On a scaling limit of the parabolic Anderson model
with exclusion interaction
Dirk Erhard
University of Warwick
Joint work in progress with Martin Hairer
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
1 / 21
Setup
We consider equations of the type
∂t u = ∆u + F (u, ξ),
where F is non-linear in u, affin in ξ and ξ is an irregular input.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
2 / 21
Setup
We consider equations of the type
∂t u = ∆u + F (u, ξ),
where F is non-linear in u, affin in ξ and ξ is an irregular input.
Equation often ill-posed.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
2 / 21
Setup
We consider equations of the type
∂t u = ∆u + F (u, ξ),
where F is non-linear in u, affin in ξ and ξ is an irregular input.
Equation often ill-posed.
Naïve approach: Look at
∂t uε = ∆uε + F (uε , ξε ),
ξε smoothened version of ξ,
−→ often does not converge as ε ↓ 0.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
2 / 21
Setup
We consider equations of the type
∂t u = ∆u + F (u, ξ),
where F is non-linear in u, affin in ξ and ξ is an irregular input.
Equation often ill-posed.
Naïve approach: Look at
∂t uε = ∆uε + F (uε , ξε ),
ξε smoothened version of ξ,
−→ often does not converge as ε ↓ 0.
Solution: Use regularity structures to renormalise the equation.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
2 / 21
Directions within regularity structures: White noise
Look at convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
ξε mollified white noise.
0
Recall: ξ ∈ S (Rd ) is called white noise if ξ is Gaussian and
E[ξ(x)ξ(y )] = δx (y ).
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
3 / 21
Directions within regularity structures: White noise
Look at convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
ξε mollified white noise.
0
Recall: ξ ∈ S (Rd ) is called white noise if ξ is Gaussian and
E[ξ(x)ξ(y )] = δx (y ).
Goal: Find an algorithmic way to deal with a large class of
equations of the above type.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
3 / 21
Directions within regularity structures: White noise
Look at convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
ξε mollified white noise.
0
Recall: ξ ∈ S (Rd ) is called white noise if ξ is Gaussian and
E[ξ(x)ξ(y )] = δx (y ).
Goal: Find an algorithmic way to deal with a large class of
equations of the above type.
Example: Generalized KPZ (Bruned, Hairer, Zambotti)
∂t u = ∆u + f (u)(∂x u)2 + k (u)∂x u + h(u) + g(u)ξ
−→ 120 terms need to be controlled ⇒ not doable by hands.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
3 / 21
Directions within regularity structures: non gaussian
approximation
Look at the convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
where ξε is a non gaussian, smooth, strongly mixing field that
approximates white noise.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
4 / 21
Directions within regularity structures: non gaussian
approximation
Look at the convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
where ξε is a non gaussian, smooth, strongly mixing field that
approximates white noise.
−→ Gaussian tools break down, renormalisation more complicated.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
4 / 21
Directions within regularity structures: non gaussian
approximation
Look at the convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
where ξε is a non gaussian, smooth, strongly mixing field that
approximates white noise.
−→ Gaussian tools break down, renormalisation more complicated.
Examples:
KPZ (Shen/Hairer)
∂t uε = ∆uε + (∂x uε )2 + ξε
"generalised" Φ43 equation (Shen/Xu)
∂t uε = ∆uε + V (uε ) + ξε ,
for some suitable polynomial V .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
4 / 21
Directions within regularity structures: the discrete
case
Look at the convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
where time and space may be discrete.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
5 / 21
Directions within regularity structures: the discrete
case
Look at the convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
where time and space may be discrete.
−→ The theory of regularity structures needs to be adapted to the
discrete setting.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
5 / 21
Directions within regularity structures: the discrete
case
Look at the convergence of the sequence of equations
∂t uε = ∆uε + F (uε , ξε ),
where time and space may be discrete.
−→ The theory of regularity structures needs to be adapted to the
discrete setting.
Successfully dealt with by Hairer/Matetski who studied
∂t uε (x, t) = ∆uε (x, t) − uε (x, t)3 + ξε (x, t),
where x ∈ (Z/εZ)3 , t ≥ 0, and ξε is a gaussian approximation of
white noise.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
5 / 21
The PAM equation

 ∂u (x, t) = ∆d u(x, t) + ξ(x, t)u(x, t),
∂t

u(x, 0) = u0 (x).
∆d u(x, t) =
P
y :|y −x|=1 [u(y , t)
Dirk Erhard (University of Warwick)
x ∈ Zd , t ≥ 0
− u(x, t)] is the discrete Laplacian.
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
6 / 21
The PAM equation

 ∂u (x, t) = ∆d u(x, t) + ξ(x, t)u(x, t),
∂t

u(x, 0) = u0 (x).
∆d u(x, t) =
P
y :|y −x|=1 [u(y , t)
x ∈ Zd , t ≥ 0
− u(x, t)] is the discrete Laplacian.
The solution is given via the Feynman Kac formula
Z t
u(x, t) = Ex exp
ξ(X (s), t − s) ds u0 (X (t)) ,
0
where X is a simple random walk that starts in x under Ex .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
6 / 21
The PAM equation

 ∂u (x, t) = ∆d u(x, t) + ξ(x, t)u(x, t),
∂t

u(x, 0) = u0 (x).
∆d u(x, t) =
P
y :|y −x|=1 [u(y , t)
x ∈ Zd , t ≥ 0
− u(x, t)] is the discrete Laplacian.
The solution is given via the Feynman Kac formula
Z t
u(x, t) = Ex exp
ξ(X (s), t − s) ds u0 (X (t)) ,
0
where X is a simple random walk that starts in x under Ex .
Goal: Investigate the above equation under an appropriate space-time
scaling.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
6 / 21
The simple symmetric exclusion process
The simple symmetric exclusion process is a Markov process on
d
d
{0, 1}Z whose generator acts on local function f : {0, 1}Z → R
via
X
(Lf )(η) =
η(u)[1 − η(v )][f (η u,v ) − f (η)],
||u−v ||=1
where
Dirk Erhard (University of Warwick)


η(z)
u,v
η (z) = η(u)


η(v )
if z 6= u, v ,
if z = v ,
if z = u.
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
7 / 21
The simple symmetric exclusion process
The simple symmetric exclusion process is a Markov process on
d
d
{0, 1}Z whose generator acts on local function f : {0, 1}Z → R
via
X
(Lf )(η) =
η(u)[1 − η(v )][f (η u,v ) − f (η)],
||u−v ||=1
where


η(z)
u,v
η (z) = η(u)


η(v )
if z 6= u, v ,
if z = v ,
if z = u.
N
Fact: Let ρ ∈ (0, 1), then νρ = x∈Zd Ber(ρ) is an invariant and
reversible measure for η. We always start η from νρ .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
7 / 21
The simple symmetric exclusion process
The simple symmetric exclusion process is a Markov process on
d
d
{0, 1}Z whose generator acts on local function f : {0, 1}Z → R
via
X
(Lf )(η) =
η(u)[1 − η(v )][f (η u,v ) − f (η)],
||u−v ||=1
where


η(z)
u,v
η (z) = η(u)


η(v )
if z 6= u, v ,
if z = v ,
if z = u.
N
Fact: Let ρ ∈ (0, 1), then νρ = x∈Zd Ber(ρ) is an invariant and
reversible measure for η. We always start η from νρ .
ξ = η − ρ.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
7 / 21
Graphical representation of the exclusion process
t
qy
↑
←
0
Dirk Erhard (University of Warwick)
↑←
→↑
↑
q
x
Zd
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
8 / 21
Graphical representation of the exclusion process
t
qy
↑
←
0
↑←
→↑
↑
q
x
Zd
Let X y be a particle starting at time t at position y following the
arrows downwards. Then,
ηt (y ) = η0 (Xty ).
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
8 / 21
Graphical representation of the exclusion process
t
qy
↑
←
0
↑←
→↑
↑
q
x
Zd
Let X y be a particle starting at time t at position y following the
arrows downwards. Then,
ηt (y ) = η0 (Xty ).
Remark: The law of X y is that of a simple random walk!
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
8 / 21
Graphical representation of the exclusion process
t
qy
↑
←
0
↑←
→↑
↑
q
x
Zd
Let X y be a particle starting at time t at position y following the
arrows downwards. Then,
ηt (y ) = η0 (Xty ).
Remark: The law of X y is that of a simple random walk!
=⇒
E[ξ(x, 0)ξ(y , t)] = E[(η0 (x) − ρ)(η0 (Xty ) − ρ)]
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
8 / 21
Graphical representation of the exclusion process
t
qy
↑
←
0
↑←
→↑
↑
q
x
Zd
Let X y be a particle starting at time t at position y following the
arrows downwards. Then,
ηt (y ) = η0 (Xty ).
Remark: The law of X y is that of a simple random walk!
=⇒
E[ξ(x, 0)ξ(y , t)] = E[(η0 (x) − ρ)(η0 (Xty ) − ρ)]
X
=
pt (y , z)E[(η0 (x) − ρ)(η0 (z) − ρ)]
= pt (y , x)ρ(1 − ρ).
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
8 / 21
Fluctuations of the exclusion process
With the help of the Markov property we may even deduce that
E[ξ(x, s)ξ(y , t)] = ρ(1 − ρ)pt−s (x, y ).
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
9 / 21
Fluctuations of the exclusion process
With the help of the Markov property we may even deduce that
E[ξ(x, s)ξ(y , t)] = ρ(1 − ρ)pt−s (x, y ).
Moreover,
p22N (t−s) (2N x, 2N y ) ≈ (4π22N (t − s))−d/2 e−||x−y ||
Dirk Erhard (University of Warwick)
2 /(t−s)
.
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
9 / 21
Fluctuations of the exclusion process
With the help of the Markov property we may even deduce that
E[ξ(x, s)ξ(y , t)] = ρ(1 − ρ)pt−s (x, y ).
Moreover,
p22N (t−s) (2N x, 2N y ) ≈ (4π22N (t − s))−d/2 e−||x−y ||
2 /(t−s)
.
Reasonable (and correct) guess:
2Nd/2 ξ(x2N , t22N ) converges to a Gaussian process Φ such that
2
E[Φ(x, s)Φ(y , t)] = ρ(1 − ρ)(4π(t − s))−d/2 e−||x−y || /(t−s) .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
9 / 21
Need for renormalisation
Scaling time by 22N , space by 2N and ξ by 2Nd/2 we obtain
Z t
N
2N
Nd/2
N
2N
2N
u(2 x, 2 t) ≈ E2N x exp 2
ξ(2 X (2 s), 2 s) ds
0
Z t
ξ(2N X (22N s), 22N s) ds
= 1 + 2Nd/2 E2N
0
Nd−1
+2
Dirk Erhard (University of Warwick)
Z
E2N x
2
Y
N
ξ(2 X (2
2N
si ), 2
2N
si )dsi + · · ·
0≤s1 ,s2 ≤t i=1
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
10 / 21
Need for renormalisation
Scaling time by 22N , space by 2N and ξ by 2Nd/2 we obtain
Z t
N
2N
Nd/2
N
2N
2N
u(2 x, 2 t) ≈ E2N x exp 2
ξ(2 X (2 s), 2 s) ds
0
Z t
ξ(2N X (22N s), 22N s) ds
= 1 + 2Nd/2 E2N
0
Nd−1
+2
Z
E2N x
2
Y
N
ξ(2 X (2
2N
si ), 2
2N
si )dsi + · · ·
0≤s1 ,s2 ≤t i=1
Taking expectation with respect to ξ of the last term we get
approximately
Z
1
−||X (2N s1 )−X (2N s2 )||2 /(s1 −s2 )
ds1 ds2
E2N x
e
d/2
0≤s1 ,s2 ≤t |s1 − s2 |
−→ problematic as soon as d ≥ 2.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
10 / 21
The result
Let TdN = [0, 2N )d ∩ Zd and define ξN (x, t) = 2Nd/2 ξ(2N x, 22N t) for
x ∈ TdN 2−N . Let uN be the solution to
∂uN
(x, t) = 22N ∆d uN (x, t) + [ξN (x, t) − CN ]uN (x, t)
∂t
on TdN = [0, 2N )d ∩ Zd .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
11 / 21
The result
Let TdN = [0, 2N )d ∩ Zd and define ξN (x, t) = 2Nd/2 ξ(2N x, 22N t) for
x ∈ TdN 2−N . Let uN be the solution to
∂uN
(x, t) = 22N ∆d uN (x, t) + [ξN (x, t) − CN ]uN (x, t)
∂t
on TdN = [0, 2N )d ∩ Zd .
Main result. Let d ∈ {2, 3}. There is a sequence of constants CN
tending to infinity and T > 0 such that wN converges in distribution in
C α,α/2 (Td × [0, T ]) with α < 2 − d/2. The limit w formally satisfies
∂w
(x, t) = ∆w(x, t) + (Φ(x, t) − ∞)w(x, t).
∂t
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
11 / 21
Introduction to cumulants I
Given an index set A and a collection of random variables
{Xa }a∈A . We write for B ⊆ A
Y
Xa .
XB = {Xa : a ∈ B} and X B =
a∈B
P(B) denotes the set of all partitions of B.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
12 / 21
Introduction to cumulants I
Given an index set A and a collection of random variables
{Xa }a∈A . We write for B ⊆ A
Y
Xa .
XB = {Xa : a ∈ B} and X B =
a∈B
P(B) denotes the set of all partitions of B.
Definition
Fix a finite subset B ⊆ A. We define the cumulant Ec (XB ) via
X
Y
Ec (XB ) =
(|π| − 1)!(−1)|π|−1
E(X B̄ ).
π∈P(B)
Dirk Erhard (University of Warwick)
B̄∈π
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
12 / 21
Introduction to cumulants II
Ec (XB ) =
X
(|π| − 1)!(−1)|π|−1
π∈P(B)
Dirk Erhard (University of Warwick)
Y
E(X B̄ ).
B̄∈π
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
13 / 21
Introduction to cumulants II
Ec (XB ) =
X
(|π| − 1)!(−1)|π|−1
π∈P(B)
Y
E(X B̄ ).
B̄∈π
Examples:
Ec [X{1,2} ] = E[X1 X2 ] − EX1 EX2 .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
13 / 21
Introduction to cumulants II
Ec (XB ) =
X
(|π| − 1)!(−1)|π|−1
π∈P(B)
Y
E(X B̄ ).
B̄∈π
Examples:
Ec [X{1,2} ] = E[X1 X2 ] − EX1 EX2 .
Q
P
Q
Q
Ec [X{1,2,3} ] = E[ 3i=1 Xi ] − 3i=1 EXi E[ j6=i Xj ] + 2 3i=1 EXi .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
13 / 21
Introduction to cumulants II
Ec (XB ) =
X
(|π| − 1)!(−1)|π|−1
π∈P(B)
Y
E(X B̄ ).
B̄∈π
Examples:
Ec [X{1,2} ] = E[X1 X2 ] − EX1 EX2 .
Q
P
Q
Q
Ec [X{1,2,3} ] = E[ 3i=1 Xi ] − 3i=1 EXi E[ j6=i Xj ] + 2 3i=1 EXi .
Remark:
Cumulants are a mean to measure joint interaction of all random
variables involved.
If the Xi ’s are gaussian, then Ec (XB ) = 0 unless |B| ≤ 2.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
13 / 21
Introduction to Wick products
Definition
The Wick product : XA : is recursively defined via : X∅ : = 1 and
X
X Y
Ec (XB̄ ).
XA =
: XB :
B⊆A
Dirk Erhard (University of Warwick)
π∈P(A\B) B̄∈π
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
14 / 21
Introduction to Wick products
Definition
The Wick product : XA : is recursively defined via : X∅ : = 1 and
X
X Y
Ec (XB̄ ).
XA =
: XB :
B⊆A
π∈P(A\B) B̄∈π
Examples:
: X : = X − EX , and if the Xi ’s have mean zero,
: X1 X2 : = X1 X2 − EX1 X2 ,
Q
Q
P
Q
: X1 X2 X3 : = 3i=1 Xi − E 3i=1 Xi − 3i=1 Xi E j6=i Xj .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
14 / 21
Introduction to Wick products
Definition
The Wick product : XA : is recursively defined via : X∅ : = 1 and
X
X Y
Ec (XB̄ ).
XA =
: XB :
B⊆A
π∈P(A\B) B̄∈π
Examples:
: X : = X − EX , and if the Xi ’s have mean zero,
: X1 X2 : = X1 X2 − EX1 X2 ,
Q
Q
P
Q
: X1 X2 X3 : = 3i=1 Xi − E 3i=1 Xi − 3i=1 Xi E j6=i Xj .
Remark: Expectations of Wick products can be expressed in terms of
cumulants.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
14 / 21
Application to the PAM equation
Recall that
Nd−1
2
Z
E E2N
2
Y
N
ξ(2 X (2
2N
si ), 2
2N
!
si )dsi
0≤s1 ,s2 ≤t i=1
causes problems as soon as d ≥ 2.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
15 / 21
Application to the PAM equation
Recall that
Nd−1
2
Z
E E2N
2
Y
N
ξ(2 X (2
2N
si ), 2
2N
!
si )dsi
0≤s1 ,s2 ≤t i=1
causes problems as soon as d ≥ 2.
Q
Idea: Replace 2i=1 ξ(2N X (22N si ), 22N si ) by its Wick product.
−→ need to control cumulants of ξ.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
15 / 21
How to obtain cumulants of ξ I
t
qy
↑
←
0
Dirk Erhard (University of Warwick)
↑←
→↑
↑
q
x
Zd
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
16 / 21
How to obtain cumulants of ξ I
t
qy
↑
←
0
↑←
→↑
↑
q
x
Zd
Let X y1 , . . . , X yn be a collection of random walks that jump
according to the exponential clocks from the graphical
construction, then (ηt (y1 ), . . . , ηt (yn )) = (η0 (Xty1 ), . . . , η0 (Xtyn )).
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
16 / 21
How to obtain cumulants of ξ I
qy
t
↑
←
↑←
→↑
↑
q
x
0
Zd
Let X y1 , . . . , X yn be a collection of random walks that jump
according to the exponential clocks from the graphical
construction, then (ηt (y1 ), . . . , ηt (yn )) = (η0 (Xty1 ), . . . , η0 (Xtyn )).
Given a measurable function f , then for any initial state η0 ,
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
= Eη0 [f (η0 (Xty1 ), . . . , η0 (Xtyn )]
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
16 / 21
How to obtain cumulants of ξ I
qy
t
↑
←
↑←
→↑
↑
q
x
0
Zd
Let X y1 , . . . , X yn be a collection of random walks that jump
according to the exponential clocks from the graphical
construction, then (ηt (y1 ), . . . , ηt (yn )) = (η0 (Xty1 ), . . . , η0 (Xtyn )).
Given a measurable function f , then for any initial state η0 ,
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
= Eη0 [f (η0 (Xty1 ), . . . , η0 (Xtyn )]
X y ,...,y
=
pt 1 n (z1 , . . . , zn )Eη0 [f (η0 (z1 ), . . . , η0 (zn ))].
z1 ,...,zn
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
16 / 21
How to obtain cumulants of ξ II
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
X y ,...,y
=
pt 1 n (z1 , . . . , zn )Eη0 [f (η0 (z1 ), . . . , η0 (zn ))].
(1)
z1 ,...,zn
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
17 / 21
How to obtain cumulants of ξ II
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
X y ,...,y
=
pt 1 n (z1 , . . . , zn )Eη0 [f (η0 (z1 ), . . . , η0 (zn ))].
(1)
z1 ,...,zn
Application of (1):
4
hY
i
E
ξ(xi , ti )
i=1
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
17 / 21
How to obtain cumulants of ξ II
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
X y ,...,y
=
pt 1 n (z1 , . . . , zn )Eη0 [f (η0 (z1 ), . . . , η0 (zn ))].
(1)
z1 ,...,zn
Application of (1):
3
4
i
hY
hY
i
ξ(xi , ti )Eηt3 [ξ(x4 , t4 )]
E
ξ(xi , ti ) = E
i=1
Dirk Erhard (University of Warwick)
i=1
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
17 / 21
How to obtain cumulants of ξ II
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
X y ,...,y
=
pt 1 n (z1 , . . . , zn )Eη0 [f (η0 (z1 ), . . . , η0 (zn ))].
(1)
z1 ,...,zn
Application of (1):
3
4
i
hY
hY
i
ξ(xi , ti )Eηt3 [ξ(x4 , t4 )]
E
ξ(xi , ti ) = E
i=1
i=1
=
X
3
hY
i
ptx44−t3 (z 3 )E
ξ(xi , ti )ξ(z 3 , t3 )]
i=1
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
17 / 21
How to obtain cumulants of ξ II
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
X y ,...,y
=
pt 1 n (z1 , . . . , zn )Eη0 [f (η0 (z1 ), . . . , η0 (zn ))].
(1)
z1 ,...,zn
Application of (1):
3
4
i
hY
hY
i
ξ(xi , ti )Eηt3 [ξ(x4 , t4 )]
E
ξ(xi , ti ) = E
i=1
i=1
=
X
3
hY
i
ptx44−t3 (z 3 )E
ξ(xi , ti )ξ(z 3 , t3 )]
i=1
2
hY
i
X x
ξ(xi , ti )Eηt2 [ξ(x3 , t3 )ξ(z 3 , t3 )]
=
pt44−t3 (z 3 )E
i=1
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
17 / 21
How to obtain cumulants of ξ II
Eη0 [f (ηt (y1 ), . . . , ηt (yn ))]
X y ,...,y
=
pt 1 n (z1 , . . . , zn )Eη0 [f (η0 (z1 ), . . . , η0 (zn ))].
(1)
z1 ,...,zn
Application of (1):
3
4
i
hY
hY
i
ξ(xi , ti )Eηt3 [ξ(x4 , t4 )]
E
ξ(xi , ti ) = E
i=1
i=1
=
X
3
hY
i
ptx44−t3 (z 3 )E
ξ(xi , ti )ξ(z 3 , t3 )]
i=1
2
hY
i
X x
ξ(xi , ti )Eηt2 [ξ(x3 , t3 )ξ(z 3 , t3 )]
=
pt44−t3 (z 3 )E
i=1
=
X
2
hY
,z 3 2 2
ptx44−t3 (z 3 )ptx33−t
(z1 , z2 )E
2
i
ξ(xi , ti )ξ(z12 , t2 )ξ(z22 , t2 )]
i=1
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
17 / 21
How to obtain cumulants of ξ III
We obtain:
4
i
hY
ξ(xi , ti )
E
i=1
=
X
2 2
,z 3 2 2 x2 ,z1 ,z2
1 1 1
ptx44−t3 (z 3 )ptx33−t
(z
,
z
)p
1
2
t2 −t1 (z1 , z2 , z3 )E
2
h
ξ(x1 , t1 )
3
Y
i
ξ(zi1 , t1 ) .
i=1
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
18 / 21
How to obtain cumulants of ξ III
We obtain:
4
i
hY
ξ(xi , ti )
E
i=1
=
X
2 2
,z 3 2 2 x2 ,z1 ,z2
1 1 1
ptx44−t3 (z 3 )ptx33−t
(z
,
z
)p
1
2
t2 −t1 (z1 , z2 , z3 )E
2
h
ξ(x1 , t1 )
3
Y
i
ξ(zi1 , t1 ) .
i=1
Since the initial configuration is νρ =
N
x∈Zd
Ber(ρ), only four terms
contribute to the above sum:
1
x1 = z11 = z21 = z31 : ptx44−t3 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ),
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
18 / 21
How to obtain cumulants of ξ III
We obtain:
4
i
hY
ξ(xi , ti )
E
i=1
=
X
2 2
,z 3 2 2 x2 ,z1 ,z2
1 1 1
ptx44−t3 (z 3 )ptx33−t
(z
,
z
)p
1
2
t2 −t1 (z1 , z2 , z3 )E
2
h
ξ(x1 , t1 )
3
Y
i
ξ(zi1 , t1 ) .
i=1
Since the initial configuration is νρ =
N
x∈Zd
Ber(ρ), only four terms
contribute to the above sum:
1
x1 = z11 = z21 = z31 : ptx44−t3 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ),
2
x1 = z11 and z21 = z31 : ptx44−t3 (x3 )ptx22−t1 (x1 ),
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
18 / 21
How to obtain cumulants of ξ III
We obtain:
4
i
hY
ξ(xi , ti )
E
i=1
=
X
2 2
,z 3 2 2 x2 ,z1 ,z2
1 1 1
ptx44−t3 (z 3 )ptx33−t
(z
,
z
)p
1
2
t2 −t1 (z1 , z2 , z3 )E
2
h
ξ(x1 , t1 )
3
Y
i
ξ(zi1 , t1 ) .
i=1
Since the initial configuration is νρ =
N
x∈Zd
Ber(ρ), only four terms
contribute to the above sum:
1
2
3
x1 = z11 = z21 = z31 : ptx44−t3 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ),
x1 = z11 and z21 = z31 : ptx44−t3 (x3 )ptx22−t1 (x1 ),
P x4
,z
x1 = z21 and z11 = z31 :
pt4 −t3 (z)ptx33−t
(z̄, x2 )ptz̄2 −t1 (x1 )
2
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
18 / 21
How to obtain cumulants of ξ III
We obtain:
4
i
hY
ξ(xi , ti )
E
i=1
=
X
2 2
,z 3 2 2 x2 ,z1 ,z2
1 1 1
ptx44−t3 (z 3 )ptx33−t
(z
,
z
)p
1
2
t2 −t1 (z1 , z2 , z3 )E
2
h
ξ(x1 , t1 )
3
Y
i
ξ(zi1 , t1 ) .
i=1
Since the initial configuration is νρ =
N
x∈Zd
Ber(ρ), only four terms
contribute to the above sum:
1
2
3
4
x1 = z11 = z21 = z31 : ptx44−t3 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ),
x1 = z11 and z21 = z31 : ptx44−t3 (x3 )ptx22−t1 (x1 ),
P x4
,z
x1 = z21 and z11 = z31 :
pt4 −t3 (z)ptx33−t
(z̄, x2 )ptz̄2 −t1 (x1 ) and
2
P
,z
x1 = z31 and z11 = z21 :
ptx44−t3 (z)ptx33−t
(x2 , z̄)ptz̄2 −t1 (x1 ).
2
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
18 / 21
How to obtain cumulants of ξ IV
1
2
3
4
ptx44−t3 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ),
ptx44−t3 (x3 )ptx22−t1 (x1 ),
P x4
,z
pt4 −t3 (z)ptx33−t
(z̄, x2 )ptz̄2 −t1 (x1 )
2
P x4
,z
pt4 −t3 (z)ptx33−t
(x2 , z̄)ptz̄2 −t1 (x1 ).
2
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
19 / 21
How to obtain cumulants of ξ IV
1
2
3
4
ptx44−t3 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ),
ptx44−t3 (x3 )ptx22−t1 (x1 ),
P x4
,z
pt4 −t3 (z)ptx33−t
(z̄, x2 )ptz̄2 −t1 (x1 )
2
P x4
,z
pt4 −t3 (z)ptx33−t
(x2 , z̄)ptz̄2 −t1 (x1 ).
2
Since ξ has mean zero,
Ec (ξ(xi , ti ), i ∈ {1, . . . , 4})
4
hY
i X h Y
i h Y
i
=E
ξ(xi , ti ) −
E
ξ(xk , tk ) E
ξ(xk , tk )
i=1
Dirk Erhard (University of Warwick)
i<j
k ∈{i,j}
k ∈{i,j}
/
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
19 / 21
How to obtain cumulants of ξ IV
1
2
3
4
ptx44−t3 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ),
ptx44−t3 (x3 )ptx22−t1 (x1 ),
P x4
,z
pt4 −t3 (z)ptx33−t
(z̄, x2 )ptz̄2 −t1 (x1 )
2
P x4
,z
pt4 −t3 (z)ptx33−t
(x2 , z̄)ptz̄2 −t1 (x1 ).
2
Since ξ has mean zero,
Ec (ξ(xi , ti ), i ∈ {1, . . . , 4})
4
hY
i X h Y
i h Y
i
=E
ξ(xi , ti ) −
E
ξ(xk , tk ) E
ξ(xk , tk )
i=1
i<j
k ∈{i,j}
k ∈{i,j}
/
−→ The first term above survives. The second term perfectly cancels
out. The third and fourth term would cancel out if
ptx1 ,x2 (y1 , y2 ) = ptx1 (y1 )ptx2 (y2 ) for all xi , yi .
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
19 / 21
factorising the transition probabilities I
Goal: Write ptx1 ,x2 (y1 , y2 ) = ptx1 (y1 )ptx2 (y2 ) + "stuff" and characterise
the "stuff".
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
20 / 21
factorising the transition probabilities I
Goal: Write ptx1 ,x2 (y1 , y2 ) = ptx1 (y1 )ptx2 (y2 ) + "stuff" and characterise
the "stuff".
Let x, y , z ∈ Zd and define fy (z) = 1l{y = z} and denote the exclusion
particle started at x by X x , then there is a martingale Mtx (y ) such that
1l{Xtx = y } = 1l{X0x = y } +
Z
0
t
(Lfy )(Xsx ) ds + Mtx (y ).
Here, L is the generator of the exclusion process.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
20 / 21
factorising the transition probabilities I
Goal: Write ptx1 ,x2 (y1 , y2 ) = ptx1 (y1 )ptx2 (y2 ) + "stuff" and characterise
the "stuff".
Let x, y , z ∈ Zd and define fy (z) = 1l{y = z} and denote the exclusion
particle started at x by X x , then there is a martingale Mtx (y ) such that
1l{Xtx = y } = 1l{X0x = y } +
Z
0
t
(Lfy )(Xsx ) ds + Mtx (y ).
Here, L is the generator of the exclusion process. Solving this equation
with Duhamel’s principle shows that
1l{Xtx
= y} =
Dirk Erhard (University of Warwick)
ptx (y )
+
Z tX
0
y
pt−s
(z) dMsx (z).
z
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
20 / 21
factorising the transition probabilities I
Goal: Write ptx1 ,x2 (y1 , y2 ) = ptx1 (y1 )ptx2 (y2 ) + "stuff" and characterise
the "stuff".
Let x, y , z ∈ Zd and define fy (z) = 1l{y = z} and denote the exclusion
particle started at x by X x , then there is a martingale Mtx (y ) such that
1l{Xtx = y } = 1l{X0x = y } +
Z
0
t
(Lfy )(Xsx ) ds + Mtx (y ).
Here, L is the generator of the exclusion process. Solving this equation
with Duhamel’s principle shows that
1l{Xtx
Note that Mrx,t :=
= y} =
Rr P
0
Dirk Erhard (University of Warwick)
z
ptx (y )
+
Z tX
0
y
pt−s
(z) dMsx (z).
z
y
pt−s
(z) dMsx (z) is a zero mean martingale in r .
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
20 / 21
factorising the transition probabilities II
We conclude that
ptx1 ,x2 (y1 , y2 ) = E
2
hY
2
i
hY
i
xi
1l{Xtxi = yi } = E
(ptxi (yi ) + Mt,t
)
i=1
i=1
=
2
Y
i=1
Dirk Erhard (University of Warwick)
ptxi (yi ) + E
2
hY
i
xi
Mt,t
(yi ) ,
i=1
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
21 / 21
factorising the transition probabilities II
We conclude that
ptx1 ,x2 (y1 , y2 ) = E
2
hY
2
i
hY
i
xi
1l{Xtxi = yi } = E
(ptxi (yi ) + Mt,t
)
i=1
i=1
=
2
Y
i=1
ptxi (yi ) + E
2
hY
i
xi
Mt,t
(yi ) ,
i=1
−→ Thus "stuff" is given by the expectation of the product of two
martingales. We are able to handle that.
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
21 / 21
factorising the transition probabilities II
We conclude that
ptx1 ,x2 (y1 , y2 ) = E
2
hY
2
i
hY
i
xi
1l{Xtxi = yi } = E
(ptxi (yi ) + Mt,t
)
i=1
i=1
=
2
Y
i=1
ptxi (yi ) + E
2
hY
i
xi
Mt,t
(yi ) ,
i=1
−→ Thus "stuff" is given by the expectation of the product of two
martingales. We are able to handle that.
Thus, the fourth cumulants consists of a term of the form
px4 (x3 )ptx33−t2 (x2 )ptx22−t1 (x1 ) and two terms of the form
Pt4 −tx34
pt4 −t3 (z)"stuff"ptz̄2 −t1 (x1 ).
Dirk Erhard (University of Warwick)
On a scaling limit of the parabolic Anderson model with exclusion
December
interaction
8, 2015
21 / 21