· 16 random setting in the family of all geometric equations, F(p,x) = |p|+ < V ·p >, satisfying...

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homogenization of the G-equation in random environments

Panagiotis E. SouganidisUniversity of Chicago

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• basic mathematical model in combustion

thin front moving with normal velocity vn = 1 + V (x/ε)

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• G-equation is the level set initial value problem

uεt = |Duε|+ < V (x/ε), Duε > in R

d × (0, ∞) uε(·, 0) = u0

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d × (0, ∞) uε(·, 0) = u0

• behavior as ε → 0

“averaged” thin front moves with normal velocity vn = H (n)

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d × (0, ∞) uε(·, 0) = u0

• behavior as ε → 0

“averaged” thin front moves with normal velocity vn = H (n)

• uε → u

ut = H (u) in Rd × (0, ∞) u(·, 0) = u0

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• another model turbulent reaction-diffusion equation

uεt = ε∆uε+ < V (x/εα), Duε > +(1/ε)f (x/εα, uε) in R

d × (0, ∞)

f KPP-like with equilibria 0 and 1

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d × (0, ∞)


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d × (0, ∞)


• as ε → 0

uε → 0 in Z < 0 and uε → 1 in IntZ = 0

max(Zt − H (DZ), Z) = 0 in Rd × (0, ∞)

H comes up in the homogenization of

uεt = ε∆uε+ < V (x/εα), Duε > +fu(x/εα, 0)

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examples of “self averaging” environments

• periodic, quasiperiodic, almost periodic

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examples of “self averaging” environments

• periodic, quasiperiodic, almost periodic

• random

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some more random examples

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cluster (Poisson) processes – random clouds of points

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cluster (Poisson) processes – random clouds of points

FUNDAMENTAL DIFFERENCE

periodic/almost periodic compactrandom not compact

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random setting

in the family of all geometric equations, F(p, x) = |p|+ < V · p >, satisfyingsome common uniform properties, like common Lip bounds, ... concentrateon a subfamily F(p, x, ω) appearing with some “frequency” described bythe fact that ω belongs to a probability space (Ω,F,P) such that

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random setting


• if F(p, ·, ω) is in the family and we change position to y,then F(p, · + y, ω) is also in the family, i.e.,F(p, · + y, ω) = F(p, ·, τyω), and the frequency of appearance isindependent of the change of the location, i.e.,τy : Ω → Ω is measure preserving stationarity

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random setting


• if F(p, ·, ω) is in the family and we change position to y,then F(p, · + y, ω) is also in the family, i.e.,F(p, · + y, ω) = F(p, ·, τyω), and the frequency of appearance isindependent of the change of the location, i.e.,τy : Ω → Ω is measure preserving stationarity

• under translations, the operators “repeat themselves”, i.e.,

if P(A) > 0, then P(∪y∈Rn τyA) = 1 ergodicity

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a heuristic picture

• different equations have the same “frequency” no matter where they arelocated

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a heuristic picture


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a heuristic picture


• but they mix (are ergodic)

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a heuristic picture


• but they mix (are ergodic)

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the G-equation


d × (0, ∞)

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the G-equation


d × (0, ∞)

• ‖V ‖ < 1 =⇒ problem is coercive

homogenization follows from results of

Lions, Papanicolaou, Varadhan (periodic), Ishii (almost periodic),Souganidis (stationary ergodic)

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the G-equation


d × (0, ∞)

• ‖V ‖ < 1 =⇒ problem is coercive

homogenization follows from results of

Lions, Papanicolaou, Varadhan (periodic), Ishii (almost periodic),Souganidis (stationary ergodic)

• ‖V ‖ ≥ 1 =⇒ problem is not coercive

homogenization may not be possible

the uε’s may not converge locally uniformly

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• an example


d × (0, ∞) and uε(·, 0) = u0

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• an example


d × (0, ∞) and uε(·, 0) = u0

V = V (x) is 1-periodic and in Q1 = (−1/2, 1/2)d

V (x) = −10x if |x| < 1/6 and V (x) = 0 if |x| ≥ 1/3

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• an example


d × (0, ∞) and uε(·, 0) = u0


V (x) = −10x if |x| < 1/6 and V (x) = 0 if |x| ≥ 1/3

variational (control) representation formula ⇒ uǫ(x, t) = sup u0(X(x, t))

sup over all Lipschitz paths X with Xx(0) = x and

Xx(s) = V (Xx(s)) = α for controls |α| ≤ 1

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• an example


d × (0, ∞) and uε(·, 0) = u0


V (x) = −10x if |x| < 1/6 and V (x) = 0 if |x| ≥ 1/3




if u0(x) =< p, x > with |p| > 0, then, for all t > 0,

limε→0 |uε(0, t)| = 0

while

lim infε→0 uε(xε, t) > 0 for any |xε| = ε/2

the vector field V traps the trajectories starting at the lattice points

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• an example


d × (0, ∞) and uε(·, 0) = u0


V (x) = −10x if |x| < 1/6 and V (x) = 0 if |x| ≥ 1/3




if u0(x) =< p, x > with |p| > 0, then, for all t > 0,

limε→0 |uε(0, t)| = 0

while

lim infε→0 uε(xε, t) > 0 for any |xε| = ε/2

the vector field V traps the trajectories starting at the lattice points

to homogenize need an assumption on V (other than coercivity) thatallows the controls to overcome the traps

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the G-equation in the periodic setting


d × (0, ∞) and uε(·, 0) = u0

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d × (0, ∞) and uε(·, 0) = u0

V bounded, Lip. continuous, periodic and with “small divergence ”, i.e.

cI ‖ div V ‖Ld (Q) ≤ 1 (cI is the isoperimetric constant in the cube Q)

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d × (0, ∞) and uε(·, 0) = u0



Theorem (convergence (Nolen, Cardaliaguet and Souganidis))

there exists a positively homogeneous (of degree one), Lipschitz continuous,convex, nonnegative H such that if ut = H (Du) = 0 in R

d × (0, ∞) andu(·, 0) = u0, then, as ε → 0, uε → u locally uniformly in R

d × (0, ∞)

special case by Xin and Yu

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d × (0, ∞) and uε(·, 0) = u0



Theorem (convergence (Nolen, Cardaliaguet and Souganidis))

there exists a positively homogeneous (of degree one), Lipschitz continuous,convex, nonnegative H such that if ut = H (Du) = 0 in R

d × (0, ∞) andu(·, 0) = u0, then, as ε → 0, uε → u locally uniformly in R

d × (0, ∞)

special case by Xin and Yu

Theorem (enhancement (Nolen, Cardaliaguet and Souganidis))

(i) H (p) ≥ |p|(1 − cI ‖divV ‖Ld(Q))+ < E(V + x div V ), p >

(ii) if EV = 0 and div V = 0, then H (p) > |p| if and only if< V (x), p >= 0 for all x

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the G-equation in random environments


d × (0, ∞) and uε(·, 0) = u0

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d × (0, ∞) and uε(·, 0) = u0

V bounded, Lip. cont., stationary ergodic and EV = 0 and div V = 0

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d × (0, ∞) and uε(·, 0) = u0

V bounded, Lip. cont., stationary ergodic and EV = 0 and div V = 0

Theorem (convergence (Cardaliaguet and Souganidis))

there exists a positively homogeneous (of degree one), Lip. continuous,convex, nonnegative H such that H (p) ≥ |p| and, if ut = H (Du) = 0 inR

d × (0, ∞) and u(·, 0) = u0, then, as ε → 0, uε → u locally uniformly inR

d × (0, ∞)

Nolen and Novikov studied d = 2 under the additional assumption that V is the

gradient of a bounded potential

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the role of coersivity and a brief review

the formal expansion uε(x, t) = u(x, t) + εw(x/ε) leads to the cell problem

• for each p ∈ Rd there exists a unique constant H (p) such that

|Dw + p|+ < V (y), Dw + p >= H (p) in Rd

has a solution (corrector) w satisfying w(y)/|y| → 0 as |y| → ∞

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|Dw + p|+ < V (y), Dw + p >= H (p) in Rd


• existence of correctors for coercive problemsapproximate δwδ = |Dwδ + p|+ < V (y), Dw + p >

estimate δ‖wδ‖ + ‖Dwδ‖ ≦ C

normalize wδ(y) = wδ(y) − wδ(0), (‖Dwδ‖ + |wδ(y)|/|y|) ≦ C

δwδ + δwδ(0) = |Dwδ + p|+ < V (y), Dwδ + p >

δ → 0 ⇒ wδ → w, lim δwδ(0) = c = |Dw + p|+ < V , Dw + p >

if w is strictly sublinear c = H (p) and w is a corrector

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|Dw + p|+ < V (y), Dw + p >= H (p) in Rd








in the periodic setting w periodic and, hence, bounded

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|Dw + p|+ < V (y), Dw + p >= H (p) in Rd








in the periodic setting w periodic and, hence, bounded

• strict sublinearity is a problem in stationary environments(counterexamples)

• no gradient bounds in noncoercive settings

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remarks about the cell problem

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approximate cell problem

δwδ = H (Dwδ + p, y) = |Dwδ + p|+ < V , Dwδ + p >

enough to prove δwδ → H uniformly in balls of radious 1/δ

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• it is the best one can hope

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wδ(x) = δwδ(x/δ) solves wδ = H (Dwδ + p, x/δ) in Rd

if the problem homogenizes, then the wδ’s converge locally uniformly to w

solving w = H (Dw + p) in Rd and, hence, w = H (p)

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• δwδ’s in balls of radius 1/δ ⇒ H is unique

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in periodic environments above convergence is equivalent to uniform andfollows if appropriate estimates are available

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in periodic environments above convergence is equivalent to uniform andfollows if appropriate estimates are available

in random media is not known how to obtain it without proving firsthomogenization – the problem is not luck of estimates but the luck ofcompactness

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noncoercive periodic G-equation

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δwδ = |Dwδ + p|+ < V (y), Dwδ + p >

the Lipschitz bounded is replaced by

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δwδ = |Dwδ + p|+ < V (y), Dwδ + p >


• osc(δwδ) ≤ Cδ for some C = C (d, |p|)

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sketch of the proof of

|δwδ < θ ∩ Q| < 1/2 ⇒ δwδ ≥ θ − Cδ

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|δwδ < θ ∩ Q| < 1/2 ⇒ δwδ ≥ θ − Cδ

zδ = δwδ solves −δC ≤ |Dzδ |+ < V , Dzδ >≤ Cδ

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|δwδ < θ ∩ Q| < 1/2 ⇒ δwδ ≥ θ − Cδ


ρδ(θ) = |zδ < θ ∩ Q| ≥ (δC )−1´


δC ρδ ≥´


d−1(zδ < θ ∩ Q)−

´



δ ⇒

ρδ ≥ δ−1γρd−1/d

δ ⇒ ρδ(θ1) ≤ (ρδ(θ2) − cδ−1)d+ (θ2 > θ1)

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|δwδ < θ ∩ Q| < 1/2 ⇒ δwδ ≥ θ − Cδ


ρδ(θ) = |zδ < θ ∩ Q| ≥ (δC )−1´


δC ρδ ≥´


d−1(zδ < θ ∩ Q)−

´



δ ⇒

ρδ ≥ δ−1γρd−1/d

δ ⇒ ρδ(θ1) ≤ (ρδ(θ2) − cδ−1)d+ (θ2 > θ1)

• main difficulty in extending this argument to the stationary ergodicenvironment is the luck of an isoperimetric inequality

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homogenization of convex superlinear random HJ-equations

uεt = H (Duε, x/ε, ω) in R

d × (0, ∞) uε(·, 0) = u0

H stationary ergodic, convex, superlinear (coercive)

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d × (0, ∞) uε(·, 0) = u0


Lax-Oleinik formula ⇒

uε(x, t, ω) = supy∈Rd u0(y) − Lε(x, y, t, 0, ω)

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d × (0, ∞) uε(·, 0) = u0




Lε(x, y, t, s, ω) = εL(x/ε, y/ε, t/ε, s/ε, ω)

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d × (0, ∞) uε(·, 0) = u0





(i) stationary L((t + h)p, (s + h)p, t + h, s + h, ω) = L(tp, sp, t, s, τhpω)

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d × (0, ∞) uε(·, 0) = u0






(ii) sub-additive

L(tp, sp, t, s, ω) ≤ L(tp, ρp, t, ρ, ω) + L(ρp, sp, ρ, s, ω) (s < ρ < t)

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d × (0, ∞) uε(·, 0) = u0






(ii) sub-additive


(iii) bounded |L(tp, sp, t, s, ω)| ≤ C (t − s) (C = C (p))

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d × (0, ∞) uε(·, 0) = u0






(ii) sub-additive


(iii) bounded |L(tp, sp, t, s, ω)| ≤ C (t − s) (C = C (p))

(iv) Lipschitz continuous in balls of radius t

|L(t(p + q), tp, t, 0, ω) − L(tp, t, 0, ω)| ≤ Ct|q| for p bounded

(iii) and (iv) are due to the superlinearity/coercivity while (ii) follows form convexity

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d × (0, ∞) uε(·, 0) = u0


Lε stationary, sub-additive, bounded, Lip. continuous

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d × (0, ∞) uε(·, 0) = u0



• subadditive ergodic theorem ⇒

Lε(x, y, t, s, ω) → tL(t−1(x − y)) a.s. in ω and uniformly for x − y bounded

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d × (0, ∞) uε(·, 0) = u0





it follows that, as ε → 0 a.s. in ω and locally uniformly in x, t,

uε(x, t, ω) → u(x, t) = supy∈Rd u0(y) = supy∈Rd u0(y) − tL(t−1(x − y))

ut = H (Du) in Rd × (0, ∞) and H (p) = supq∈Rd < p, q > −L(q)

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d × (0, ∞) uε(·, 0) = u0








main difficulty for the G-equation is the lack of coercivity

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d × (0, ∞) uε(·, 0) = u0








main difficulty for the G-equation is the lack of coercivity

the sub-additive ergodic theorem

µ : B × Ω → R stationary, subadditive, integrable (µ((0, 1), ·) ∈ L1(Ω))

⇒ there exists µ : Ω → R such that, as t → ∞ and a.s. in ω,

t−1µ((0, t), ω) → µ(ω)

if setting is ergodic, then µ is independent of ω

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noncoercive G-equation in stationary random media

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d × (0, ∞) uε(·, 0) = u0

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d × (0, ∞) uε(·, 0) = u0


uε(x, t, ω) = supα∈A u0(Xα,ε,ωx (t)) = supy∈Rε(x,t,ω) u0(y)

= supy∈Rd u0(y) − Lε(x, y, t, ω)

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d × (0, ∞) uε(·, 0) = u0




A = α : L∞((0, ∞);Rd) with ‖α‖ ≤ 1

Xα,εx (s) = V (Xα,ε

x (s)/ε, ω) + α(s) Xα,εx (0) = x

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d × (0, ∞) uε(·, 0) = u0




A = α : L∞((0, ∞);Rd) with ‖α‖ ≤ 1


x (s)/ε, ω) + α(s) Xα,εx (0) = x

Rε(x, t, ω) the reachable set of x at time t

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d × (0, ∞) uε(·, 0) = u0




A = α : L∞((0, ∞);Rd) with ‖α‖ ≤ 1


x (s)/ε, ω) + α(s) Xα,εx (0) = x


Lε(x, y, t, ω) = 0 if y ∈ Rε(x, t, ω) and − ∞ if not

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d × (0, ∞) uε(·, 0) = u0




A = α : L∞((0, ∞);Rd) with ‖α‖ ≤ 1


x (s)/ε, ω) + α(s) Xα,εx (0) = x



the behavior of Lε(x, y, t, ω) = εL(x/ε, y/ε, t/ε, ω) is controlled by the

behavior of Rε(x, t, ω) = εR(x/ε, t/ε, ω)

L and R are as above but for ε = 1

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d × (0, ∞) uε(·, 0) = u0




A = α : L∞((0, ∞);Rd) with ‖α‖ ≤ 1


x (s)/ε, ω) + α(s) Xα,εx (0) = x



the behavior of Lε(x, y, t, ω) = εL(x/ε, y/ε, t/ε, ω) is controlled by the

behavior of Rε(x, t, ω) = εR(x/ε, t/ε, ω)

L and R are as above but for ε = 1

the long time behavior of R is determined by the minimal time function

θ(x, y, ω) = mint ≥ 0 : y ∈ R(x, t, ω)

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Xαx (s) = V (Xα

x (s), ω) + α(s) Xα,x (0) = x (α ∈ A)

R(x, t, ω) is the reachable set of x at time t

θ(x, y, ω) minimal time to reach y starting at x

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Xαx (s) = V (Xα

x (s), ω) + α(s) Xα,x (0) = x (α ∈ A)



Theorem

there exists a positively homogeneous, convex L : Rd → R such that|q| ≤ L(q) ≤ (1 + ‖V ‖) and, a.s. in ω and for all R > 0,

limt→∞ sup|p|≤R,|x|≤R |t−1θ(tx, t(x + p), ω) − L(p)| = 0

and

limt→∞ t−1R(x, t, ω) = y ∈ R

d : L(x − y) ≤ 1

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Xαx (s) = V (Xα

x (s), ω) + α(s) Xα,x (0) = x (α ∈ A)



Theorem

there exists a positively homogeneous, convex L : Rd → R such that|q| ≤ L(q) ≤ (1 + ‖V ‖) and, a.s. in ω and for all R > 0,

limt→∞ sup|p|≤R,|x|≤R |t−1θ(tx, t(x + p), ω) − L(p)| = 0

and

limt→∞ t−1R(x, t, ω) = y ∈ R

d : L(x − y) ≤ 1

averaged Hamiltonian H (p) = sup< p, q >: L(q) = 1

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the asymptotic behavior of the minimal time

natural setting for the subadditive ergodic theorem

• µ((a, b), ω) = θ(aq, bq, ω) (a, b) ⊂ R, p ∈ Rd

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• subadditivity

µ((a, c), ω) = θ(aq, cq, ω) ≤ θ(aq, bq, ω) + θ(bq, cq, ω) =µ((a, b), ω) + µ((b, c), ω)

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• subadditivity


• stationarity

µ(c + (a, b), ω) = θ((a + c)q, (b + c)q, ω) = θ((aq, bq, τcqω) =µ((a, b), τcqω)

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• subadditivity


• stationarity

µ(c + (a, b), ω) = θ((a + c)q, (b + c)q, ω) = θ((aq, bq, τcqω) =µ((a, b), τcqω)

• L1– bound there is a nontrivial difficulty here

for all ε ∈ (0, ε0) and a.s. in ω, there exists T (ω, ε) > 0 such that

θ(x, y, ω) ≤ T (ω, ε) + ε|x| + (1 + ε)|y − x| for all x, y ∈ Rd

not known whether T (·, ε) ∈ L1(Ω)

cannot use subadditive ergodic thm

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new strategy

for each q ∈ Rd with |q| < 1 find α ∈ A so that the soln X0,0,α,ω of

X0,0,α,ω(s) = V (X0,0,α,ω(s), ω) + α(s) with X0,0,α,ω(0) = 0

(i) has, a.s. in ω and as t → ∞, an average t−1X0,0,α,ω(t) close to q

(ii) the subadditive ergodic thm can be applied to the minimal timeθ(X0,0,α,ω(s), X0,0,α,ω(t)) yielding a random variable γ such that, ast → ∞ and a.s. in ω, t−1θ(0, X0,0,α,ω(t), ω) → γ(ω)

87

new strategy





fix q, α; write Xs for X0,0,α,ω(s); recall |Xs| ≤ C (1 + | V ‖)t

subadditivity, reachability estimate ⇒

θ(0, tq, ω) ≤ θ(0, Xt , ω) + θ(Xt, tq, ω) ≤

θ(0, Xt, ω) + T (ω, ε) + ε|Xt| + (1 + ε)|Xt − tq|

88

new strategy







θ(0, tq, ω) ≤ θ(0, Xt , ω) + θ(Xt, tq, ω) ≤


properties of Xt ⇒ lim supt→∞ t−1θ(0, tq, ω) ≤ γ(ω) + εC

89

new strategy







θ(0, tq, ω) ≤ θ(0, Xt , ω) + θ(Xt, tq, ω) ≤



ε → 0 ⇒ limt→∞ t−1θ(0, tq, ω) = γ(ω) a.s. in ω

90

new strategy







θ(0, tq, ω) ≤ θ(0, Xt , ω) + θ(Xt, tq, ω) ≤




θ stationary ⇒ limt→∞ t−1θ(0, tq, ω) is independent of ω a.s.

91

new strategy







θ(0, tq, ω) ≤ θ(0, Xt , ω) + θ(Xt, tq, ω) ≤




θ stationary ⇒ limt→∞ t−1θ(0, tq, ω) is independent of ω a.s.

how can we find such a control?

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the special control

for each q ∈ Rd with |q| ≤ 1 there exists α ∈ A so that the solution

Xt = X0,0,α,ω(t) of the ode

X0,0,α,ω(s) = V (X0,0,α,ω(s), ω) + α(s) X0,0,α,ω(0) = 0

has, as t → ∞ and a.s. in ω, the properties

(i) t−1Xt − q is small

(ii) t−1θ(0, Xt, ω) has a limit γ(ω)

93

the special control







• key idea

construct a random control as an independent Markov process and show,using the Kakutani and the subadditive ergodic thms, that (i) and (ii)hold in the augmented probability space

94

the special control







• key idea

construct a random control as an independent Markov process and show,using the Kakutani and the subadditive ergodic thms, that (i) and (ii)hold in the augmented probability space

when d = 2, there is a simpler construction

95

the path

X0,0,a,ωs = V (X0,0,a,ω

s , ω) + a X0,0,a,ω0 = 0 a constant control

div(V + a) = 0 ⇒ t → T at ω = τ

X0,0,a,ω

t

ω is measure preserving

96

the path

X0,0,a,ωs = V (X0,0,a,ω


div(V + a) = 0 ⇒ t → T at ω = τ

X0,0,a,ω

t


Fa σ-algebra of T a

t -invariant sets

97

the path

X0,0,a,ωs = V (X0,0,a,ω


div(V + a) = 0 ⇒ t → T at ω = τ

X0,0,a,ω

t



t -invariant sets

ergodic thm ⇒ t−1´ t

0(V (X0,0,a,ω

s , ω) + a)ds → Z a = E[V + a|Fa] a.s

Θa(s, t, ω) = θ(X0,0,a,ωs , X0,0,a,ω

t , ω) is subadditive, stationary and

integrable, since 0 ≤ Θa(s, t, ω) ≤ (t − s)

98

the path

X0,0,a,ωs = V (X0,0,a,ω


div(V + a) = 0 ⇒ t → T at ω = τ

X0,0,a,ω

t



t -invariant sets


0(V (X0,0,a,ω

s , ω) + a)ds → Z a = E[V + a|Fa] a.s

Θa(s, t, ω) = θ(X0,0,a,ωs , X0,0,a,ω



subadditive erg thm ⇒ t−1Θa(0, t, ω) has a random limit a.s.

99

the path

X0,0,a,ωs = V (X0,0,a,ω


div(V + a) = 0 ⇒ t → T at ω = τ

X0,0,a,ω

t



t -invariant sets


0(V (X0,0,a,ω

s , ω) + a)ds → Z a = E[V + a|Fa] a.s

Θa(s, t, ω) = θ(X0,0,a,ωs , X0,0,a,ω




W a = w ∈ Rd : P|Z a − w| < ε > 0 for ε > 0 essential support of Z a

w ∈ W a =⇒ limt→∞ t−1θ(0, tw, ω) exists a.s.

100

the path

X0,0,a,ωs = V (X0,0,a,ω


div(V + a) = 0 ⇒ t → T at ω = τ

X0,0,a,ω

t



t -invariant sets


0(V (X0,0,a,ω

s , ω) + a)ds → Z a = E[V + a|Fa] a.s

Θa(s, t, ω) = θ(X0,0,a,ωs , X0,0,a,ω




W a = w ∈ Rd : P|Z a − w| < ε > 0 for ε > 0 essential support of Z a

w ∈ W a =⇒ limt→∞ t−1θ(0, tw, ω) exists a.s.

d = 2 =⇒ W a ⊂ Ra

in general not true in higher dimensions

101

an additional random setting

fix a ∈ Rd with |a| < 1 and δ > 0

(Ω, F, P) probability space of sequences Z = ((tn , kn))n∈Z of iid randomvalues on ∆ = (0, 2δ) × 1, ..., 2d with law Q such that

E(t0) = δ E|t − δ| ≤ δ2

ak ∈ Rd with |ak | ≤ 1 and E(ak0 ) = a

102




E(t0) = δ E|t − δ| ≤ δ2


(T z)z∈∆ with T zω = τX

0,0,ak ,ω

t

ω is measure preserving and ergodic on Ω

103




E(t0) = δ E|t − δ| ≤ δ2



0,0,ak ,ω

t


right shift τ : Ω → Ω is measure preserving and (Zn)n∈Z stationary wrt τ

104




E(t0) = δ E|t − δ| ≤ δ2



0,0,ak ,ω

t


right shift τ : Ω → Ω is measure preserving and (Zn)n∈Z stationary wrt τ

Kakutani ergodic thm:

if g ∈ L1(Ω), then, a.s. and in L1 wrt P ⊗ P,

limn→+∞ n−1∑n−1

i=0g(T Zi(ω) · · · T Z0(ω)ω) = E(g)

105

the path in all dimensions

the random control α = α(ω) is defined by

α(ω)(t) = akn (ω) for t ∈ [σn(ω), σn+1(ω)] where σn(ω) =∑n

i=0ti(ω)

106




i=0ti(ω)

Kakutani erg thm, E(V ) = 0, T Zi(ω) · · · T Z0(ω)ω = τX

0,0,α(ω),ω

σi(ω)

ω =⇒


i=0V (X

0,0,α(ω),ω

σi (ω), ω) = E(V ) = 0

107




i=0ti(ω)


0,0,α(ω),ω

σi(ω)

ω =⇒


i=0V (X

0,0,α(ω),ω

σi (ω), ω) = E(V ) = 0

=⇒

lim supt→+∞ |t−1X0,0,α(ω),ωt − a| ≤ Cδ a.s.

108




i=0ti(ω)


0,0,α(ω),ω

σi(ω)

ω =⇒


i=0V (X

0,0,α(ω),ω

σi (ω), ω) = E(V ) = 0

=⇒

lim supt→+∞ |t−1X0,0,α(ω),ωt − a| ≤ Cδ a.s.

=⇒

limn→+∞ n−1θ(0, X0,0,α(ω),ω

σn (ω)) → Γ(ω, ω) a.s.

109

the proof of lim supt→+∞|t−1

X0,0,α(ω),ω

t − a| ≤ Cδ a.s.

for t > 0 there is some n st σn ≤ t ≤ σn+1 and σn+1 − σn ≤ 2δ and Vbounded ⇒ enough

lim supn |σ−1n Xα

σn− a| ≤ Cδ a.s.

110


X0,0,α(ω),ω

t − a| ≤ Cδ a.s.




V Lipschitz continuous and ti ≤ δ =⇒

σ−1n Xα

σn= σ−1

n

∑n−1

i=0(Xα

σi+1− Xα

σi)

= σ−1n

∑n−1

i=0ti(V (Xα

σi) + aki ) + O(σ−1

n nδ2)

111


X0,0,α(ω),ω

t − a| ≤ Cδ a.s.





σ−1n Xα

σn= σ−1

n

∑n−1

i=0(Xα

σi+1− Xα

σi)

= σ−1n

∑n−1

i=0ti(V (Xα

σi) + aki ) + O(σ−1

n nδ2)

σ−1n Xα

σn=

δσ−1n

∑n−1

i=0V (Xα

σi)+σ−1

n

∑n−1

i=0(ti −δ)V (Xα

σi)+σ−1

n

∑n−1

i=0ti aki +O(σ−1

n nδ2)

112


X0,0,α(ω),ω

t − a| ≤ Cδ a.s.





σ−1n Xα

σn= σ−1

n

∑n−1

i=0(Xα

σi+1− Xα

σi)

= σ−1n

∑n−1

i=0ti(V (Xα

σi) + aki ) + O(σ−1

n nδ2)

σ−1n Xα

σn=

δσ−1n

∑n−1

i=0V (Xα

σi)+σ−1

n

∑n−1

i=0(ti −δ)V (Xα

σi)+σ−1

n

∑n−1

i=0ti aki +O(σ−1

n nδ2)

the law of large number =⇒

n−1σn → E(σ0) = δ; n−1∑n−1

i=0ti aki → δa;

n−1∑n−1

i=0|ti − δ| → E(|t0 − δ|) ≤ δ2

· 16 random setting in the family of all geometric equations, F(p,x) = |p|+ < V ·p >, satisfying...

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Transcript of · 16 random setting in the family of all geometric equations, F(p,x) = |p|+ < V ·p >, satisfying...