Random Drift Field

Bálint Tóth

(Budapest University of Technology and Economics)

B ´ ALINT T ´ OTH

(TU Budapest and U of Bristol)

Relaxed Sector Condition and

Random Walk in Divergence Free Random Drift Field

KR ´ AMLI-FEST, Szeged, 26-27 July 2013

Notation:

(Ω, π, τ_z : z ∈ Z^d) probabability space

with ergodic Z^d-action E = {e ∈ Z^d : |e| = 1} possible steps of the rw v_e : Ω → [−1,1], e ∈ E

◦ v_e(ω) + v_−e(τ_eω) ≡ 0 vector field

◦ ^X

e∈E

v_e(ω) ≡ 0 divergence-free

◦

Z

Ω v_e(ω)dπ(ω) = 0, no overall drift

V_e(ω, x) := v_e(τ_xω) the vector field over Z^d

Helmholtz’s Theorem:

d ^{= 2: The} height function: Z²_∗ := Z² + (1/2,1/2).

There exists H : Ω × Z²∗ → R scalar field with stationary incre- ments such that

V = curl H, V_e(x) = H(x + e + e_e

2 ) − H(x + e − e_e 2 )

d ^{= 3: The} stream field: Z³_∗ := Z² + (1/2,1/2,1/2).

There exists H_e : Ω × Z³_∗ → R, e ∈ E, vector field with stationary increments such that

V = curl H, V_e(ω, x) = . . . explain in plain words

Examples and essentially different cases:

◦ H stationary + ergodic + bounded [S.M. Kozlov (1985)]:

h ∈ L^∞, H(ω, x) = h(τ_xω) − h(ω)

◦ H stationary + ergodic + unbounded + curlH bounded:

h ∈ L² \ L^∞, H(ω, x) = h(τ_xω) − h(ω)

this is the case discussed today. H₋₁-condition

X

◦ H {stationary + ergodic} increments (but not stationary).

+ curlH bounded. No H₋₁-condition, superdiffusive.

◦◦ randomly oriented Manhattan-lattice

◦◦ six-vertex / square ice (d = 2)

◦◦ dimer tiling (d = 2)

The random walk in random environment: 0 < ε < 1/(2d), fixed

P_ω^X_n+1 ^{= x + eX}₀^{n, X}_n ^{= x = 1}

2d + εV_e(ω, x)

The environment process:

η_n := τ_Xnω

Stationary and ergodic Markov process on (Ω, π). (Due to div- freeness.)

Some operators on the Hilbert space L²(Ω, π):

L²(Ω, π)-gradient : ∇_ef(ω) := f(τ_eω) − f(ω)

∇^∗_e = ∇_−e

L²(Ω, π)-Laplacian : ∆f(ω) := 1 d

X

e∈E

(f(τ_eω) − f(ω))

∆^∗ = ∆ ≤ 0

multiplication ops. : M_ef(ω) := v_e(ω)f(ω) M_e^∗ = M_e

A commutation relation – due to div-freeness of v:

X

e∈E

M_e∇_e + ^X

e∈E

∇_eM_−e = 0

The transition operator / infinitesimal generator of the en- vironment process:

L = P − I = 1

2∆ + ε ^X

e∈E

M_e∇_e =: −S + A

Martingale decomposition of the displacement:

ϕ : Ω → R^d, ϕ(ω) := ^X

e∈E

v_e(ω)e

Then:

Y_n := X_n − ε

n−1 X

k=0

ϕ(η_k)

is a martingale with stationary+ergodic increments.

X_n = Y_n + ε

n−1 X

k=0

ϕ(η_k)

Gaol: understand diffusive behaviour of ^Pn−1_k=0ϕ(η_k).

Relaxed Sector Condition [I. Horv´ath, B. T´oth, B. Vet˝o (2012)]

Theorem: Efficient martingale approximation (a la Kipnis-Varadhan) holds for ^Pn−1

k=0 ϕ(η_k) if

(1) ” S^−1/2AS^−1/2 ” is skew self-adjoint

(not just skew symmetric).

(2) ϕ ∈ Ran(S^−1/2) H₋₁-condition Remarks:

(1) Extends Varadhan et al.’s Graded Sector Condition.

(2) Proof: partly reminiscent of Trotter-Kurtz.

Two possible definitions of ” S^−1/2AS^−1/2 ”:

B := ^X

e∈E

(−∆)^−1/2∇_e M_e (−∆)^−1/2 = ” S^−1/2AS^−1/2 ”

on C := Dom(−∆)^−1/2 = Ran(−∆)^1/2 Be := (−∆)^{−1/2 X}

e∈E

M_e (−∆)^−1/2∇_e = ” S^−1/2AS^−1/2 ”

on C^e := {f ∈ L² : ^X

e∈E

M_e (−∆)^−1/2∇_ef ∈ Dom(−∆)^−1/2} Facts (easy): (1) B is skew symmetric on C.

(2) C ⊂ C^e and B^e

C = B.

(3) B^e = B^e and B^e = −B^∗.

Wanted: B = B^e, or, equivalently B = −B^∗

What is missing from skew self-adjointmess of ” S^−1/2AS^−1/2 ”?

von Neumann’s criterion:





B skew symmetric, and Ran(B ± I) = H



⇔





B essentially skew self-adjoint





Needed:

X

e∈E

M_e (−∆)^−1/2∇_eψ = (−∆)^1/2ψ ⇒ ψ = 0.

Warning: Formal manipulation deceives: ψ /∈ Dom(−∆)^−1/2!

Raise it to the lattice Z^d: change of notation: from now on:

∇,∆, . . . = lattice gradient, lattice Laplacian, . . . Wanted:

NO nontrivial scalar field Ψ : Ω × Z^d → R with stationary increment, and E ^{Ψ = 0} solves the PDE

∆Ψ = V · ∇Ψ. (1)

” Ψ = (−∆)^−1/2ψ ”, ∇Ψ(ω, x) = (−∆)^−1/2∇ψ(τ_xω)

Note similarity: No sublinearly growing harmonic function on Z^d.

H₋₁ ^assumed:

The height function/stream field is stationary, L². Hence

n→∞lim n⁻¹E X_n² < ∞.

Let Ψ be solution of (1). Then n 7→ R_n := Ψ(X_n) is a martingale with stationary and ergodic increments.

ρ² := E(R_n+1 − R_n)² ,

(1 − 2dε)kψ k² ≤ ρ² ≤ (1 + 2dε)kψ k² and

n^−1/2R_n ⇒ N(0, ρ²)

IF

lim

|x|→∞|x|⁻¹ |Ψ(x) | = 0 a.s. (2) then ρ = 0,

X

Na¨ıve guess: (2) holds for 0-mean fields with stationary incre- ments. (d = 1: ergodic theorem. Generally no true in d ≥ 2.)

Proof of (2) in d = 2:

Maximum principle: max

x∈Λ |Ψ(x) | = max

x∈∂Λ|Ψ(x) | By ergodic thm . . . : N⁻¹ max

x∈∂[−N,N]²

|Ψ(x) | −→^P 0,

Thus: N⁻¹ max

x∈[−N,N]²

|Ψ(x) | −→^P 0,

Altogether:

P |R_n | > ε√ n

= P |R_n | > ε√

n, |X_n | > K√

n + P |R_n | > ε√

n, |X_n | ≤ K√ n

≤ P |X_n | > K√

n + P ^max |Ψ(x) | > ε√

n → 0