Branching Brownian motion: extremal process and
ergodic theorems
Anton Bovier
with Louis-Pierre Arguin and Nicola Kistler
RCS&SM, Venezia, 06.05.2013
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Plan
1
BBM
2
Maximum of BBM
3
The Lalley-Sellke conjecture
4
The extremal process of BBM
5
Ergodic theorems
6
Universality
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
Branching Brownian Motion
Branching Brownian motion is one of the fundamental models in
probability. It combines two classical objects:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
Branching Brownian Motion
Branching Brownian motion is one of the fundamental models in
probability. It combines two classical objects:
Brownian motion
Pure random motion
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
Branching Brownian Motion
Branching Brownian motion is one of the fundamental models in
probability. It combines two classical objects:
Brownian motion
Galton-Watson process
Pure random motion
Pure random genealogy
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
Branching Brownian motion
Branching Brownian motion (BBM) combines the two processes: Each
particle of the Galton-Watson process performs Brownian motion
independently of any other. This produces an immersion of the
Galton-Watson process in space.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
Branching Brownian motion
Branching Brownian motion (BBM) combines the two processes: Each
particle of the Galton-Watson process performs Brownian motion
independently of any other. This produces an immersion of the
Galton-Watson process in space.
Picture by Matt Roberts, Bath
Maury Bramson
H. McKean
A.V. Skorokhod
J.E. Moyal
BBM is the canonical model of a spatial branching process.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
Galton-Watson tree and corresponding BBM
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
Galton-Watson tree and corresponding BBM
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
BBM as Gaussian process
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
BBM as Gaussian process
Fix GW-tree. Label individuals at time t as
i1 (t), . . . , in(t) (t)
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
BBM as Gaussian process
Fix GW-tree. Label individuals at time t as
i1 (t), . . . , in(t) (t)
d(i` (t), ik (t)) ≡ time of most recent
common ancestor of i` (t) and ik (t)
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
BBM as Gaussian process
Fix GW-tree. Label individuals at time t as
i1 (t), . . . , in(t) (t)
d(i` (t), ik (t)) ≡ time of most recent
common ancestor of i` (t) and ik (t)
BBM is Gaussian process with covariance
Exk (t)x` (s) = d(ik (t), i` (s))
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
BBM as Gaussian process
Fix GW-tree. Label individuals at time t as
i1 (t), . . . , in(t) (t)
d(i` (t), ik (t)) ≡ time of most recent
common ancestor of i` (t) and ik (t)
BBM is Gaussian process with covariance
Exk (t)x` (s) = d(ik (t), i` (s))
BBM special case of models where
Exk (t)x` (t) = tA t −1 d(ik (t), i` (t))
A. Bovier ()
for A : [0, 1] → [0, 1].
Branching Brownian motion: extremal process and ergodic theorems
R
Branching Brownian motion
BBM as Gaussian process
Fix GW-tree. Label individuals at time t as
i1 (t), . . . , in(t) (t)
d(i` (t), ik (t)) ≡ time of most recent
common ancestor of i` (t) and ik (t)
BBM is Gaussian process with covariance
Exk (t)x` (s) = d(ik (t), i` (s))
BBM special case of models where
Exk (t)x` (t) = tA t −1 d(ik (t), i` (t))
for A : [0, 1] → [0, 1].
⇒ GREM models of spin-glasses.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
First question: how big is the biggest?
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
First question: how big is the biggest?
To compare:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
First question: how big is the biggest?
To compare:
Single Brownian motion:
2
Z x
h
√ i
1
z
dz
P X (t) ≤ x t = √
exp −
2
2π −∞
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
First question: how big is the biggest?
To compare:
Single Brownian motion:
2
Z x
h
√ i
1
z
dz
P X (t) ≤ x t = √
exp −
2
2π −∞
e t = En(t) independent Brownian motions:
√
√
√
1
− 2x
P max t xk (t) ≤ t 2 − √ ln t + x → e − 4πe
k=1,...,e
2 2
BBM
A. Bovier ()
Many BMs
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
The KPP-F equation
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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Maximum of BBM
The KPP-F equation
One of the simplest reaction-diffusion equations is the
Fisher-Kolmogorov-PetrovskyPiscounov (F-KPP) equation:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
The KPP-F equation
One of the simplest reaction-diffusion equations is the
Fisher-Kolmogorov-PetrovskyPiscounov (F-KPP) equation:
1
∂t v (x, t) = ∂x2 v (x, t)+v −v 2
2
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
The KPP-F equation
One of the simplest reaction-diffusion equations is the
Fisher-Kolmogorov-PetrovskyPiscounov (F-KPP) equation:
1
∂t v (x, t) = ∂x2 v (x, t)+v −v 2
2
A. Bovier ()
Fisher
Kolmogorov
Petrovsky
Branching Brownian motion: extremal process and ergodic theorems
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Maximum of BBM
The KPP-F equation
One of the simplest reaction-diffusion equations is the
Fisher-Kolmogorov-PetrovskyPiscounov (F-KPP) equation:
1
∂t v (x, t) = ∂x2 v (x, t)+v −v 2
2
Fisher
Kolmogorov
Petrovsky
Fischer used this equation to model the evolution of biological
populations. It accounts for:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
The KPP-F equation
One of the simplest reaction-diffusion equations is the
Fisher-Kolmogorov-PetrovskyPiscounov (F-KPP) equation:
1
∂t v (x, t) = ∂x2 v (x, t)+v −v 2
2
Fisher
Kolmogorov
Petrovsky
Fischer used this equation to model the evolution of biological
populations. It accounts for:
birth: v ,
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
The KPP-F equation
One of the simplest reaction-diffusion equations is the
Fisher-Kolmogorov-PetrovskyPiscounov (F-KPP) equation:
1
∂t v (x, t) = ∂x2 v (x, t)+v −v 2
2
Fisher
Kolmogorov
Petrovsky
Fischer used this equation to model the evolution of biological
populations. It accounts for:
birth: v ,
death: −v 2 ,
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
The KPP-F equation
One of the simplest reaction-diffusion equations is the
Fisher-Kolmogorov-PetrovskyPiscounov (F-KPP) equation:
1
∂t v (x, t) = ∂x2 v (x, t)+v −v 2
2
Fisher
Kolmogorov
Petrovsky
Fischer used this equation to model the evolution of biological
populations. It accounts for:
birth: v ,
death: −v 2 ,
diffusive migration: ∂x2 v .
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
KPP-F equation and the maximum of of BBM
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
KPP-F equation and the maximum of of BBM
u(t, x) ≡ P
A. Bovier ()
max
k=1...n(t)
xk (t) ≤ x
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
KPP-F equation and the maximum of of BBM
u(t, x) ≡ P
max
k=1...n(t)
xk (t) ≤ x
1 − u solves the F-KPP equation, i.e.
(
1
if x ≥ 0
1 2
2
∂t u = ∂x u + u − u, u(0, x) =
2
0
if x < 0
McKean, 1975:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
KPP-F equation and the maximum of of BBM
u(t, x) ≡ P
max
k=1...n(t)
xk (t) ≤ x
1 − u solves the F-KPP equation, i.e.
(
1
if x ≥ 0
1 2
2
∂t u = ∂x u + u − u, u(0, x) =
2
0
if x < 0
McKean, 1975:
Bramson, 1978:
u(t, x+m(t)) → ω(x),
A. Bovier ()
m(t) =
√
3
2t− √ ln t
2 2
Branching Brownian motion: extremal process and ergodic theorems
R
Maximum of BBM
KPP-F equation and the maximum of of BBM
u(t, x) ≡ P
max
k=1...n(t)
xk (t) ≤ x
1 − u solves the F-KPP equation, i.e.
(
1
if x ≥ 0
1 2
2
∂t u = ∂x u + u − u, u(0, x) =
2
0
if x < 0
McKean, 1975:
Bramson, 1978:
u(t, x+m(t)) → ω(x),
.
m(t) =
√
3
2t− √ ln t
2 2
where ω(x) solves
√
1 2
2
2 ∂x ω + 2∂x ω + ω − ω = 0
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The Lalley-Sellke conjecture
The derivative martingale
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The Lalley-Sellke conjecture
The derivative martingale
Lalley-Sellke, 1987: ω(x) is random shift of Gumbel-distribution
√ i
h
− 2x
,
ω(x) = E e −C Ze
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The Lalley-Sellke conjecture
The derivative martingale
Lalley-Sellke, 1987: ω(x) is random shift of Gumbel-distribution
√ i
h
− 2x
,
ω(x) = E e −C Ze
(d)
Z = limt→∞ Z (t), where Z (t) is the derivative martingale,
√ √
X √
Z (t) =
{ 2t − xk (t)}e − 2{ 2t−xk (t)}
k≤n(t)
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The Lalley-Sellke conjecture
The derivative martingale
Lalley-Sellke, 1987: ω(x) is random shift of Gumbel-distribution
√ i
h
− 2x
,
ω(x) = E e −C Ze
(d)
Z = limt→∞ Z (t), where Z (t) is the derivative martingale,
√ √
X √
Z (t) =
{ 2t − xk (t)}e − 2{ 2t−xk (t)}
k≤n(t)
Lalley-Sellke conjecture: P-a.s., for any x ∈ R,
1
lim
T ↑∞ T
A. Bovier ()
Z
0
T
1Inmaxn(t) x
k=1 k (t)−m(t)≤x
o
√ = exp −CZ e − 2x
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
Looking at BBM from the top
Closer look at the extremes: Zooming into the top
Can we describe the asymptotic structure of the largest points, and their
genealogical structure?
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
Classical Poisson convergence for many BMs
From classical extreme values statistics one knows:
Let Xi (t), i ∈ N, iid Brownian motions. Then, the point process
t
Pt ≡
e
X
i=1
δXi (t)−√2t+
1
√
2 2
ln t
→ PPP
√
4πe −x dx ,
where PPP(µ) is Poisson point process with intensity measure µ.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
Extensions to correlated processes
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
Extensions to correlated processes
GREM [Derrida ’82]: Recall Exi` (t) xik (t) = tA t −1 d(i` (t), ik (t)) .
A increasing step function.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
Extensions to correlated processes
GREM [Derrida ’82]: Recall Exi` (t) xik (t) = tA t −1 d(i` (t), ik (t)) .
A increasing step function.
Extreme behaviour relatively insensitive to correlations: If
A(x) < x, ∀x ∈ (0, 1), then no change in the extremal process.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
Extensions to correlated processes
GREM [Derrida ’82]: Recall Exi` (t) xik (t) = tA t −1 d(i` (t), ik (t)) .
A increasing step function.
Extreme behaviour relatively insensitive to correlations: If
A(x) < x, ∀x ∈ (0, 1), then no change in the extremal process.
Poisson cascades: If A takes only finitely many values, and A(x) > x; for
some x ∈ (0, 1), the extremal process is known (Derrida, B-Kurkova) and
given by Poisson cascade process.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
Extensions to correlated processes
GREM [Derrida ’82]: Recall Exi` (t) xik (t) = tA t −1 d(i` (t), ik (t)) .
A increasing step function.
Extreme behaviour relatively insensitive to correlations: If
A(x) < x, ∀x ∈ (0, 1), then no change in the extremal process.
Poisson cascades: If A takes only finitely many values, and A(x) > x; for
some x ∈ (0, 1), the extremal process is known (Derrida, B-Kurkova) and
given by Poisson cascade process.
Borderline: If A takes only finitely many values, and A(x) ≤ x, for all
x ∈ [0, 1], but A(x) = x, for some x ∈ (0, 1), the extremal process is again
Poisson, but with reduced intensity (B-Kurkova).
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
Extensions to correlated processes
GREM [Derrida ’82]: Recall Exi` (t) xik (t) = tA t −1 d(i` (t), ik (t)) .
A increasing step function.
Extreme behaviour relatively insensitive to correlations: If
A(x) < x, ∀x ∈ (0, 1), then no change in the extremal process.
Poisson cascades: If A takes only finitely many values, and A(x) > x; for
some x ∈ (0, 1), the extremal process is known (Derrida, B-Kurkova) and
given by Poisson cascade process.
Borderline: If A takes only finitely many values, and A(x) ≤ x, for all
x ∈ [0, 1], but A(x) = x, for some x ∈ (0, 1), the extremal process is again
Poisson, but with reduced intensity (B-Kurkova).
What happens at the natural border A(x) = x??
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The life of BBM
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The life of BBM
General principle: follow history of the leading particles!
There are three phases with distinct properties and effects:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The life of BBM
General principle: follow history of the leading particles!
There are three phases with distinct properties and effects:
the early years
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The life of BBM
General principle: follow history of the leading particles!
There are three phases with distinct properties and effects:
the early years
midlife
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The life of BBM
General principle: follow history of the leading particles!
There are three phases with distinct properties and effects:
the early years
midlife
before the end
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The life of BBM
General principle: follow history of the leading particles!
There are three phases with distinct properties and effects:
the early years
midlife
before the end
Let us look at them.......
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The early years...
Randomness persists for all times from what happened in the early history:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The early years...
Randomness persists for all times from what happened in the early history:
Both
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The early years...
Randomness persists for all times from what happened in the early history:
and
Both
occur with positive probability, independent of t!
In the second case, all particles at time r have the same chance to have
offspring that is close to the maximum.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
The early years...
Randomness persists for all times from what happened in the early history:
and
Both
occur with positive probability, independent of t!
In the second case, all particles at time r have the same chance to have
offspring that is close to the maximum.
Two consequences:
the random variable Z , the “derivative martingale”
particles near the maximum at time t can have common ancestors at
finite, t-independent times (when t ↑ ∞).
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
...Midlife...
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
...Midlife...
Key fact: The function m(t) is convex:
The function m(t)
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
...Midlife...
Key fact: The function m(t) is convex:
Descendants of a particle maximal at time
0 s t cannot be maximal at time t!
The function m(t)
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
...Midlife...
Key fact: The function m(t) is convex:
Descendants of a particle maximal at time
0 s t cannot be maximal at time t!
Particles realising the maximum at time t
have ancestors at times s that are selected
from the very many particles that are a lot
below the maximum at time s.
A. Bovier ()
The function m(t)
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
...Midlife...
Key fact: The function m(t) is convex:
Descendants of a particle maximal at time
0 s t cannot be maximal at time t!
Particles realising the maximum at time t
have ancestors at times s that are selected
from the very many particles that are a lot
below the maximum at time s.
The function m(t)
Offspring of the selected particles is atypical!
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
...Midlife...
Key fact: The function m(t) is convex:
Descendants of a particle maximal at time
0 s t cannot be maximal at time t!
Particles realising the maximum at time t
have ancestors at times s that are selected
from the very many particles that are a lot
below the maximum at time s.
The function m(t)
Offspring of the selected particles is atypical!
Only one descendant of a selected particle at
times 0 s t can be at finite distance
from the maximum at time t.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
...just before the end
Any particle the arrives close to the maximum at time t can have
produced offspring shortly before. These will be only a finite amount
smaller then their brothers.
Hence, particles near the maximum come in small families.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
The extremal process
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The extremal process
Et ≡
Let
n(t)
X
δxi (t)−m(t)
i=1
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Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
The extremal process
Et ≡
Let
n(t)
X
δxi (t)−m(t)
i=1
Let Z be the limit of the derivative martingale, and set
√
X
PZ =
δpi ≡ PPP CZe − 2x dx
i∈N
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The extremal process
Et ≡
Let
n(t)
X
δxi (t)−m(t)
i=1
Let Z be the limit of the derivative martingale, and set
√
X
PZ =
δpi ≡ PPP CZe − 2x dx
i∈N
√ Let L(t) ≡ maxj≤n(t) xj (t) > 2t and
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
The extremal process
Et ≡
Let
n(t)
X
δxi (t)−m(t)
i=1
Let Z be the limit of the derivative martingale, and set
√
X
PZ =
δpi ≡ PPP CZe − 2x dx
i∈N
√ Let L(t) ≡ maxj≤n(t) xj (t) > 2t and
∆(t) ≡
X
δxk (t)−maxj≤n(t) xj (t)
conditioned on L(t).
k
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The extremal process
Et ≡
Let
n(t)
X
δxi (t)−m(t)
i=1
Let Z be the limit of the derivative martingale, and set
√
X
PZ =
δpi ≡ PPP CZe − 2x dx
i∈N
√ Let L(t) ≡ maxj≤n(t) xj (t) > 2t and
∆(t) ≡
X
δxk (t)−maxj≤n(t) xj (t)
conditioned on L(t).
k
Law of ∆(t) under P (·|L(t)) converges to law of point process, ∆. Let
(i)
∆(i) be iid copies of ∆, with atoms ∆j .
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
The extremal process
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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The extremal process of BBM
The extremal process
Theorem ( Arguin-B-Kistler, 2011 (PTRF 2013))
With the notation above, the point process Et converges in law to a point
process E, given by
X
E≡
δp +∆(i)
i
i,j∈N
A. Bovier ()
j
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The extremal process
Theorem ( Arguin-B-Kistler, 2011 (PTRF 2013))
With the notation above, the point process Et converges in law to a point
process E, given by
X
E≡
δp +∆(i)
i
i,j∈N
A. Bovier ()
j
Branching Brownian motion: extremal process and ergodic theorems
R
The extremal process of BBM
The extremal process
Theorem ( Arguin-B-Kistler, 2011 (PTRF 2013))
With the notation above, the point process Et converges in law to a point
process E, given by
X
E≡
δp +∆(i)
i
i,j∈N
j
Similar result obtained independently by Aı̈dékon, Brunet, Berestycki, and Shi.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
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Ergodic theorems
Ergodic theorem for the max
Alternative look: what happens if we consider time averages?
Naively one might expect a law of large numbers:
1
lim
T ↑∞ T
Z
T
0
1Inmaxn(t) x
k=1 k (t)−m(t)≤x
o
√ = E exp −CZ e − 2x
a.s.
But this cannot be true!
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Ergodic theorem for the max
Alternative look: what happens if we consider time averages?
Naively one might expect a law of large numbers:
1
lim
T ↑∞ T
Z
T
0
1Inmaxn(t) x
k=1 k (t)−m(t)≤x
o
√ = E exp −CZ e − 2x
a.s.
But this cannot be true! Lalley and Sellke conjectured a random version:
Theorem (Arguin, B, Kistler, 2012 (EJP 18))
P-a.s., for any x ∈ R,
1
lim
T ↑∞ T
A. Bovier ()
Z
0
T
1Inmaxn(t) x
k=1 k (t)−m(t)≤x
o
√ = exp −CZ e − 2x
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Ergodic theorem for the extremal process
We prove this conjecture and extend it to the entire extremal process:
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Ergodic theorem for the extremal process
We prove this conjecture and extend it to the entire extremal process:
Theorem (Arguin, B, Kistler (2012))
Et converges P-almost surely weakly under time-average to the Poisson
cluster process EZ . That is, P-a.s., ∀f ∈ Cc+ (R),
Z
Z
Z
1
exp − f (y )Et,ω (dy ) dt → E exp − f (y )EZ (dy )
T
Here EZ is the process E for given value Z of the derivative martingale, E
is w.r.t. the law of that process, given Z .
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Elements of the proof
For ε > 0 and RT T , decompose
Z
Z
1 T
exp − f (y )Et,ω (dy ) dt
T 0
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Elements of the proof
For ε > 0 and RT T , decompose
Z
Z
Z
Z
1 εT
1 T
exp − f (y )Et,ω (dy ) dt =
exp − f (y )Et,ω (dy ) dt
T 0
T
| 0
{z
}
(I ):vanishes as ↓ 0
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Elements of the proof
For ε > 0 and RT T , decompose
Z
Z
Z
Z
1 εT
1 T
exp − f (y )Et,ω (dy ) dt =
exp − f (y )Et,ω (dy ) dt
T 0
T
| 0
{z
}
(I ):vanishes as ↓ 0
Z
Z
1 T
+
E exp − f (y )Et,ω (dy ) FRT dt
T εT
{z
}
|
(II ):what we want
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Elements of the proof
For ε > 0 and RT T , decompose
Z
Z
Z
Z
1 εT
1 T
exp − f (y )Et,ω (dy ) dt =
exp − f (y )Et,ω (dy ) dt
T 0
T
| 0
{z
}
(I ):vanishes as ↓ 0
Z
Z
1 T
+
E exp − f (y )Et,ω (dy ) FRT dt
T εT
{z
}
|
(II ):what we want
Z
1 T
+
Yt (ω)dt
T εT
|
{z
}
(III ): needs LLN
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Elements of the proof: the LLN
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Elements of the proof: the LLN
Z
(III ) ≡ exp −
T
Z
f (y )Et,ω (dy ) − E exp −
T
T
T
f (y )Et,ω (dy ) FRT
should vanish by a law of large numbers.
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
Elements of the proof: the LLN
Z
(III ) ≡ exp −
T
Z
f (y )Et,ω (dy ) − E exp −
T
T
T
f (y )Et,ω (dy ) FRT
should vanish by a law of large numbers.
We use a criterion which is an adaptation of the theorem due to Lyons:
Lemma
Let {Ys }s∈R+ be a.s. uniformly bounded and E[Ys ] = 0 for all s. If
Z T
∞
2 X
1
1
E Ys ds < ∞,
T
T 0
T =1
then
1
T
A. Bovier ()
Z
T
Ys ds → 0, a.s.
0
Branching Brownian motion: extremal process and ergodic theorems
R
Ergodic theorems
LLN
Requires covariance estimate:
Lemma
√
Let Ys from (III). For RT = o( T ) with limT →∞ RT = +∞, there exists
κ > 0, s.t.
κ
E[Ys Ys 0 ] ≤ Ce −RT for any s, s 0 ∈ [εT , T ] with |s − s 0 | ≥ RT .
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Universality
Universality
The new extremal process of BBM should not be limited to BBM:
Branching random walk (Aı̈dekon, Madaule)
Gaussian free field in d = 2 [Bolthausen, Deuschel, Giacomin,
Bramson, Zeitouni,Biskup and Louisdor (!) ....]
Cover times of random walks [Ding, Zeitouni, Sznitman,....]
Spin glasses with log-correlated potentials [Fyodorov, Bouchaud,..]
and building block for further models:
Extensions to stronger correlations: beyond the borderline
[Fang-Zeitouni ’12]...
Extension back to spin glasses: some of the observations made give
hope.... see Louis-Pierre’s talk
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R
Universality
References
L.-P. Arguin, A. Bovier, and N. Kistler, The genealogy of extremal particles of branching Brownian motion, Commun.
Pure Appl. Math. 64, 1647–1676 (2011).
L.-P. Arguin, A. Bovier, and N. Kistler, Poissonian statistics in the extremal process of braching Brownian motion, Ann.
Appl. Probab. 22, 1693–1711 (2012).
L.-P. Arguin, A. Bovier, and N. Kistler, The extremal process of branching Brownian motion, to appear in Probab.
Theor. Rel. Fields (2012).
L.-P. Arguin, A. Bovier, and N. Kistler, An ergodic theorem for the frontier of branching Brownian motion, EJP 18
(2013).
L.-P. Arguin, A, Bovier, and N. Kistler, An ergodic theorem for the extremal process of of branching Brownian motion,
arXiv:1209.6027, (2012).
E. Aı̈dékon, J. Berestyzki, É. Brunet, Z. Shi, Branching Brownian motion seen from its tip, to appear in Probab. Theor.
Rel. Fields (2012).
Thank you for your attention!
A. Bovier ()
Branching Brownian motion: extremal process and ergodic theorems
R