The Pascal Principle for a Particle Among Sub-diffusive Mobile Traps
Lung-Chi Chen 1
Rongfeng Sun 2
March 6, 2012
Abstract
We study a trapping problem on Zd with mobile traps, where a particle starting from the origin
is killed when it first encounters a trap. The traps are distributed as a Poisson point process
on Zd , each moving independently as a simple symmetric random walk with i.i.d. holding times,
where heavy-tailed holding times give rise to sub-diffusive trap motion. It is accepted in the physics
literature that among all deterministic trajectories the particle may follow up to time t, the constant
trajectory maximizes the survival probability. This is known as the Pascal principle. Previously,
the Pascal principle has been verified rigorously for exponential holding times. In this note, we
extend it to holding times with a general continuous distribution. As a byproduct, we find that
the expected number of sites visited by an n-step simple symmetric random walk can only increase
if deterministic jumps are inserted in the path of the walk. We conjecture that this holds for all
symmetric random walks on Zd .
AMS 2010 subject classification: 60K37, 60K35, 82C22.
Keywords. Pascal principle, trapping problem, random walk range.
1
Introduction
We consider a particle moving among a Poisson field of mobile traps on Zd , defined as follows. At
time 0, there is Ny number of traps at each y ∈ Zd , where {Ny }y∈Zd are distributed as i.i.d. Poisson
random variables with mean 1. Each trap then moves independently as a random walk on Zd with i.i.d.
holding times, with increment distribution p(·) on Zd for the jumps and holding time distribution µ(·)
on (0, ∞) for the time between successive jumps. For each y ∈ Zd and 1 ≤ j ≤ Ny , let Yjy := (Yjy (t))t≥0
denote the time-evolution of the j-th trap starting from y at time 0. Then at each time t ≥ 0, the
trap configuration is determined by
X
ξ(t, x) :=
1{Yjy (t)=x} ,
x ∈ Zd .
(1.1)
y∈Zd ,1≤j≤Ny
A particle X := (X(t))t≥0 moving on Zd is then killed at the first time
τX,ξ := inf{t ≥ 0 : ξ(t, X(t)) ≥ 1},
(1.2)
when the particle first meets a trap. The particle motion X may be either deterministic or random.
We are interested in the probability St that X survives up to time t, when the randomness in X
and the trap field {Yjy }y∈Zd ,1≤j≤Ny have been averaged out. In what follows, Eξ and Pξ will denote
expectation and probability for the traps {Yjy }y∈Zd ,1≤j≤Ny .
The trapping problem described above has a long history in the physics and mathematics literature.
It originated in the modeling of a diffusion influenced chemical reaction A + B → C, and has since
1
Department of Mathematics, Fu-Jen Catholic University, 510 Chung Cheng Road, Hsinchuang , Taipei County 24205,
Taiwan. E-mail: [email protected]
2
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076 Singapore. Email:
[email protected]
1
found applications in many other contexts (see e.g. [MOBC04, YOLBK08, DGRS12] and the references
therein). When the traps are immobile, namely ξ(t, ·) = ξ(0, ·) for all t ≥ 0, and the particle motion
X is that of a simple symmetric random walk on Zd with exponential holding times, Donsker and
Varadhan’s large deviation analysis of the range of a random walk [DV79] shows that the survival
d
probability St decays asymptotically as exp{−(c + o(1))t d+2 }. The analogue in continuous space,
also known as a Brownian motion among Poissonian obstacles, has been treated comprehensively by
Sznitman in [S98]. On the other hand, when the particle is immobile, i.e. X(·) ≡ 0, and the traps
follow simple symmetric random walks with exponential holding times, then it can be calculated that
(see e.g. [DGRS12])

√

d = 1,
(c1 + o(1)) t



t
d = 2,
− log St = (c2 + o(1))
(1.3)

log t



(cd + o(1))t
d ≥ 3.
When both the particle and the traps follow simple symmetric random walks with exponential holding
times, Bramson and Lebowitz’s result in [BL91] suggest that − log St is of the same order as in
(1.3). For a discrete time version of the trapping problem, this was established rigorously by Moreau
et al in [MOBC03, MOBC04], and the continuous time result then follows easily by discrete time
approximation, as shown in [DGRS12]. Furthermore, it was found that in dimensions 1 and 2, even
the constant pre-factors coincide with c1 and c2 in (1.3), where X(·) ≡ 0.
The central idea in [MOBC03, MOBC04] is what the authors call the Pascal principle, which is
the statement that: among all deterministic trajectories X may follow up to time t, the constant
trajectory X(·) ≡ 0 maximizes the survival probability St . It was named the Pascal principle because
Pascal once made the philosophical assertion that “all misfortune of man comes from the fact that
he does not stay peacefully in his room”. In [MOBC03, MOBC04], the Pascal principle was proved
rigorously for a discrete time version of the trapping problem via an ingenious calculation, which leads
to an upper bound for St that is independent of the realization of X. In dimensions 1 and 2, such
an upper bound turns out to coincide with a matching lower bound up to the leading order with the
same pre-factor.
More recently, there have been studies in the physics literature of trapping problems where the
trap motions and the particle evolve sub-diffusively (see e.g. [YOLBK08, BAY09]). The sub-diffusive
motion is modeled by a random walk with i.i.d. holding times, which is sometimes called a continuous
time random walk, and it is non-Markovian unless the holding times are exponentially distributed.
More precisely, let (Zn )n≥0 be a discrete time random walk on Zd with increment distribution p(·), and
let (σi )i≥1 be i.i.d. holding times with distribution µ. Then we can define a continuous time random
walk (Y (t))t≥0 with
Y (t) = Zk
for t ∈ [Tk , Tk+1 ),
with Tk :=
k
X
σi for k ≥ 0.
(1.4)
i=1
If p(·) has finite second moments, and µ([x, ∞)) = c+o(1)
as x → ∞ so that µ is in the domain of
xα
attraction of an α-stable subordinator for some α ∈ (0, 1), then (Y (t))t≥0 is sub-diffusive in the sense
α
α
that Y (t) is of the order t 2 . Furthermore, as A → ∞, (Y (At)/(At) 2 )t≥0 converges in distribution to
a Brownian motion time-changed by the inverse of an α-stable subordinator (see e.g. [MS04]). Such a
subordinated Brownian motion is also known as a fractional kinetic process, which can arise naturally
from diffusion in a random medium (see e.g. [BaC06]). Assuming that the trap motions are subdiffusive with exponent α ∈ (0, 1) and the particle motion X is diffusive, it was found in [YOLBK08]
d
2
, − log St is of the order t d+2 , which is the
that there is a dynamic phase transition: when 0 < α < d+2
2
same as for the case when the traps are immobile; and when d+2
≤ α ≤ 1, − log St is of the order αd
2
in dimensions 1 and 2 (modulo a logarithmic correction in dimension 2), which is the same as for the
case when X(·) ≡ 0, while the exact order in dimensions d ≥ 3 is still undetermined. In [BAY09], the
2
analysis was extended to the case where both the traps and the particle are sub-diffusive, in which
case − log St is of the same order as for the case X(·) ≡ 0.
In the analysis of [YOLBK08, BAY09] for the trapping problem with sub-diffusive trap motion,
the Pascal principle was assumed to hold, which gives a bound on the survival probability by the case
where X(·) ≡ 0. However a rigorous proof of the Pascal principle was lacking in this case, because the
discrete time trapping problem, for which the Pascal principle was proved in [MOBC03, MOBC04],
cannot be used to approximate a continuous time model with non-Markovian trap motion. The
purpose of this note is therefore to give a rigorous proof of the Pascal principle in this case:
Theorem 1.1 [Pascal Principle for continuous time trapping] Let ξ be defined as in (1.1),
where the traps’ increment distribution p(·) is that of a simple symmetric random walk, and the holding
time distribution µ(·) is continuous. Then for any t ≥ 0 and any X(·) : [0, ∞) → Zd with locally finitely
many jumps, we have
St (X) := Pξ (τX,ξ > t) ≤ St (0) := Pξ (τ0,ξ > t),
(1.5)
where τX,ξ is defined in (1.2), with τ0,ξ for the case X(·) ≡ 0.
Our strategy is to convert the problem back into a discrete time trapping problem, albeit in a different
form from the trapping problem considered in [MOBC03, MOBC04]. As an intermediate step, Theorem 1.1 can be deduced from the following monotonicity result for the range of a simple symmetric
random walk.
Theorem 1.2 [Range of a random walk with insertion perturbation] Let (Z̄n )n≥0 be random
path on Zd such that (Z̄2k )k≥0 is a simple symmetric random walk, and Z̄2k = Z̄2k+1 for all k ≥ 0. Let
Rn (Z̄) := {Z̄i : 0 ≤ i ≤ n} denote the range of Z̄ up to time n, and let |Rn (Z̄)| denote its cardinality.
Then for any path (fn )n≥0 on Zd with f2k−1 = f2k for all k ∈ N, we have
E[|R2n (Z̄)|] ≤ E[|R2n (Z̄ + f )|]
for all n ∈ N.
(1.6)
Note that f and Z̄ never jump at the same time, therefore (1.6) amounts to the statement that: the
expected cardinality of the range of a simple symmetric random walk (up to time n) can only increase,
if deterministic jumps are inserted in its path.
Remark 1.3 For p(·) with p(0) ≥ 21 and symmetric in the sense that p(x) = p(−x) for all x ∈ Zd ,
Theorems 1.1–1.2 already follow from the results in [MOBC03, MOBC04], which we will explain in
Section 2. Theorem 1.1 addresses the important case of a simple symmetric random walk for which
p(0) = 0, while Theorem 1.2 is of independent interest. In dimension 1, we will prove Theorems 1.1–1.2
for the following more general class: p(·) is symmetric, p(n) ≥ p(n + 1) for all n ∈ N, and p(0) ≥ p(3).
We conjecture that Theorems 1.1–1.2 hold for all symmetric p(·) on Zd .
To give further background on the mathematical literature for the trapping problem with mobile
traps, we note that [DGRS12] gives an overview of results for the trapping problem on Zd where the
traps and the particle follow simple symmetric random walks with exponential holding times. The
Pascal principle of [MOBC03, MOBC04] was reviewed therein, and new results were obtained for the
quenched survival probability of the particle in a typical realization of the trap field. In continuous
space, the trapping problem with traps following independent Brownian motions has been studied
in [PSSS11] as a detection problem in a mobile communication network. The Pascal principle for
Brownian traps was subsequently established in [PS11] in a more general form, which was further
extended in [DSS11] to traps following general Lévy motions.
The rest of the paper is organized as follows. In Section 2, we will show how Theorem 1.1 follows
from Theorem 1.2, which can then be further reduced to a discrete time trapping problem that can
be compared to the setting in [MOBC03, MOBC04]. Theorem 1.2 will then be proved in Section 3 for
the simple symmetric random walk on Zd , and proved in Section 4 for a class of symmetric random
walks on Z.
3
2
Reduction to Random Walk Range and Discrete Time Trapping
Let us fix a realization of the particle motion X(·) : [0, ∞) → Zd with locally finitely many jumps, as
in Theorem 1.1. By integrating out the Poisson field ξ, we have
o
n X
Y
PYy (τX ≤ t) ,
(2.1)
exp{−1 + PYy (τX > t)} = exp −
St (X) = Pξ (τX,ξ > t) =
y∈Zd
y∈Zd
where PYy denotes probability for a trap Y starting at y ∈ Zd at time 0, and
τX := τX (Y ) := τ0 (Y − X) := inf{t ≥ 0 : Y (t) − X(t) = 0}.
(2.2)
Therefore (1.5) reduces to
X
PYy (τX ≤ t) ≥
X
PYy (τ0 ≤ t).
(2.3)
y∈Zd
y∈Zd
By translation invariance, this can be rewritten in terms of a single trap starting from the origin:
X
X
PY0 (τ−y (Y ) ≤ t),
PY0 (τ−y (Y − X) ≤ t) ≥
y∈Zd
y∈Zd
which is equivalent to
EY0 [|Rt (Y − X)|] ≥ EY0 [|Rt (Y )|],
(2.4)
where Rt (Y − X) := {Y (s) − X(s) : 0 ≤ s ≤ t} is the range of Y − X up to time t. By conditioning
on the times at which Y jumps, and using the fact that the jumps of Y and X almost surely do not
occur at the same time because the holding time distribution µ(·) is continuous, we observe that (2.4)
would follow if we show that the expected cardinality of the range of a simple symmetric random walk
up to any finite time n can only increase with the insertion of deterministic jumps. This is precisely
the content of Theorem 1.2, where the random walk jumps at even times and deterministic jumps are
inserted at odd times.
To give a unified treatment of Theorems 1.1–1.2, we reformulate Theorem 1.2 in terms of a discrete
time trapping problem. Alternatively we could derive this directly from (2.3) by conditioning on the
jump times of the traps. Note that analogous to the derivation of (2.4), we can rewrite (1.6) as
X
X
PZ̄
PZ̄
(2.5)
x (τ−f (Z̄) ≤ 2n) ≥
x (τ0 (Z̄) ≤ 2n),
x∈Zd
x∈Zd
where
τ−f := τ−f (Z̄) := min{n ≥ 0 : Z̄n = −fn }.
(2.6)
We can reverse the roles of traps and particle and think of the trajectory −f as a trap, with a particle
starting from every site on Zd . The LHS of (2.5) is then the expected number of particles killed by the
trap −f by time 2n (this is sometimes called a target problem in the physics literature, with −f being
the target). We can reformulate (2.5) in terms of simple symmetric random walks by contracting the
time intervals [2k, 2k + 1], k ≥ 0, into single time points. This results in a new trap field where at
each time i ≥ 0, there could be two sites in Zd which act as traps (see Figure 1 (b)). More precisely,
the new time-space trap field is given by
{(i, −f2i ), (i, −f2i+1 ) : i ≥ 0}.
Denote φi = −f2i . Since f2i+1 = f2i+2 , the trap field equals {(i, φi ), (i, φi+1 ) : i ≥ 0}. For a path
Z := (Zn )n≥0 in Zd , denote
τ̃φ := τ̃φ (Z) := min{n ≥ 0 : Zn = φn or φn+1 }.
Then we note that (2.5), and hence Theorem 1.2, is equivalent to the following:
4
(2.7)
(a)
(b)
Figure 1: The same function φ := (φi )i≥0 gives rise to two trap fields: (a) the model studied
in [MOBC03, MOBC04]; (b) the model considered in Prop. 2.1.
Proposition 2.1 [Pascal principle for discrete time trapping] Let PZ
x denote probability for a
simple symmetric random walk Z on Zd with Z0 = x. Then for any φ := (φi )i≥0 ∈ Zd , we have
X
X
PZ
for all n ∈ N.
(2.8)
PZ
x (τ̃0 ≤ n)
x (τ̃φ ≤ n) ≥
x∈Zd
x∈Zd
We now compare Proposition 2.1 with Moreal et al’s result in [MOBC03, MOBC04], where the
discrete time trapping problem considered therein corresponds to a time-space trap field {(i, φi ) : i ≥
0}, so that τ̃φ in (2.8) is replaced by τφ , defined as in (2.6). The analogue of (2.8) is then proved for any
random walk Z whose increment distribution p(·) is symmetric with p(0) ≥ 12 , and the corresponding
monotonicity result for the random walk range is (see e.g. [DGRS12, Sec. 2.4] for details)
Z
EZ
0 [|Rn (Z − φ)|] ≥ E0 [|Rn (Z)|].
(2.9)
Note that the perturbation on the random walk path Z is by adding a deterministic path −φ, instead
of inserting jumps as in Theorem 1.2. Since τ̃φ ≤ τφ and τ̃0 = τ0 , (2.8) with τ̃φ replaced by τφ gives a
stronger result, and therefore Theorems 1.1–1.2 must also hold for increment distributions p(·) which
are symmetric with p(0) ≥ 21 . This explains the first part of Remark 1.3. As pointed out in [MOBC04],
(2.9) cannot hold for the simple symmetric random walk on Zd because of periodicity. Indeed, if d = 1
n (Z−φ)|
and we let (φn )n≥0 take values alternately between 0 and 1, then limn→∞ |R|R
= 12 almost surely,
n (Z)|
because Z − φ can only visit even lattice sites.
3
The Simple Symmetric Random Walk on Zd
In this section, we will prove Proposition 2.1, from which Theorems 1.2 and 1.1 then follow. Let
Z := (Zn )n≥0 be a random walk on Zd with increment distribution p(·). Let pn (·) denote the n-step
increment distribution, i.e., pn (x) := PZ
0 (Zn = x), with p0 (x) = δ0 (x), and we set p−1 (x) := 0 for all
d
d
x ∈ Z . For n ≥ −1 and x ∈ Z , we denote
pn,n+1 (x) := pn (x) + pn+1 (x).
Proposition 2.1 then follows from the following two lemmas.
Lemma 3.1 If Z is a random walk whose transition probabilities satisfy
pn,n+1 (0) ≥ pn+1,n+2 (0)
pn,n+1 (0) ≥ pn,n+1 (x)
then (2.8) holds.
5
∀ n ≥ −1,
∀ n ≥ −1, x ∈ Zd ,
(3.1)
(3.2)
Lemma 3.2 If Z is a simple symmetric random walk on Zd , then (3.1)–(3.2) hold.
Proof of Lemma 3.1. Our proof of (2.8) is based on a modification of the argument in [MOBC03,
MOBC04] for the analogue of (2.8), where τ̃φ is replaced by τφ . The argument in [MOBC03, MOBC04]
is based a simple, but ingenious calculation, which we now briefly recall.
To prove
X
X
PZ
for all n ∈ N,
(3.3)
PZ
x (τ0 ≤ n)
x (τφ ≤ n) ≥
x∈Zd
x∈Zd
Moreau et al require the random walk to satisfy
pn (0) ≥ pn+1 (0)
∀ n ≥ 0,
(3.4)
d
pn (0) ≥ pn (x)
∀ n ≥ 0, x ∈ Z ,
(3.5)
which is easily seen to hold by Fourier transform if p(·) is symmetric and p(0) ≥ 12 . They then use the
identity
n X
X
X
Z
Px (Zn = φn ) =
PZ
(3.6)
1=
x (τφ = i)pn−i (φn − φi ),
i=0 x∈Zd
x∈Zd
where the event {Zn = φn } has been decomposed according to the value of τφ (note that Zn = φn
implies τφ ≤ n). Applying (3.5) in (3.6), one obtains
n X
X
PZ
x (τφ = i)pn−i (0) ≥ 1 =
n X
X
PZ
x (τ0 = i)pn−i (0).
i=0 x∈Zd
i=0 x∈Zd
P
Define Wφ (−1) := 0, and Wφ (i) := Wφ (i − 1) + x∈Zd PZ
x (τφ = i) for i ≥ 0. Define W0 (·) analogously.
After rearranging terms, the above inequality can then be rewritten as
Wφ (n) − W0 (n) ≥
n−1
X
pn−i−1 (0) − pn−i (0)
Wφ (i) − W0 (i) .
(3.7)
i=0
Since pn−i−1 (0) − pn−i (0) ≥ 0 for all 0 ≤ i ≤ n − 1 by (3.4), one obtains Wφ (n) − W0 (n) ≥ 0 for all
n ≥ 0 by induction, which is precisely (3.3).
Note that the simple symmetric random walk does not satisfy (3.4) due to periodicity, and the
above argument fails, as does (3.3). However, we will show that (2.8) holds. To remedy the above
argument and take into account the periodicity of the simple symmetric random walk, we note that in
the trapping problem we are considering, each trap φi is present at both time i and i − 1. Therefore
instead of only considering the constraint Zn = φn as in the decomposition (3.6), we can also consider
the constraint Zn−1 = φn . The event Zn−1 = φn implies τ̃φ ≤ n − 1, which allows us to obtain
decompositions analogous to (3.6):
X
1=
PZ
x (Zn = φn ) =
x∈Zd
1=
X
PZ
x (Zn−1
= φn ) =
n X
X
EZ
x 1{τ̃φ =i} pn−i (φn − Zi ) ,
i=0 x∈Zd
n−1
XX
(3.8)
EZ
x
1{τ̃φ =i} pn−1−i (φn − Zi ) .
i=0 x∈Zd
x∈Zd
Note that given τ̃φ = i, Zi ∈ {φi , φi+1 }. Recall that p−1 (·) := 0, adding the two equalities then gives
2=
n X
X
EZ
x 1{τ̃φ =i} pn−1−i (φn − Zi ) + pn−i (φn − Zi ) .
i=0 x∈Zd
This identity holds in particular for the case φ ≡ 0. We can now apply condition (3.2) to obtain
n X
X
pn−i−1,n−i (0)PZ
x (τ̃φ
= i) ≥ 2 =
i=0 x∈Zd
n X
X
i=0 x∈Zd
6
pn−i−1,n−i (0)PZ
x (τ̃0 = i).
(3.9)
P
Define W̃φ (−1) := 0, and W̃φ (i) := W̃φ (i − 1) + x∈Zd PZ
x (τ̃φ = i) for i ≥ 0. Define W̃0 (·) analogously.
Rearranging terms in the above inequality then gives the following analogue of (3.7):
W̃φ (n) − W̃0 (n) ≥
n−1
X
pn−i−2,n−i−1 (0) − pn−i−1,n−i (0) W̃φ (i) − W̃0 (i) .
(3.10)
i=0
By assumption (3.1), pn−i−2,n−i−1 (0) − pn−i−1,n−i (0) ≥ 0 for all 0 ≤ i ≤ n − 1, which implies that
W̃φ (n) − W̃0 (n) ≥ 0 for all n ≥ 0 by induction. This is precisely (2.8).
ik·Z1 ] =
Proof of Lemma 3.2. For k = (k1 , · · · , kd ) ∈ [−π, π]d , let ψ(k) := EZ
0 [e
Z
1
e−ik·x ψ n (k)dk
∀ x ∈ Zd .
pn (x) =
(2π)d [−π,π]d
1
d
Pd
i=1 cos ki .
Then
(3.11)
By periodicity, pn (0) = 0 for all n odd, while the above identity shows that p2n (0) is decreasing in n.
These facts readily imply (3.1).
Using (3.11), we can easily deduce (3.2) for the case n is even. However when n is odd, (3.2)
does not seem to admit a simple proof using characteristic functions. Instead we will proceed via a
P
coupling argument. Assume that n ∈ N is odd. If x = (x1 , · · · , xd ) ∈ Zd is such that |x|1 := di=1 |xi |
is even, then pn,n+1 (x) = pn+1 (x) ≤ pn+1 (0) = pn,n+1 (0), where the inequality can be deduced from
(3.11). If |x|1 is odd, then pn,n+1 (x) = pn (x). Also note that pn,n+1 (0) = pn+1 (0) = pn (e1 ), where
e1 = (1, 0, · · · , 0) ∈ Zd . Therefore to complete the proof of (3.2), it only remains to show that
∀ n ∈ N, x ∈ Zd with n and |x|1 both odd.
pn (x) ≤ pn (e1 )
(3.12)
Y
By the symmetry of the random walk, pn (x) = PX
x (Xn = 0) and pn (e1 ) = Pe1 (Yn = 0) for two simple
symmetric random walks X and Y , starting respectively at x and e1 . We will construct a coupling of
X and Y such that when Xn = 0, we also have Yn = 0.
By symmetry, we may assume without loss of generality that xi ≥ 0 for all 1 ≤ i ≤ d. Since |x|1 is
odd, we may further assume that xi is odd for 1 ≤ i ≤ 2m − 1 for some m ∈ N, and xi is even for 2m ≤
(i)
(i)
i ≤ d. We will group (xi )2≤i≤2m−1 into m − 1 pairs: (x2 , x3 ), · · · , (x2m−2 , x2m−1 ). Let Xk and Yk
(1)
(d)
denote respectively the i-th component of Xk and Yk ∈ Zd . We will couple X = (Xk , · · · , Xk )k≥0
(1)
(d)
and Y = (Yk , · · · , Yk )k≥0 in such a way that:
(i)
(i)
(1) at each time k ≥ 0, |Xk | ≥ |Yk | for all 1 ≤ i ≤ d;
(i)
(i)
(i)
(i)
(i)
(i)
(2) if Xk = Yk , then Xk0 = Yk0 for all k 0 ≥ k;
(i)
(i)
(mod 2), then Xk0 ≡ Yk0 (mod 2) for all k 0 ≥ k;
(3) if Xk ≡ Yk
(2i)
(4) for each 1 ≤ i ≤ m − 1, Xk
(2i)
+ Yk
(2i+1)
≡ Xk
(2i+1)
+ Yk
(mod 2) for all k ≥ 0.
Y
Such a coupling clearly would imply PX
x (Xn = 0) ≤ Pe1 (Yn = 0). In words, our coupling will be such
(i)
(i)
that: if Xk = Yk for some 1 ≤ i ≤ d, then we couple the jumps of X and Y in the i-th coordinate
(i)
(i)
(i)
(i)
so that X· = Y· for all later times; if Xk 6= Yk but they have the same parity, then we couple the
(i)
jumps in the i-th coordinate so that X·
first time
(i)
X·
(i)
= Y· ; if
(i)
Xk
and
i0
(i)
Yk
and Y·
(i)
(i)
move as mirror images across
(i)
Xk +Yk
2
, until the
do not have the same parity, then 2 ≤ i ≤ 2m − 1, and by (4),
(i0 )
(i0 )
we can find another coordinate such that Xk and Yk also have different parities, and we couple
the jumps in the i-th and i0 -th coordinates such that the jump of X in the i-th coordinate coincides
with the jump of Y in the i0 -th coordinate and vice versa. The precise formulation is as follows.
Assume that X and Y have been coupled up to time k, and properties (1)–(4) have not been
(i)
(i)
violated. Then Xk ≡ Yk (mod 2) for i = 1 and 2m ≤ i ≤ d. Given Xk+1 − Xk ∈ {ei , −ei } for some
1 ≤ i ≤ d:
7
(i)
(i)
(i)
(i)
(i)
(i)
(a) if Xk = Yk , then we set Yk+1 := Yk + (Xk+1 − Xk );
(b) if Xk ≡ Yk
(i)
(i)
(mod 2) and Xk 6= Yk , then we set Yk+1 := Yk − (Xk+1 − Xk );
(c) if Xk 6≡ Yk (mod 2), then i ∈ {2j, 2j + 1} for some 1 ≤ j ≤ m − 1. Let i0 := {2j, 2j + 1}\{i}.
Then we set Yk+1 := Yk + ei0 hei , Xk+1 − Xk i where h, i denotes inner product on Rd .
Note that Y is distributed as a simple symmetric random walk. Conditions (2)–(4) remain un-violated
at time k +1. We now check that (1) also holds at time k +1. Indeed, jumps as specified in (a) does not
(i)
(i)
(i)
(i)
violate (1). For a jump as specified in (b), by our assumptions |Xk | ≥ |Yk | and Xk ≡ Yk (mod 2),
(i)
(i)
(i)
(i)
(i)
(i)
either Xk = −Yk , in which case Xk+1 = −Yk+1 ; or |Xk | ≥ |Yk | + 2, in which case we must have
(i)
(i)
(i)
(i)
|Xk+1 | ≥ |Yk+1 |. For a jump as specified in (c), we must have |Xk | ≥ 1 + |Yk |, and by (4), also
(i0 )
(i0 )
|Xk | ≥ 1 + |Yk |. Since X and Y do not make jumps in the same coordinate, (1) must hold at time
k + 1 as well. This verifies that the coupling given in (a)–(c) satisfies conditions (1)–(4) at all times,
which concludes the proof of the proposition.
4
A Class of Symmetric Random Walks on Z
In this section, we prove Proposition 2.1 for a class of symmetric random walks on Z, which includes
the simple symmetric random walk. Theorems 1.1–1.2 then also hold for this class of walks.
Proposition 4.1 Let Z be a random walk on Z with increment distribution p(·), which satisfies p(k) =
p(−k) and p(k) ≥ p(k + 1) for all k ∈ N, and p(0) ≥ p(3). Then (2.8) holds.
Note that when p(0) > 0, (3.1) fails for n = −1, therefore Proposition 4.1 cannot be deduced from
Lemma 3.1. Our proof of Prop. 4.1 will be based on induction together with a suitable notion of
symmetric domination of measures on Z. The argument is inspired by the one used in [DSS11] to
prove a rearrangement inequality for Lévy processes, which can be regarded as a generalized version
of the Pascal principle for a trapping problem, where traps follow independent Lévy motions. The
main difference here is that we use a weaker notion of symmetric domination, which allows us to prove
Prop. 4.1 for p(·) with p(0) < p(1). Unfortunately such an argument does not seem to work on the
lattice in higher dimensions, although we conjecture that (2.8), and hence Theorems 1.1–1.2, hold for
all symmetric p(·), in all dimensions.
Proof of Prop. 4.1. For n ≥ 0 and x ∈ Zd , define
X
uφn (x) := 1 − vnφ (x) :=
PZ
z (Zn = x, τ̃φ > n),
(4.1)
z∈Z
and let u0n (x) = 1 − vn0 (x) denote the case φ· ≡ 0. We can interpret uφn (x) as the expected number of
particles alive at time n at position x, if initially one particle starts at every site in Z and gets killed
when it encounters the time-space trap field {(i, φi ), (i, φi+1 ) : i ≥ 0}. Then (2.8) is equivalent to
X
X
vn0 (x) ≤
vnφ (x).
(4.2)
x∈Z
x∈Z
We will prove (4.2) by proving that vnφ symmetrically dominates vn0 , denoted by vnφ vn0 , in the sense
that
X
X
vn0 (x) ≤
vnφ (x0 + x)
∀ k ≥ 0, x0 ∈ Z.
(4.3)
|x|≥k
|x|≥k
We will prove vnφ vn0 , for all n ≥ 0, by induction. It is easily verified that v0φ v00 . Now let us
φ
0 .
assume that vnφ vn0 for some n ≥ 0 and try to prove that vn+1
vn+1
8
Note that we have the following recursion relation for uφn :


0
if x ∈ {φn+1 , φn+2 },

X
φ
uφn+1 (x) =
un (y)p(x − y)
if x ∈
/ {φn+1 , φn+2 }.


(4.4)
y∈Z
Since uφn (φn+1 ) = 0, we have
φ
vn+1
(x)
=1−
uφn+1 (x)
=



if x ∈ {φn+1 , φn+2 },
1
p(x − φn+1 ) +


X
vnφ (y)p(x − y)
if x ∈
/ {φn+1 , φn+2 }.
(4.5)
y6=φn+1
Similarly,
0
vn+1
(x)
=



p(x) +


1
X
if x = 0,
vn0 (y)p(x − y)
(4.6)
if x 6= 0.
y6=0
φ
0 , with k = 1, which admits special cancelations
We first verify the analogue of (4.3) for vn+1
and vn+1
and explains why we do not need to assume p(0) ≥ p(1). The case k = 0 will be treated along the
way.
φ
Note that the second line in (4.5) gives a lower bound on vn+1
(x) for all x ∈ Z, therefore
X
φ
(φn+1 + x) ≥
vn+1
|x|≥1
X
0
vn+1
(x)
|x|≥1
=
X
p(x) +
XX
|x|≥1
|x|≥1 y6=x
X
XX
p(x) +
|x|≥1
p(y)vnφ (φn+1 + x − y) = F1 (0) +
X
F1 (z)vnφ (φn+1 + z),
z6=0
p(y)vn0 (x
− y) = F1 (0) +
|x|≥1 y6=x
X
F1 (z)vn0 (z),
z6=0
where we made the change of variable z = x − y, and introduced the notation
X
X
X
X
F1 (z) :=
p(z + x) =
p(z − x) =
p(y) =
p(y).
|x|≥1
|x|≥1
|y−z|≥1
|y+z|≥1
By the symmetry of p(·), F1 (z) = F1 (−z), and by the assumption that p(y) is decreasing in y ≥ 1, we
see that F1 (z) is increasing in z ≥ 1. Indeed, for z ≥ 1,
X
X
F1 (z + 1) − F1 (z) =
p(y) −
p(y) = p(z) − p(z + 1) ≥ 0.
|y−z−1|≥1
|y−z|≥1
By a layer-cake representation for F1 , i.e., writing F1 (z) = F1 (1) +
X φ
X
0
vn+1 (φn+1 + x) −
vn+1
(x)
|x|≥1
≥
X
P|z|
i=2 (F1 (i)
− F1 (i − 1)), we have
|x|≥1
F1 (z)(vnφ (φn+1 + z) − vn0 (z))
(4.7)
z6=0
= F1 (1)
X
(vnφ (φn+1 + z) − vn0 (z)) +
∞
X
X
(F1 (i) − F1 (i − 1))
(vnφ (φn+1 + z) − vn0 (z)),
i=2
|z|≥1
|z|≥i
which is non-negative because vnφ vn0 by the induction assumption.
φ
0 (0) = 1, (4.7) implies
Note that since vn+1
(φn+1 ) = vn+1
X
φ
vn+1
(x) ≥
x
X
x
9
0
vn+1
(x),
which in turn implies that
X
φ
vn+1
(x0 + x) ≥
X
0
vn+1
(x)
∀ x0 ∈ Z,
(4.8)
|x|≥1
|x|≥1
φ
0 (0) = 1. This verifies the analogue of (4.3) for v φ
0
because vn+1
(x0 ) ≤ vn+1
n+1 and vn+1 , with k = 0, 1.
For k ≥ 2 and x0 ∈ Z, by (4.5) and a change of variable z := x − y, we have
XX
X
X φ
vn+1 (x0 + x) ≥
p(y)vnφ (x0 + x − y) =
Fk (z)vnφ (x0 + z),
|x|≥k y∈Z
|x|≥k
z∈Z
where
X
Fk (z) :=
X
p(y) =
p(y).
|y−z|≥k
|y+z|≥k
Similarly, by (4.6) and the fact that vn0 (0) = 1, we have
X
X
0
vn+1
(x) =
Fk (z)vn0 (z).
z∈Z
|x|≥k
By the symmetry of p(·), Fk (z) = Fk (−z), and for z ≥ 0,
X
X
Fk (z + 1) − Fk (z) =
p(y) −
p(y) = p(z + 1 − k) − p(z + k).
|y−z−1|≥k
|y−z|≥k
If z + 1 − k 6= 0, then 1 ≤ |z + 1 − k| < z + k, and hence the above difference is non-negative by
the assumption that p(·) is symmetric and p(y) is decreasing in y ≥ 1. If z + 1 − k = 0, then the
above difference is still non-negative because we are considering the case k ≥ 2 and we assumed that
p(0) ≥ p(3). Therefore Fk is increasing in z ≥ 0. As in (4.7), we can write
X φ
X
0
vn+1 (x0 + x) −
vn+1
(x)
|x|≥k
≥
X
|x|≥k
Fk (z)(vnφ (x0 + z) − vn0 (z))
(4.9)
z∈Z
= Fk (0)
X
z∈Z
(vnφ (x0 + z) − vn0 (z)) +
∞
X
(Fk (i) − Fk (i − 1))
i=1
X
(vnφ (x0 + z) − vn0 (z)),
|z|≥i
which again is non-negative because vnφ vn0 by the induction assumption. This completes the
φ
0
vn+1
defined as in (4.3). Proposition 4.1 then follows.
induction proof that vn+1
Acknowledgement L.-C. Chen is supported by research grant 99-2115-M-030-004-MY3 from the
National Science Council of Taiwan. He thanks the National University of Singapore for support
during research visits. R. Sun is supported by research grant R-146-000-119-133 from the National
University of Singapore. He thanks Academia Sinica of Taiwan for support during research visits.
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