Outline
Introduction
Sketch of the proofs
Open problems
Persistence exponent for random processes in
Brownian scenery
Nadine Guillotin-Plantard
joint work with F. Castell (I2M) and F. Watbled (LMBA)
Persistence probabilities and related fields – Darmstadt
July 17, 2014
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
1
Introduction
Definition of the discrete model
Motivation
Historical overview
Our result
2
Sketch of the proofs
Lower bound
Upper bound: The main idea
3
Open problems
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Definition
S = {Sn , n ∈ N} a random walk in Zd .
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Definition
S = {Sn , n ∈ N} a random walk in Zd .
ξ = {ξx , x ∈ Zd } i.i.d. random variables (the random scenery).
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Definition
S = {Sn , n ∈ N} a random walk in Zd .
ξ = {ξx , x ∈ Zd } i.i.d. random variables (the random scenery).
The random walk in random scenery:
Zn :=
n
X
ξSk , n ≥ 1.
k=1
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Definition
S = {Sn , n ∈ N} a random walk in Zd .
ξ = {ξx , x ∈ Zd } i.i.d. random variables (the random scenery).
The random walk in random scenery:
Zn :=
n
X
ξSk , n ≥ 1.
k=1
P the product law of ξ, S.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
An example
d = 2, ξx ∈ {+1, −1}, S nearest-neighbour.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Motivation
Links to other problems and models:
1. Energy function of a polymer in a random medium;
2. Random walks in randomly directed lattices;
3. Anomalous diffusion in layered random media (Matheron- De
Marsily model in hydrology)
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Motivation
Links to other problems and models:
1. Energy function of a polymer in a random medium;
2. Random walks in randomly directed lattices;
3. Anomalous diffusion in layered random media (Matheron- De
Marsily model in hydrology)
Question:
What is the limiting distribution of (Zn )n≥1 ? Or what is its
continuous counterpart?
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Assumptions
Assumptions on the random scenery:
E [ξ0 ] = 0 and E ξ02 = 1.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Assumptions
Assumptions on the random scenery:
E [ξ0 ] = 0 and E ξ02 = 1.
Assumptions on the random walk:
S is the simple random walk in Zd .
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Connection to the self-intersection local time of S
Nn (x) :=
Pn
k=1 1{Sk =x}
the local time of S at x up to time n.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Connection to the self-intersection local time of S
Nn (x) :=
Pn
k=1 1{Sk =x}
Zn =
the local time of S at x up to time n.
Pn
k=1 ξSk
N. Guillotin
=
P
x∈Zd
ξx Nn (x).
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Connection to the self-intersection local time of S
Nn (x) :=
Pn
k=1 1{Sk =x}
Zn =
the local time of S at x up to time n.
Pn
k=1 ξSk
=
P
x∈Zd
ξx Nn (x).
P
P
E Zn2 |S = x∈Zd Nn2 (x) = 1≤i,j≤n 1{Si =Sj } =: In
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Connection to the self-intersection local time of S
Nn (x) :=
Pn
k=1 1{Sk =x}
Zn =
the local time of S at x up to time n.
Pn
k=1 ξSk
=
P
x∈Zd
ξx Nn (x).
P
P
E Zn2 |S = x∈Zd Nn2 (x) = 1≤i,j≤n 1{Si =Sj } =: In
In is the self-intersection local time of S up to time n.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Connection to the self-intersection local time of S
Nn (x) :=
Pn
k=1 1{Sk =x}
Zn =
the local time of S at x up to time n.
Pn
k=1 ξSk
=
P
x∈Zd
ξx Nn (x).
P
P
E Zn2 |S = x∈Zd Nn2 (x) = 1≤i,j≤n 1{Si =Sj } =: In
In is the self-intersection local time of S up to time n.

d = 1,
 Cn3/2
Var(Zn ) = E [In ] ∼
Cn log n d = 2,

Cn
d ≥ 3.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Limit theorems
Z[nt]
⇒ ∆t under P ?
an
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Limit theorems
Z[nt]
⇒ ∆t under P ?
an
d = 1 : Kesten-Spitzer ’79, and Borodin ’79.
an = n3/4 ,
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Limit theorems
Z[nt]
⇒ ∆t under P ?
an
d = 1 : Kesten-Spitzer ’79, and Borodin ’79.
an = n3/4 , {W (x); x ∈ R} a two-sided real Brownian motion.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Limit theorems
Z[nt]
⇒ ∆t under P ?
an
d = 1 : Kesten-Spitzer ’79, and Borodin ’79.
an = n3/4 , {W (x); x ∈ R} a two-sided real Brownian motion.
{Y (t); t ≥ 0} a Brownian motion, independent of W .
{Lt (x), x ∈ R, t ≥ 0} local time of Y .
Z
∆t :=
Lt (x)dW (x).
R
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Limit theorems
Z[nt]
⇒ ∆t under P ?
an
d = 1 : Kesten-Spitzer ’79, and Borodin ’79.
an = n3/4 , {W (x); x ∈ R} a two-sided real Brownian motion.
{Y (t); t ≥ 0} a Brownian motion, independent of W .
{Lt (x), x ∈ R, t ≥ 0} local time of Y .
Z
∆t :=
Lt (x)dW (x).
R
Remark: (∆t )t≥0 is neither Gaussian, nor Markovian.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Limit theorems
Z[nt]
⇒ ∆t under P ?
an
d = 1 : Kesten-Spitzer ’79, and Borodin ’79.
an = n3/4 , {W (x); x ∈ R} a two-sided real Brownian motion.
{Y (t); t ≥ 0} a Brownian motion, independent of W .
{Lt (x), x ∈ R, t ≥ 0} local time of Y .
Z
∆t :=
Lt (x)dW (x).
R
Remark: (∆t )t≥0 is neither Gaussian, nor Markovian.
√
d = 2 : an = n log n, ∆ Brownian motion (Bolthausen ’89).
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Limit theorems
Z[nt]
⇒ ∆t under P ?
an
d = 1 : Kesten-Spitzer ’79, and Borodin ’79.
an = n3/4 , {W (x); x ∈ R} a two-sided real Brownian motion.
{Y (t); t ≥ 0} a Brownian motion, independent of W .
{Lt (x), x ∈ R, t ≥ 0} local time of Y .
Z
∆t :=
Lt (x)dW (x).
R
Remark: (∆t )t≥0 is neither Gaussian, nor Markovian.
√
d = 2 : an = n log n, ∆ Brownian motion (Bolthausen ’89).
√
d ≥ 3 : an = n , ∆ Brownian motion (Spitzer’s Book ’76).
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
What can we say about the persistence exponent of the processes
(Zn )n≥1 and (∆t )t≥0 ?
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
What can we say about the persistence exponent of the processes
(Zn )n≥1 and (∆t )t≥0 ?
Redner and Majumdar conjectured (using physical arguments,
simulations and comparison with the FBM) that the persistence
exponent of ∆t should be equal to 1/4.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
What can we say about the persistence exponent of the processes
(Zn )n≥1 and (∆t )t≥0 ?
Redner and Majumdar conjectured (using physical arguments,
simulations and comparison with the FBM) that the persistence
exponent of ∆t should be equal to 1/4.
This was solved in (up to logarithmic factors):
Castell, F.; Guillotin-Plantard, N.; Pène, F.; Schapira, Br.
On the one-sided exit problem for stable processes in random
scenery. Electron. Commun. Probab. 18(33):1–7, 2013.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
What can we say about the persistence exponent of the processes
(Zn )n≥1 and (∆t )t≥0 ?
Redner and Majumdar conjectured (using physical arguments,
simulations and comparison with the FBM) that the persistence
exponent of ∆t should be equal to 1/4.
This was solved in (up to logarithmic factors):
Castell, F.; Guillotin-Plantard, N.; Pène, F.; Schapira, Br.
On the one-sided exit problem for stable processes in random
scenery. Electron. Commun. Probab. 18(33):1–7, 2013.
The proof heavily depends on the increments independence of the
process Y .
Is it possible to compute the persistence exponent without it? for
more general processes Y ?
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
W = {W (x); x ∈ R} a standard two-sided real Brownian motion
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
W = {W (x); x ∈ R} a standard two-sided real Brownian motion
Y = {Y (t); t ≥ 0} a self-similar process of index γ ∈ (0, 2), with
stationary increments.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
W = {W (x); x ∈ R} a standard two-sided real Brownian motion
Y = {Y (t); t ≥ 0} a self-similar process of index γ ∈ (0, 2), with
stationary increments. W and Y are independent.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
W = {W (x); x ∈ R} a standard two-sided real Brownian motion
Y = {Y (t); t ≥ 0} a self-similar process of index γ ∈ (0, 2), with
stationary increments. W and Y are independent.
There exists a jointly continuous version {Lt (x); x ∈ R, t ≥ 0} of
the local time of Y .
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
W = {W (x); x ∈ R} a standard two-sided real Brownian motion
Y = {Y (t); t ≥ 0} a self-similar process of index γ ∈ (0, 2), with
stationary increments. W and Y are independent.
There exists a jointly continuous version {Lt (x); x ∈ R, t ≥ 0} of
the local time of Y .
The random process in Brownian scenery {∆t ; t ≥ 0}
Z
∆t =
Lt (x) dW (x).
R
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
W = {W (x); x ∈ R} a standard two-sided real Brownian motion
Y = {Y (t); t ≥ 0} a self-similar process of index γ ∈ (0, 2), with
stationary increments. W and Y are independent.
There exists a jointly continuous version {Lt (x); x ∈ R, t ≥ 0} of
the local time of Y .
The random process in Brownian scenery {∆t ; t ≥ 0}
Z
∆t =
Lt (x) dW (x).
R
The process ∆ is itself a h-self-similar process with stationary
increments, with
γ
h := 1 − .
2
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Our result
Let
Z
V1 :=
L21 (x) dx ” = ”
1Z 1
Z
δ0 (Yt − Ys ) dsdt
0
R
0
be the self-intersection local time of Y .
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Our result
Let
Z
V1 :=
L21 (x) dx ” = ”
1Z 1
Z
δ0 (Yt − Ys ) dsdt
0
R
0
be the self-intersection local time of Y .
(H1) ∃ α > 1, C > 0, c > 0 s.t. for any t ≥ 0,
P[V1 ≥ t] ≤ C exp(−ct α ).
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Our result
Let
Z
V1 :=
L21 (x) dx ” = ”
1Z 1
Z
δ0 (Yt − Ys ) dsdt
0
R
0
be the self-intersection local time of Y .
(H1) ∃ α > 1, C > 0, c > 0 s.t. for any t ≥ 0,
P[V1 ≥ t] ≤ C exp(−ct α ).
(H2) ∃ β > 0, C > 0, c > 0 s.t. for any t > 0,
P[V1 ≤ t] ≤ C exp(−ct −β ).
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Our result
Let
Z
V1 :=
L21 (x) dx ” = ”
1Z 1
Z
δ0 (Yt − Ys ) dsdt
0
R
0
be the self-intersection local time of Y .
(H1) ∃ α > 1, C > 0, c > 0 s.t. for any t ≥ 0,
P[V1 ≥ t] ≤ C exp(−ct α ).
(H2) ∃ β > 0, C > 0, c > 0 s.t. for any t > 0,
P[V1 ≤ t] ≤ C exp(−ct −β ).
Theorem (F. Castell, N. G-P, F. Watbled, 2014)
Assume (H1) and (H2) hold. ∃c > 0, s.t. for T large enough,
h
i
T −γ/2 (ln T )−c ≤ P sup ∆t ≤ 1 ≤ T −γ/2 (ln T )+c .
t∈[0,T ]
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Examples of Processes Y satisfying our assumptions
1. Stable Lévy process with index δ ∈ (1, 2] with γ = 1δ , α = δ,
δ
.
and β = 2δ−1
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Examples of Processes Y satisfying our assumptions
1. Stable Lévy process with index δ ∈ (1, 2] with γ = 1δ , α = δ,
δ
.
and β = 2δ−1
2. Fractional Brownian motion with Hurst index H ∈ (0, 1) with
γ = H, α = 1/H, and β = 2.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Definition of the discrete model
Motivation
Historical overview
Our result
Examples of Processes Y satisfying our assumptions
1. Stable Lévy process with index δ ∈ (1, 2] with γ = 1δ , α = δ,
δ
.
and β = 2δ−1
2. Fractional Brownian motion with Hurst index H ∈ (0, 1) with
γ = H, α = 1/H, and β = 2.
3. Iterated Brownian motion with γ = 1/4, α = β = 4/3.
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Lower bound
Upper bound: The main idea
Aurzada and Molchan’s approach for FBM
Theorem (Molchan, 1999 + Aurzada, 2011)
Let {Xt ; t ≥ 0} be a continuous process, self-similar with index
h > 0, with stationary increments s.t. for every θ > 0,
h
i
E exp θ max |Xt | < +∞.
t∈[0,1]
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Lower bound
Upper bound: The main idea
Aurzada and Molchan’s approach for FBM
Theorem (Molchan, 1999 + Aurzada, 2011)
Let {Xt ; t ≥ 0} be a continuous process, self-similar with index
h > 0, with stationary increments s.t. for every θ > 0,
h
i
E exp θ max |Xt | < +∞.
t∈[0,1]
Then, there exists c > 0 s.t. for T large enough,
h
i
T −(1−h) (ln T )−c ≤ P sup Xt ≤ 1 .
t∈[0,T ]
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Lower bound
Upper bound: The main idea
Given Y , the process {∆t , t ≥ 0} is a centered Gaussian process
with covariance matrix
Z
E[∆s ∆t |Y ] =
Ls (x)Lt (x) dx ≥ 0.
R
Moreover, for any 0 < s < t,
Z
E[∆s (∆t − ∆s )|Y ] =
Ls (x)(Lt (x) − Ls (x)) dx ≥ 0.
R
Use Slepian’s Lemma conditionally to Y and then integrate with
respect to the law of Y .
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Open problems
• Persistence exponent of the discrete model
Zn =
n
X
ξSk , n ≥ 1
k=1
with ξx ∈ {±1}, {Sn , n ≥ 0} the simple random walk in Z?
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Open problems
• Persistence exponent of the discrete model
Zn =
n
X
ξSk , n ≥ 1
k=1
with ξx ∈ {±1}, {Sn , n ≥ 0} the simple random walk in Z?
Conjecture 1: As n → +∞,
P[ max Zk ≤ 1] ∼ C n−1/4 .
1≤k≤n
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Open problems
• Persistence exponent of the discrete model
Zn =
n
X
ξSk , n ≥ 1
k=1
with ξx ∈ {±1}, {Sn , n ≥ 0} the simple random walk in Z?
Conjecture 1: As n → +∞,
P[ max Zk ≤ 1] ∼ C n−1/4 .
1≤k≤n
• What can we say if {W (x), x ∈ R} is not a Brownian motion but
a two-sided β-stable Lévy process, with β ∈ (1, 2)?
N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Open problems
• Persistence exponent of the discrete model
Zn =
n
X
ξSk , n ≥ 1
k=1
with ξx ∈ {±1}, {Sn , n ≥ 0} the simple random walk in Z?
Conjecture 1: As n → +∞,
P[ max Zk ≤ 1] ∼ C n−1/4 .
1≤k≤n
• What can we say if {W (x), x ∈ R} is not a Brownian motion but
a two-sided β-stable Lévy process, with β ∈ (1, 2)?
Conjecture 2: The persistence exponent should be equal to
h=
N. Guillotin
β−1
.
αβ
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
• Matheron, G.; de Marsily G. Is transport in porous media always
diffusive? A counterexample.
Water Resources Res. 16:901–907, 1980.
• Redner, S. Invited Symposium Contribution: Superdiffusion in
Random Velocity Fields. Proceedings of the Bar-Ilan Conference in
Condensed-Matter Physics, Physica A 168, 1990.
• Redner, S. Survival Probability in a Random Velocity Field.
Phys. Rev., E56, 4967, 1997.
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N. Guillotin
Persistence exponent for RP in BS
Outline
Introduction
Sketch of the proofs
Open problems
Thank you for your attention!
N. Guillotin
Persistence exponent for RP in BS