Non-homogeneous random walks
Andrew Wade
Department of Mathematics and Statistics
University of Strathclyde
November 2010
Joint work with Iain MacPhee, Mikhail Menshikov (Durham)
and Marina Vachkovskaia (University of Campinas)
1 Talk outline
2 Classical (spatially homogeneous) random walks
Model 1: Pearson–Rayleigh walk
Model 2: Simple random walk
The general homogeneous random walk
Recurrence/transience dichotomy
3 Non-homogeneous random walks
General setting
Unruly behaviour
Asymptotically zero drift regime
Model: Centrally biased walks
Lamperti’s recurrence theorem
Angular asymptotics
Rate of escape
4 Some ideas of proofs
5 Concluding remarks
Outline
• The random walk is a fundamental model (or class of
models) in probability theory and beyond. It displays deep
mathematical properties and enjoys broad application
across the sciences.
• Generally speaking, a random walk is a stochastic process
modelling the motion of a particle (or random walker) in
space. The particle’s motion is modelled by a series of
random jumps at discrete instants in time.
• Fundamental questions for these models involve the
long-time asymptotic behaviour of the walker.
Outline (cont.)
• For the purposes of this talk, we interpret ‘random walk’
quite broadly, to include any discrete-time Markov process
on Rd with some sense of ‘locality’ (that is, the jumps are
not too big). We won’t be more precise till later; we start by
recalling two classical random walk models.
• These two models, and much of classical random walk
theory in general, assume spatial homogeneity, i.e., the
distribution of the jumps of the walk does not depend on
the present position of the walker.
Outline (cont.)
• Most of this talk will concentrate on the case where this
assumption does not hold, where the walks are
non-homogeneous. Such models are often much more
realistic for applications, but also present harder
mathematical problems.
• We discuss some fundamental work of J. Lamperti from
the early 1960s and then some more recent work with my
collaborators I. MacPhee and M. Menshikov (University of
Durham).
Karl Pearson and Lord Rayleigh
• Pearson (1857–1936; left) was a statistical pioneer, who
founded Biometrika and coined “histogram”. . .
• Rayleigh (J.W. Strutt, 1842–1919; right) was a
Nobel-prize-winning physicist, who explained why the sky
is blue (Rayleigh scattering). . .
Karl Pearson
• Pearson published mainly on statistics and biometrics, but
also on elasticity, aether physics, Germanic literature,. . . .
• Credited with the first use of the phrase “random walk” in a
letter to Nature in 1905.
Pearson’s letter to Nature
Rayleigh’s response
Pearson–Rayleigh random walk
We may as well take the step size ℓ = 1. Then the
Pearson–Rayleigh random walk is the stochastic process
X0 , X1 , X2 , . . . on R2 defined as follows.
• Start at X0 = 0, the origin of R2 ;
• Given Xn , take Xn+1 to be uniformly distributed on the
unit-radius circle with centre Xn .
This is a Markov process. The model generalizes readily to
higher (or lower!) dimensions: take Xn+1 to be uniform on the
unit sphere in Rd centred at Xn .
Pearson–Rayleigh random walk (cont.)
Figure: Jump of the Pearson–Rayleigh walk.
Random migration
Pearson’s interest in the random walk was as a model of
“random migration”, or the evolution of territories of e.g. insect
colonies. His investigations led to a 1906 monograph:
Microbe locomotion
The Pearson–Rayleigh walk has also been used to model the
locomotion of microscopic organisms.
Simulation of 104 steps of the Pearson–Rayleigh walk on R2 .
George Pólya
• While walking in a Zurich
park in 1914, Pólya
encountered the same
couple several times on
his walk.
• He asked: was this, after
all, so unlikely?
• Some time later Pólya
• George Pólya
(1887–1985).
published his paper on an
idealized version of the
problem, now known as
simple random walk
(SRW).
Simple random walk
Consider a random walker on the d-dimensional integer lattice
Zd .
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
b
At each step, the walker jumps
to one of the 2d neighbouring
sites of the lattice, choosing
uniformly at random from
each.
Pólya’s question: What is the
probability that the walker
eventually returns to his
starting point?
Pólya’s question
Simulations are not that helpful! Here is a simulation of 104
steps of SRW on Z2 .
Pearson–Rayleigh and SRW
Both the Pearson–Rayleigh walk and the SRW can be
formulated as processes defined as partial sums of i.i.d.
random vectors: that is
Xn =
n
X
Yi ,
i=1
where Y1 , Y2 , . . . are i.i.d. and are the jumps of the process.
• For the Pearson–Rayleigh walk, Y1 is uniform on the unit
sphere in Rd .
• For SRW, Y1 uniformly distributed on {±e1 , . . . , ±ed },
where ei are the orthonormal basis vectors for Rd .
Each walk has zero drift in the sense that
E[Xn+1 − Xn | Xn ] = E[Yn+1 ] = 0.
The general homogeneous random walk
Let Y1 , Y2 , . . . be i.i.d. random vectors on Rd . We can define a
general homogeneous random walk Xn by taking the partial
sums
n
X
Yi .
Xn =
i=1
The mean drift of the walk is
µ = E[Xn+1 − Xn | Xn ] = E[Yn+1 ] = E[Y1 ],
which we assume is well defined; in particular the drift does not
depend on the current position.
Recurrence and transience
A fundamental property of a random walk is whether it is
recurrent or transient.
What is the probability that the walker eventually returns to
(some neighbourhood of) his starting point? Call this probability
p0 .
The random walk is transient if p0 < 1 and recurrent if p0 = 1.
Under mild conditions (irreducibility) a transient walk eventually
never returns to any compact subset of Rd , i.e., kXn k → ∞ a.s.,
while a recurrent walk has lim inf kXn k < ∞ a.s. but
lim sup kXn k = ∞ a.s..
Recurrence for homogeneous random walks
Consider a homogeneous random walk on Rd with
E[Xn+1 − Xn | Xn ] = µ. (For instance, with uniformly bounded
jumps.)
If µ 6= 0, the strong law of large numbers implies that
Xn
→ µ, a.s.,
n
and in particular, the walk is transient.
The zero drift case (µ = 0) is more subtle.
Recurrence for homogeneous random walks (cont.)
Theorem
Under mild conditions, a zero-drift homogeneous random walk
on Rd is
• recurrent for d = 1 or d = 2;
• transient for d ≥ 3.
A version of this theorem was first proved by Pólya (1921) for
simple random walk.
• For d ∈ {1, 2} the walker returns to any compact set
infinitely often.
• On the other hand, if d ≥ 3 the walker returns to any
compact set only finitely often.
“A drunk man will find his way home, but a drunk bird may get
lost forever.” —Shizuo Kakutani
Three questions
1 In the case d = 1 or d = 2, and if we allow the drift to
depend upon the current location, can we interpolate
between transience (µ 6= 0) and recurrence (µ = 0)? If so,
where is the phase transition?
2 What happens if we have zero drift but the walk is not
spatially homogeneous? Which properties of
homogeneous random walk are essential to the above
recurrence theorem?
3 How much do we need to perturb a 2-dimensional random
walk to make it transient? Or a 3-dimensional random walk
to make it recurrent? Or positive-recurrent?
Non-homogeneous random walk
The 3 questions above lead us to consider random walks that
are not spatially homogeneous. In this case the distribution of
the jump Xn+1 − Xn , given Xn = x ∈ Rd , is a function of x.
This generalizes the homogeneous case. In the general setting,
Xn is not a sum of independent random vectors.
The mean drift is now a function of the position
µ(x) = E[Xn+1 − Xn | Xn = x];
again we assume enough regularity so that this is well-defined.
Conditions for recurrence?
We are given a non-homogeneous random walk on Rd . For
simplicity, suppose that the jumps are uniformly bounded. If we
impose spatial homogeneity, then the recurrence classification
depends only on the drift µ(x) ≡ µ.
What about without spatial homogeneity?
Let’s fix attention on d = 2. Then, in the homogeneous case,
we know that the walk is recurrent if µ = 0 but transient if µ 6= 0.
Question: In the non-homogeneous case, is µ(x) = 0 sufficient
for recurrence in d = 2?
Unruly behaviour of zero-drift walks
Answer: No.
In the non-homogeneous case, the drift is no longer enough
information to decide about recurrence.
Theorem
Let Xn be a non-homogeneous random walk with zero drift, i.e.,
µ(x) = 0 for all x ∈ Rd . There exist such walks that are
• transient in d = 2;
• recurrent in d ≥ 3.
This result was explained to me by Mikhail Menshikov but may
be “well known”.
Elliptical random walk
Here is an example of the previous theorem in d = 2.
Modify the Pearson–Rayleigh
walk so that, given Xn , Xn+1 is
distributed (uniformly with
respect to the standard
parametrization) on an ellipse
centred at Xn and aligned so
that the minor axis is in the
direction of the vector Xn .
This zero-drift non-homogeneous random walk in R2 is
transient.
Elliptical random walk simulation
Conditions for recurrence? (cont.)
The extra regularity condition we can use to restore “normal”
behaviour to zero-drift non-homogeneous random walks is a
fixed covariance structure for the jumps.
Without loss of generality we can assume that Xn+1 − Xn has a
diagonal covariance structure, not depending on the current
position, i.e., for all x ∈ Rd ,
Cov[Xn+1 − Xn | Xn = x] = I,
the d-dimensional identity matrix.
We can do this because any zero-drift walk with a fixed
covariance structure can be transformed a zero-drift walk with
diagonal covariance by an affine map.
Affine map between fixed covariances
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
b
x −x
(x1 ,x2 )7→( 1 2 2 ,x2 )
−→
bc
bc
b
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
Recurrence for zero-drift random walks
Theorem (Lamperti, 1960)
Let Xn be a non-homogeneous random walk on Rd (d ≥ 1) with
zero drift and fixed covariance structure. Then Xn is
• recurrent for d ∈ {1, 2};
• transient for d ≥ 3.
In fact, Lamperti showed that the zero drift condition is much
stronger than is necessary for this result. We can allow
asymptotically zero drift: the theorem still holds if
kµ(x)k = o(kxk−1 ),
as kxk → ∞.
Asymptotically zero drift
Lamperti published a series of pioneering papers in the early
1960s investigating the asymptotically zero drift regime
(µ(x) → 0 as kxk → ∞) which is the natural setting in which to
probe the recurrence-transience transition.
A zero drift non-homogeneous random walk on Rd can always
be made recurrent or transient (whichever is desired) by an
asymptotically small perturbation of the drift field.
More precisely, changing the drift µ(x) by O(kxk−1 ) is sufficient
to achieve this.
Centrally biased random walk
A natural model has a drift field that points away from (or
towards) the origin. Specifically, suppose that
µ(x) =
c
x̂,
kxkα
where x̂ is a unit vector in the x-direction and c ∈ R, α > 0 are
fixed parameters.
The drift is asymptotically zero in that kµ(x)k → 0 as kxk → ∞.
This model is known as the centrally biased random walk. Such
walks were studied by Gillis and, more generally, by Lamperti.
What is the recurrence/transience behaviour of such a model?
Example of centrally biased random walk
Consider a modification of the
Pearson–Rayleigh walk.
Given Xn , take Xn+1 to be
uniform on the unit-radius
sphere in Rd centred not at Xn
but at Xn + ckXn k−α X̂n .
Simulations
Here is a simulation of 104 steps with d = 2, c = 1, and
α = 1/2. Scale ≈ 600.
Simulations
Here is a simulation of 104 steps with d = 2, c = 2, and α = 1.
Scale ≈ 200.
Simulations
Here is a simulation of 104 steps with d = 2, c = −1/2, and
α = 1. Scale ≈ 80.
Lamperti’s recurrence theorem
Lamperti (1960) showed that for α > 1, the perturbation in the
drift field is negligible, and the zero-drift recurrence theorem
holds. Thus α > 1 is known as the subcritical regime.
For α < 1, the perturbation dominates and the walk is (i)
transient if c > 0; and (ii) positive-recurrent if c < 0. So α < 1 is
known as the supercritical regime.
The case α = 1 (mean drift O(kxk−1 )) turns out to be critical.
Lamperti’s recurrence theorem (cont.)
Theorem (Lamperti 1960, 1963)
Consider a centrally biased random walk on Rd (d ≥ 1) with
mean drift ckxk−1 x̂ at x ∈ Rd . Then under mild regularity
conditions there exist c0 , c1 , c2 such that the walk is
• transient for c > c0 ;
• recurrent for c < c1 ;
• positive-recurrent for c < c2 .
Harris (1952) obtained a very special case of this theorem for
nearest-neighbour random walks on Z.
Diffusive behaviour
For all the walks covered by the previous theorem, the walk
scales at most diffusively, even in the transient case.
More precisely, a.s., for all but finitely many n,
max Xm ≤ n1/2 × log factors.
0≤m≤n
In the transient case (under some additional conditions) one
has a.s., for all but finitely many n,
max Xm ≥ n1/2 × log factors :
0≤m≤n
see Menshikov, Vachkovskaia & W. (2008), and compare the
Dvoretzky–Erdős theorem for transient Brownian motion.
In the positive-recurrent case the behaviour is sub-diffusive,
i.e., max0≤m≤n Xm is of order nγ for some γ ∈ (0, 1/2).
Angular asymptotics
The recurrence behaviour of the centrally-biased random walk
in the (critical) case α = 1, as well as the walk’s scaling
behaviour, depends crucially on the constant c in the drift.
In contrast, a property that
does not depend, to first order,
on c (or even the direction of
the drift field) is the angular
behaviour.
Let τθ be the first hitting time
by the walk of a fixed cone
with apex at 0 and half-angle
θ ∈ (0, π).
Critical case
Theorem (MacPhee, Menshikov & W. 2010)
For any non-homogeneous walk on Rd (d ≥ 2) with mean drift
at x of magnitude O(kxk−1 ), for any cone of half-angle θ,
τθ < ∞ a.s..
So although the walk can be transient, or positive-recurrent,
and can be diffusive or sub-diffusive, it will visit any cone
infinitely often.
This result is sharp in the sense that a drift field of larger
magnitude can lead to different behaviour.
Even the µ(x) = 0 case of this result extends previous work
(e.g. potential theory approach of Varopoulos).
Subcritical case: Analogue of Spitzer’s theorem
Theorem (MacPhee, Menshikov & W. 2010)
For any planar non-homogeneous walk with mean drift at
x ∈ R2 of magnitude o(kxk−1 ) and a diagonal covariance
structure, for any cone of half-angle θ, then
(
π
< ∞ if p < 4θ
p
E[τθ ]
.
π
= ∞ if p > 4θ
For zero-drift Brownian motion, this result is due to Spitzer
(1958).
The homogeneous random walk case of this result follows
(under stronger conditions) by work of Varopoulos (1999). Here
we show that the result extends to non-homogeneous random
walks, provided the drift decays rapidly enough.
Supercritical case: limiting direction
In the supercritical case α < 1, we can give the rate of escape
as well as showing that the walk has a limiting direction.
Theorem (MacPhee, Menshikov & W. 2010; Menshikov &
W. 2010)
Let Xn be a centrally biased random walk on Rd (d ≥ 1) with
mean drift ckxk−α x̂ where c > 0 and α ∈ (0, 1). Under mild
conditions, a.s.,
1
1
n− 1+α Xn → (c(1 + α)) 1+α U,
where U is a random unit vector in Rd .
Here the transience is super-diffusive
The one-dimensional version of this result strengthens and
generalizes earlier results of Lamperti (1962) and Voit (1992).
Martingale methods
Lamperti proved his recurrence theorem by considering the
one-dimensional process Zn = kXn k. Lamperti then used
martingale ideas to classify the behaviour of Zn . Note that Zn is
not in general a Markov process.
Our results also use martingale ideas, rather more complicated
than Lamperti’s since the angular asymptotics depend on
higher-dimensional behaviour, while the recurrence/transience
behaviour is essentially one-dimensional.
An advantage of the martingale approach is that the Markov
property is not essential to the proofs. The martingale
approach gives an “easy” proof of Pólya’s theorem that
generalizes broadly.
Concluding remarks
• Instead of working with random walks we could work with
continuous processes (diffusions) instead. Work in this
direction seems to have begun only recently, e.g.,
DeBlassie and Smits (2007).
• A major open problem is to study the tails of τθ for some
“reasonable” walks with kµ(x)k = O(kxk−1 ).
• Processes with drift O(x −1 ) occur more often than one
might expect. If Xn is simple random walk on Zd , then kXn k
is such a (non-Markov) process. Another example is the
stochastic billiards model of Menshikov, Vachkovskaia &
W. (2008).
References
•
D. D E B LASSIE AND R. S MITS, The influence of a power law drift on the exit time
of Brownian motion from a half-line, Stochastic Process. Appl. 117 (2007)
629–654.
•
T.E. H ARRIS, First passage and recurrence distributions, Trans. Amer. Math.
Soc. 73 (1952) 471–486.
•
J. L AMPERTI, Criteria for the recurrence or transience of stochastic processes I,
J. Math. Anal. Appl. 1 (1960) 314–330.
•
J. L AMPERTI, A new class of probability limit theorems, J. Math. Mech. 11 (1962)
749–772.
•
J. L AMPERTI, Criteria for stochastic processes II: passage-time moments, J.
Math. Anal. Appl. 7 (1963) 127–145.
•
I.M. M AC P HEE , M.V. M ENSHIKOV, AND A.R. WADE, Angular asymptotics for
multi-dimensional non-homogeneous random walks with asymptotically zero
drifts, Markov Processes Relat. Fields 16 (2010) 351–388.
•
I.M. M AC P HEE , M.V. M ENSHIKOV, AND A.R. WADE, Moments of exit times from
wedges for non-homogeneous random walks with asymptotically zero drifts.
Preprint arxiv.org/abs/0806.4561 (2010).
•
M.V. M ENSHIKOV, M. VACHKOVSKAIA , AND A.R. WADE, Asymptotic behaviour of
randomly reflecting billiards in unbounded tubular domains, J. Stat. Phys. 132
(2008) 1097–1133.
•
M.V. M ENSHIKOV AND A.R. WADE, Rate of escape and central limit theorem for
the supercritical Lamperti problem, Stochastic Process. Appl. 120 (2010)
2078–2099.
•
F. S PITZER, Some theorems concerning 2-dimensional Brownian motion, Trans.
Amer. Math. Soc. 87 (1958) 187–197.
•
N.T H . VAROPOULOS, Potential theory in conical domains, Math. Proc.
Cambridge Philos. Soc. 125 (1999) 335–384.
•
M. VOIT, Strong laws of large numbers for random walks associated with a class
of one-dimensional convolution structures, Monatsh. Math. 113 (1992) 59–74.