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Modelling transport in disordered media via diffusion on fractals
COMPUTER MODELLING
PERGAMON
Mathematical
and Computer
Modelling 31 (2000)
129-142 www.elsevier.nl/locate/mcm
Modelling Transport in Disordered Media via Diffusion on Fkactals B. HAMBLY Department of Mathematics and Statistics The University of Edinburgh, Edinburgh, U.K.
0. D. JONES Department of Mathematics, The University of Queensland Brisbane, Australia Abstract-When
viewed at an appropriate scale, a disordered medium can behave ss if it is strictly less than three-dimensional. As fractals typically have noninteger dimensions, they are natural models for disordered media, and diffusion on fractals can be used to model transport in disordered media. In particular, such diffusion processes can be used to obtain bounds on the fundemantal solution to the heat equation on a fractd. In this paper, we review the work in this area and describe how bounds on branching processes lead to bounds on heat kernels. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Disordered
media, Transport,
Fractals,
Diffusion, Heat kernels.
1. INTRODUCTION The term ‘disordered medium’ has no strict mathematical definition, but is used to describe ‘real’ structures with highly irregular geometry. In this review, the disordered media we consider are objects existing but not dense in Rd, which exhibit a similar structure at many different scales of observation,
a property termed statistical self-similarity.
Numerous constructs have
been put forward as models for disordered media. In particular, random walk paths, diffusion limited aggregation (and other growth models), and critical percolation have all received intense scrutiny. These are all examples of random fractals, which exhibit statistical self-similarity. That is, when averaged over the whole space, properties such as density scale at a fixed rate, strictly less than that of the Euclidean space they reside in. The rigorous study of diffusion on random fractals such as these is extremely difficult. However, our understanding of diffusion on random fractals is aided by our understanding of the (presumably) analogous behaviour of diffusion on regular geometric fractals, which are in general more tractable. In this paper, we aim to provide a brief survey of the current mathematical literature in this area. The study of diffusion on a fractal F, from the probabilistic point of view, is the construction of a continuous time and space stochastic process on F. The relationship with the applied mathematicians view of diffusion, as the study of the heat equation on F, is given by the following correspondence, even when there is no differentiable structure on the set. The Laplacian A on F is defined by the fact that it is the generator of the canonical diffusion process or ‘Brownian motion’ X on F. Thus, the transition density of X is the heat kernel of A, that is the fundamental
08&j-7177/CKl/$- see front matter PII: SO895-7177(00)00080-7
@ 2000 Elsevier Science Ltd. All rights reserved.
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by -G&‘&F
B. HAMBLYAND 0. D. JONES
130
solution to the heat equation -= on F. a’ Au dt ’ The first rigorous construction of diffusion on a fractal was of Brownian motion on the Sierpinski gasket [l-3]. Let
7(i,g)}
vo= { (O,O),(LO)
and
and recursively define (VI, El), (Vi, Ez), (VS, Ez), . . . by Vn+l = V, u [(2n,~) + v,J u [(2n-1,2n-1di) E n+l = En U [(2n,0) + En] U [(2,-‘,29
+ vn]
and
+ En] .
Let V = V, U [-VW] and E = E, u [-Em] and put Go = (V, E). We call Go the infinite Sierpinski graph or the pre-Sierpinski gasket. Let G, = 2-mGo and define the Sierpinski gasket G = U,“=, 2-mV.
Figure 1 gives a picture of Gc.
Figure 1. The Sierpinski gasket and Sierpinski carpet.
Let X,, be the simple nearest-neighbour random walk on G,, then as n + 00, Xn(t) := Xn(59) converges weakly to a continuous strong Markov process X on G. The time scaling is determined by requiring the expected crossing time EzEGo inf{t : Xn(t) E Go \ {cc}} to remain constant as n ---t 00. X is locally invariant w.r.t. the local isometries of G, and is accordingly termed Brownian motion on G. The X,
can in fact be nested so that for any m < n, X,
is precisely
the process obtained by observing X, only when it moves from one point in G, to another, i.e., the G,-decimation of X,. This can be used to establish the a.s. convergence of _%,. The analysis of Barlow and Perkins [3] included (among many other things) the following uniform bounds on the transition density pt (z, y) of X. There exist positive constants cl, c2, c3, c4 such that for all z, y E G and t > 0, crt-d./sexp
i-c2
(z-~]dwJ1”dw-l’)
ivt
(z,y)
< c3tKdJ2 exp
{ _c4 (
,5
y&n)
‘i’d*u-l)} ,
(‘)
Modelling Transport in Disordered Media where d, = 2 log 3/ log 5 and d, = log 5/ log 2. These scaling given the following
formal
interpretations,
eigenvalue Note that that
of the Laplacian
we write
N to mean
these exponents
(and one other)
can be gasket:
of G in a Euclidean ball radius r scales as m(r) N rdf); from the origin in time t scales as E/X(t)/ N tlldw);
(the integrated
density
of states
N(r)
= #{X
: X is an
A, X 5 r} N rdsi2).
bounded
are related
exponents
which can be made precise for the Sierpinski
df: fractal dimension (the mass m(r) d,: walk dimension (the displacement d,: spectral or fracton dimension
131
above and below by constants.
It is generally
believed
by the expression ,& = F
and this has been established The physics literature fractals
in a number
of situazons
(see [9-161 and the references
has concentrated
largely
12 - yI = T of pt(z, y). For finitely
ramified
for P(r, t), that
and lower bounds
therein)
on diffusion
P(r, t), the average
on the ‘propogator’
analogous
upper
[3-81. contained
fractals,
bounds
on geometric
(in some sense)
over
of the form (1) immediately
is, there
exist constants
such that
give for all
r,t > 0, ,st-ds/2 exp(
Physicists have estimated arguments); the so called diffusion
5 P(r,t)
--G ($)l”dw-l’}
equation
5 qd-“l2ug(
-Cg ($)“‘dw-l’}.
(3)
P(r, t) using a variety of methods: renormalization groups continuous time random walk formalism and generalizations
using fractional
derivatives.
These have lead to refinements
(scaling of the
of (3) of the form
P(r, t) = t-d*‘2S” exp {-co<‘}, for various asymptotic 0, then
LYand p, where E = r/tlidw. form of P(r, t) has p = d,/(d,-1)
appears
as a first-order
and p = d,
do better.
mathematics
literature.
One factor behind constancy behaviour is, small oscillations
correction
As yet, rigorous
Numerical simulations suggest that as < t 00 the and cy = 0. The 5” factor, with (I( = (d, -df)/?/2 < for smaller
asymptotics
<. Contrastingly, of analogous
as < + 0, values of a = 0
form have not appeared
in the
the lack of precise asymptotics in the mathematics literature is the near of the crossing time W = inf{t : X(t) E Go \ X(0) ( X(0) E Go}. That are present
in the distribution
of W [17,18]., These make exact
asymptotics
for W and thus X difficult. As yet the physics literature seems unaware of this behaviour though it is possible that it does not cross over from geometric to random fractals.
of W,
Another integrated
in the
important factor behind the problem is the fact that there can be oscillation density of states, N(X). This means that for the Sierpinski gasket, by [19],
The existence of this gap corresponds to the existence of localized eigenfunctions for the Laplacian on the gasket. For spatially homogeneous random fractals there is even worse behaviour than in the geometric case. Indeed, for the random fractals studied in [7,20], there exists dependent on the convergence rate of the environment sequence, such that ’ < ‘iyizf
N(X)
f(A)-lAd,/B
’
lim sup x+m
a function
f,
N(X) f(X)XdJ2
<
O”.
However, for random recursive fractals [21], there may be an ergodic effect which leads to more precise asymptotics. Of course, unlike fractals, disordered media are not statistically self-similar at all scales of measurement. Certainly not at the atomic scale. It is generally assumed in the physics literature that this is not a problem. This assumption has been justified to some extent in [22], where it is shown that for t 2 ~(a: - yl, the transition probabilities of X0 are of the same form (1) as the density of X.
B. HAMBLY AND 0.
132
D. JONES
2. BROWNIAN MOTION ON SELF-SIMILAR FRACTALS The construction
of Brownian
gasket to ‘nested fractals’ work of Fitzsimmons of the Laplacian geometric nested
fractals
constructing fractals.
referred
it is possible but
Brownian
the construction
gives approximations ramified).
on both nested
Other
of the Sierpinski
gasket
of Brownian
self-similar
Zhou
the class of
fractals,
and include
of a ‘harmonic
by Barlow
structure’
motion.
Brownian
and Bass [4,26,27], which
[28] have provided
and a similar
a method
of
class of infinitely-ramified
than infinitely-ramified
walks (see below).
subset can be isolated to look at a diffusion
constructions
and
fractals
Prior to the
to the construction
and hence, a Brownian
are more tractable of random
approach
the existence
carpet
and Kusuoka
of ‘nested’ sequences
one for which any bounded this means it is sufficient
a Laplacian,
on the Sierpinski
fractals
[23] from the Sierpinski
sets’, which are essentially
By assuming
to construct
motion
Finitely-ramified
a parallel
as finitely-ramified
fractals.
infinitely-ramified,
by Lindstrerm
et al. [6] to ‘affine nested fractals’.
‘p.c.f. self-similar
to by physicists
has also been constructed
is self-similar
was extended
[24,25] developed
He defined
and affine nested
on the fractal, motion
et al. [6], Kigami
on fractals.
fractals
motion
and then by Fitzsimmons
ones, as they allow
A finitely-ramified
fractal
is
by removing a finite set of points. In practice, on the fractal at such cut points. Figure 1
(finitely motion
ramified) on fractals
carpet
(infinitely
which are not usually
and Sierpinski
embedded
in Rd are given in [29,30]. All of these fractals
can be defined
as the closure
of the limiting
vertex
graphs, each a successively finer approximation to the fractal. A diffusion be constructed as the limit of a sequence of time-scaled nearest-neighbour approximating
graphs.
two-dimensional Brownian
subsets
motions
(Note that
on these sets to construct
and Zhou [28] showed that Justification
which self-similar
motion
the class of nested
on a fractal fractals
motion
on the limit.
established
However,
Kusuoka
walks in this case.) It is unclear
The uniqueness
by Sabot
of
of reflecting
is still based on local invariance
all posses to some degree.
which is not self-similar.
was recently
and a sequence
and random
of
on the fractal can then random walks on the
and Bass [26] used a sequence carpet,
motion
to use graphs
Brownian
fractals
of Barlow
the Sierpinski Brownian
it is also possible
for calling these processes
local isometries, Brownian
the construction
of R2 to approximate
set of a sequence
under
how to define
of Brownian
motion
on
[31].
The original constructions of Barlow and Perkins [3] and Lindstrom [23] use estimates of crossing times to prove the time-scaled sequence of random walks converges, and that the limit is a continuous strong Markov process. However, more successful has been the use of Dirichlet forms, developed largely in Japan [8,24,25,32,33], which give the Laplacian directly, rather than as the generator of a Brownian motion. The link between random walks and Dirichlet forms is provided by electric networks [34-361. This approach does however place some restrictions on the sort of random walks you can use. In particular, diffusion on a fractal it is still necessary Dirichlet
forms allow us to use a wealth
they must be symmetric. To construct nonsymmetric to use a sequence of random walks [37]. Also, while of analytic
techniques,
at present
we still need to know
something about crossing times to provide off diagonal bounds for the transition probabilities of the limit process. It is in the study of crossing times of diffusion on finitely-ramified self-similar fractals that branching processes appear. Define self-similar fractals in Rd as follows. For i = 1, . . . , N let & be a ‘similitude’, defined by &(z) = ai + (X - ai)/bi for ai E Rd and bi E (1,oo). For A c Rd let v(A) = @, q&(A). Hutchinson [38] proved cp has a unique fixed point F, which we call a self-similar fractal. Let W’s be the set of fixed points of the &. A point z E Ws is an essential fixed point if there exist 1 L i, j I N, and y # z E Wo such that &(lc) = +j((y). Let VObe the set of essential fixed points. Given some regularity conditions on F, in particular that it is connected and that there is no overlap of components (though their edges can meet, this is called the open-set condition), we can reconstruct F from its essential fixed points.
Modelling Transport in Disordered Media
Let h(l)...i(k)
=
h(l)
o . ” o &(k)
where 1 5 i(j)
133
5 n for all j, and define
N
v, =
U
h(l)...i(k)(h)-
i(l),...,i(k)=l
Vk then F = V,. we form graphs Fk = (vk, Ek) Clearly, VOC VI C . . . . If we put V, = UF=“=, as follows. Let EO be the edge set of the complete directed graph on VOand define for Ic > 0 E,, =
$i(q...i(k)(Eo).
U i(l),...,i(k)=l
For 1 5 i(l), . . . ,i(k) 5 IV, call C$i(l)...i(k)(F) a /c-complex. F is finitely-ramified if you can only get from one k-complex to another only via a point in vk. Specifically, for distinct sequences i(l), . . . , i(k) and j(l), . . . ,.?@), h(q...i(k)(F) f’ &(l)...j(k)(F) = h(l)...i(k)(h) f- @j(l)...j(k)(h). This property (called the nesting property) enables the definition of a nested sequence of random walks. The connected, open-set and nesting properties are essentially those required to specify p.c.f. self-similar sets, that is, finitely-ramified self-similar fractals. will assume F is such a fractal for the remainder of this section. Let X,
be a nearest-neighbour
random walk (n.n.r.w.)
inf{lc > 0 : Xn(k) E Vm} and T,,,(i + 1) = inf{lc > T,,,(i) the F, decimation of X, is given by Xm+(i) = X,(&,,(i)). will be a n.n.r.w. on F,.
If F is infinitely-ramified,
to another not directly joined by an edge in
on F,.
Unless stated othewise, we For m 5 n define T,,,(O)
=
: X,(k) E V, \ X,(2”,,,(i))}, then If F is finitely-ramified, then X,,,
then X,,, may jump from one point in V, A sequence (Xn)rzo such that for all m 5 n,
E,.
x,
% x,,, is called a nested sequence. Clearly, this is only possible for finitely-ramified selfsimilar fractals. For infinitely-ramified fractals a different approach is needed (see below). The first problem in constructing
Brownian motion on F is to find such a sequence. The correct time
scaling X, is then obtained by requiring the expected FO crossing times of Xn(t) := xcn(Xlzt) to remain constant. Clearly, Brownian motion will have to be symmetric and the restriction to any n-complex must be a scaled version of the process on F. These conditions enable us to specify that all the X,
be defined in terms of probabilities
p(x, y) = p(y,x)
for each (2,~)
E EO and
weights r(i) for 1 5 i 5 N. For any (z, y) E E, there is a unique sequence i(l), . . . ,i(n) that (z, y) = &(r~...~(~)(a, b) for (a, b) E Eo. Define the X, transition probabilities by
Pn(?Y)
m
such
da7b) 71
kElr(i(k))~
We need to know if this sequence of n.n.r.w. is nested.
For any given set of weights r(i),
the
Fe-decimation of Xi defines a map D on the set of EO transition probabilities. The fixed points of this map correspond to nested X,, sequences. Such sequences are sometimes called ‘decimation invariant’. The existence of such fixed points is proved for various classes of fractal by Fitzsimmons et al. [6] and Lindstrmm [23]. Hattori et al. [39] have shown that for some self-similar finitely-ramified fractals, no nondegenerate fixed point exists. As yet no one has characterized those fractals for which it does exist. The uniqueness of such fixed points is also still an open question for general PCF sets [31,40]. Clearly, we can look for other nested sequences of random walks. Generally, for this to be tractable some restriction needs to be made on the random walks, e.g., that they be spatially homogeneous. Such nested sequences have been constructed, and lead to diffusions which clearly are not Brownian motion. We will look at these in Section 3. The method used by Kusuoka and Zhou [28] to deal with infinitely-ramified fractals uses a different sequence of approximating graphs. Let z be an element of the interior of the convex hull
B. HAMBLY AND 0. D. JONES
134
I
I l I
I
Figure 2. Using essential fixed points, and alternatively using a point in the convex hull, to form approximating (level 2) graphs for the Sierpinski carpet.
of h, then put FL = (VI, EL> for k 2 0, where V,l = U~l~,...,i~k~=l h(l)...i(k)({~}) and if x and y are in adjacent k-complexes and the dimension of their common boundary do and df , for some do. For finitely-ramified
fractals
and we must take de = 0. For infinitely-ramified Figure 2 gives both type of graph approximation,
the boundary
(GY)
E EL
is between
can only be a single
point
fractals more complex boundaries are possible. Fz and Fl for the Sierpinski carpet.
Unfortunately, V,l $?!Vj,, for any k, nor is there any clear embedding of VL in Vi+I, so the idea of nested random walks cannot be applied. Nevertheless, it is still possible to consider the crossing times
of a sequence
of n.n.r.w.
X,
on FA and if these converge
(after appropriate
scaling),
we can hope to find a limit process. The potential theory approach also works, though analogous problems, and is the approach used by Kusuoka and Zhou [28]. For the infinitely-ramified fractals of [26,28], the uniqueness of the limiting process an open
problem.
the approximating transition density behaviour. 2.1.
sequence estimates
techniques
.An electric
Networks network
of its edges.
only give weak convergence
along
suffers is still
a subsequence
of
of processes. This has not been a problem for the computation of for the carpet [4], as all subsequential limit points have the same
In [28], the existence
Electric
sistance)
The current
then
of a self-similar
and
Dirichlet
on graph
diffusion
process on the carpet
was established.
Forms
F, is defined
in terms
Let a,(~, y) be the conductivity
C, a,(~, y). Th e D iric hl e t form associated
of the conductivities
(reciprocal
of re-
of edge (z:,y) E En and let b,(x)
with the network
=
(F,, a,) is defined for f, g : F, -+ R
by Wf,g)
= &(x7 Z,Y
Y)(f(X) - f(Y))MX)
&“( f, f) is interpreted as the energy dissipated when a potential The discrete Laplacian An on F, is defined by A”f(z)
= c
a&,
- S(Y))* f is maintained
on the network.
Y) f(Y;)$(x).
Y w h ere the inner product (., .) is w.r.t. b,, that is, and E” can be recovered from each other. A” is also the (f, 9) = C, f(x)g(x)bn(x). So, An generator of the n.n.r.w. X,, on F, defined by An and E” satisfy
&“(f,g)
P(&(i
= -(Anf,g),
4x7 Y) + 1) = y I &z(i) = x) = Pnb, Y) = b,(z).
Suppose now we have a finitely-ramified self-similar set F with approximating graphs F, as described above. Given conductivities a, for each F,, let (Xn)rzo be the corresponding sequence
Modelling Transport in Disordered Media of n.n.r.w.
The Laplacian
X m,n, is Am provided potentials
only at points
Equivalently, Kigami
corresponding
the electrical in V,,
the effective resistance
is equivalent
time-scaled
random
of Dirichlet
forms is monotone
of the Dirichlet
forms is implicit
Dirichlet
form,
to the convergence
of their Dirichlet
under the compatibility
forms is that
F. The effective
holds for graphs points, d;:
that
a, required
resistance
metric
to be the effective
dimension
process
on the fractal
them
for the
networks.
is a symmetric
regular Markov
of the Dirichlet form for analysing A, the
to the Euclidean
r(. , .) is defined
between
Scaling
of a limiting
Knowledge techniques
alternative
(r-metric)
resistance
but will not define a metric fructal
The sequence
to get compatible
the existence
the corresponding
gives rise to a natural
is, if d, > 2. Formally.
r-metric
be zero.
and hence, the limit
for which the limit is finite.
once we establish
know that
corresponding Laplace operator. The electric network approach two points
forms.
condition,
process, a fact which requires some effort to prove otherwise. of the limit process also allows access to a range of analytic
between
points
given by
we immediately
a fractal
at other
Finding a sequence of compatible random walks, and the convergence of
of nested
in the choice of conductances
of Dirichlet
of
when we specify V,
‘compatible’.
F is defined to the those f E C(F)
The advantage
is, the generator
in V, is the same for each network.
any two points
a sequence
increasing
forms is simply
where the domain Dirichlet
between
walks is equivalent
that
to (F,, a,)
let the net flow of current
of networks
to finding
of X,,
(F,, a,) is equivalent
and then
[8] calls such sequences
networks
to the F,-decimation
network
135
by setting
[41]. Note that
if the associated
metric
on
the distance this definition
process
does not hit
we can consider (the mass ,mr(r)
of F in a r-metric
ball radius
T scales
as
mT(7’) N r”?); d’,:
r-metric
walk dimension
(the displacement
The spectral dimension d, is unaffected sets, and conjectured to hold more widely
E r(X(O), X(t))
N tlld:u).
by a change in metric. It is true for p.c.f. self-similar [41], that d, = 2d’f/(d; + 1) and thus, if (2) also holds,
that d& = d; + 1.
So, we effectively d,:
resistance
(6)
lose a parameter
here, but ga.in instead
dimension
effective
(the
resistance
between
two points
x and
y scales
as
T(X, y) nJ ]Z - yldV,). Clearly,
d> = drldr
and dG = d,/d,.
Applying
this to (6) we obtain
the Einstein
relation
d, = df + d,
3. GENERALISATIONS The construction
methods
of Brownian
construct diffusion-which may be Brownian self-similar, and to construct on self-similar
motion
on self-similar
fractals
have been extended
to
motion-on geometric fractals which are not exactly fractals diffusions which certainly are not Brownian
To date, these extensions have been restricted to finitely-ramified fractals and have motion. hinged on the production of more general nested sequences of random walks, or equivalently, compatible sequences of Dirichlet forms. (Though recently Barlow et al. [42] have announced results in this area for the Sierpinski carpet.) In constructing Brownian motion on finitely-ramified self-similar fractals we look for fixed points of the map D on the space {p(a,b) = (a, b) E Ee, xca,b) p(a, b) = 1, p(a, b) = p(b, a)}, determined by decimating the random walks X, defined by equation (5). This leads to a ‘decimation invariant’ nested sequence. By relaxing the condition p(a, b) = p(b, a), Kumagai [37,43]
B. HAMBLY AND 0. D. JONES
136
obtained a nested sequence on the Sierpinski gasket which gives in the limit a diffusion which is nonsymmetric construction Hattori
but invariant under rotation of the fractal.
There is no analogous Dirichlet form
for this diffusion. et ~2. [44,45] obtained
nested sequences
by looking at trajectories
of the inverse
They have done this for the Sierpinski gasket, producing diffusion which has lo-
map D-l.
cal behaviour which is horizontal movement asymptotically, a&-gaskets,
for which no nondegenerate
and for a class of fractals
called
fixed point exists for the map D, for certain values of
a, b, c. Hattori et al. use new results on branching processes to show their nested random walks converge [45,46] ( see also [47]).
Recently
however, Hambly and Kumagai
[48] have used the
electric network framework to reproduce and extend these results and provide diagonal bounds for the associated
transition
densities.
Off diagonal bounds have yet to be produced for these
diffusions. It is also possible to consider (forward) trajectories
of the map D. These arise when considering
infinite fractals, which can be constructed by applying the inverse similitudes $il
to the fractal F.
We can specify the behaviour of an anisotropic random walk at a given (microscopic)
scale, and
then ask about its large scale (macroscopic) behaviour. It has been shown for a variety of fractals that as the scale increases you get increasingly isotropic behaviour [42,48,49]. This corresponds to the convergence of the forward trajectories
of D to a nondegenerate fixed point, representing
Brownian motion. This is a form of homogenization for operators on fractals which is induced purely by the geometry of the fractal, and not through some ergodicity of an underlying random operator. In the class of geometric fractals there are some which are not exactly self-similar, the random recursive Sierpinski gasket of [21], or the scale-irregular
gaskets of [7,20,45].
such as These
are produced by allowing some variation in the family of maps used in the construction of the fractal, and as such can be regarded as a step closer to disordered media than exactly self-similar fractals.
These examples are still finitely-ramified
however, and it is still possible to construct
nested sequences of random walks for them. For the scale-irregular gaskets the variation in the maps is spatially homogeneous and governed by an environment sequence. For random recursive fractals, which have no spatial homogeneity, we can still use the nested sequence of random walks. However, some care must be taken when choosing the ‘levels’ of approximation. This can be achived by working in the effective resistance metric,
as the probabilities
for the Brownian motion to exit a region are determined
by the
resistance. The effective resistance metric is then the appropriate metric in which to get transition density estimates. 4.
CROSSING
Let F be a finitely-ramified
TIMES
AND
BRANCHING
self-similar fractal and (F,)~&
PROCESSES
a sequence of approximating graphs
as described in Section 2. If (Xn)F& is a nested sequence of random walks on F, then the implicit branching process is constructed using the jumps of these random walks. The original ancestor is a single jump made by Xo, the first generation are the corresponding (via the decimation process) set of steps made by Xi, and so on. So, the children of any individual in the nth generation are the jumps made by Xn+ir which, when decimated, produce the given X, jump. The growth rate of the branching process immediately gives the time scaling for the random walks, and the normed limit of the branching process is the crossing time of the limiting diffusion. In general, the branching process will be multitype. Set the type of each jump to be the type of the edge (2, y) along which it travels. However, as #En -+ 00 we need some way of rationalizing the number of types before the branching process can be useful. In the case of Brownian motion, when conditoned on staying within a given n-complex, the motion of X,, depends only on {~(a, b) : (a, b) c Eo}. S0, it is sufficient to restrict ourselves to a single set of types {t(a,b) : (a, b) E Eo}, and set the type of an edge (z, y) E En to t(a, b), where
Modelling
(GY)
Transport
in Disordered
dql)..+)(a, b) forSOme 1 I i(l), . . . , i(n)
=
(a, b) E I&}
are a fixed point
Media
137
< N. In this case, the probabilities
of D, and so we get a classical
multitype
However, nested sequences corresponding to trajectories of D-l with varying environment, as the underlying transition probabilities, the number
of jumps,
the construction process
change at each level [45-47,501.
of Brownian
in a random
The crossing
environment
times
sequence
same as the time scaling on the normed
limits
process is just
of branching
of bounding
a large deviation
problem
produce branching processes and thus, the distribution of
the nested
random
X are given
the expected
used for the random
The problem
Similarly,
on a homogeneous
process.
fractal,
sequence
produced
in
gives rise to a branching
[7,51].
of the limit
The norming
process.
motion
:
{p(a,b)
branching
processes
the normed [17,18].
walks.
by the normed
limit
size of the branching
of the branching
process,
This has lead to a number
and is the
of recent
results
[17,18,47,51-541.
limit of a (supercritical)
In particular,
branching
the event of minimal
process
growth
is essentially
is of central
im-
portance. Determining the minimal growth rate is nontrivial for multitype processes [5,6], and when constructing Brownian motion on a finitely-ramified self-similar fractal, is equivalent to determining a shortest path metric on the fractal [6]. The shortest path metric is called chemical distance
in the physics
ch,(z, y) be the length
literature,
and, if it exists,
of the shortest
path
is determined
5 and y in F,.
between
grows like 1” for some 1 > 1, then we define the chemical d,:
chemical dimension
For the Sierpinski Lindsrtom
d?: ch-metric VP(r) Again, When
y)/P
we have 1 = 1.
and carpet
If the length
of this path
d, by
N ]z - y]“’ =: ch(x, y)).
[23]. As with the effective resistance
For an example metric,
with
we can redefine
1 > 1 see the the fractal
and
using this metric fractal dimension walk dimension
d, is unaffected
the relation
(the mass me(r)
(the displacement
by a change
of F in a ch-metric
ball radius
r scales as
in metric.
Ech(X(O),
X(t))
N t”“:‘).
We have d? = df/d,
and d”, = d,/d,,
so they
df and d, do.
(2) whenever
ch(. , .) exists, off diagonal
shown to hold for the Sierpinski carpet
ch,(x,
dimension
For 2, y E V,, let
N rd?);
d&: ch-metric satisfy
gasket
snowflake
walk dimensions
(lim,,,
as follows.
bounds
gasket,
for the transition
nested
fractals,
density
affine nested
of the form below have been fractals,
and the Sierpinski
[3-6,361
It is worth
noting,
that
bounds
of this form (using d; rather
t,han d,)
are not currently
seen in
the physics literature. When fractals are not exactly self-similar such tight uniform bounds on the heat kernel are generally not possible. In the case of scale-irregular gaskets, we can obtain estimates which are best possible for the diffusion. They are much as (7), but are modified by the introduction (if there is any) of the of a correlation factor [20], which depends on the speed of convergence environment sequence to its ergodic limit. For random recursive fractals [2], there are as yet no best possible bounds on the transition density, as the relationship between the short paths in the fractal and the effective resistance metric is still not well understood. However, it is clear that there will be oscillation in the bounds. It is tempting to think that by scaling differently the time taken by jumps of different type, you might obtain a different limit. From our understanding of the limiting mechanism for branching
B. HAMBLY AND 0. D. JONES
138 processes,
we can see that
of a branching for different
process
this is pointless.
is deterministic,
types of jump
the heat
kernel,
transition
with respect in Section
motion.
density
ESTIMATION
solution
to reflection
motion
in the perpendicular
of the limit
process.
of another
[47].
that
as its generator.
This means
on F, is the transition
self-similar
bisectors
compact
of the essential
fractal,
that
density
of the ideas used in the estimation
on F, an exactly
limit
FOR FRACTALS
to the heat equation
We give here an outline
for Brownian
less than
F has the Laplacian
Xt on a fractal
the fundamental
of the Brownian
types in the normed
So, scaling time differently
time scaling
rate of one type is strictly
5. HEAT KERNEL motion
between
would lead only to a deterministic
This is true even when the growth
The Brownian
The distribution
only the total size is random.
of the
symmetric
hxed points,
as defined
2.
The following
heuristic
argument
shows how the functional
form of the transition
density
arises
from the natural scaling of the process. Prom the exact self-similarity, and the definition of d,, for certain lengths 1. That we should have locally the relationship Xt = Z-IX ldwt in distribution, is P”(Xt where
BT(z)
Laplacian
E B,(z))
is the ball of radius
A on F is a compact
T about operator:
sition, with which we can construct Hausdorff measure ~1on the fractal. should
= Pll(Xld”,t 2.
E B&r)),
The transition
by Mercer’s
(8)
density,
Theorem,
pt(z,g),
A admits
will exist as the
a spectral
decompo-
the density. Note that this is a density with respect to the Now, use (8) with a change of variable, to show that ~~(2, y)
satisfy ?‘t(& Y) = ~%ldwt(&
Hence, we can deduce
that
the functional
&,
for some continuous
function
y)
VX,~EF,
l?/),
form of the density
N
t-d./2g
and determine
should
be
d(x;yJdw )
( )
g. Here, we have assumed
In order to prove this estimate
O
that
the function
d, = 2df/d,,
as in (2).
g, we begin by establishing
a bound
on pt(x, z) which is uniform in 2. Initial techniques [3-51, relied on estimating the Green density and using Tauberian theorems to turn this into estimates on the heat kernel. This use of this idea is restricted
to the case where d, < 2, when the Green
density
is bounded.
Subsequent techniques have used analytic results, which relate the time decay of the heat kernel to various inequalities for functions in F, the domain of the Dirichlet form. The Nash inequality is said to hold if there exist constants A and u such that
Ilf II;+4’”5 A (WJ) + Ilfll;) Ilfll:‘“7 This is equivalent to the following that for 0 < t < 1,
on-diagonal
heat kernel bound:
sup pt(x, y) L Cp@. WIEF
V’fEF. there exists a constant
cl such
(9)
In [6] the Nash inequality is obtained directly in the case where v = d, < 2, using the scaling in the fractal. In [48], ideas in [28] are used to get such an on-diagonal bound from a Poincare inequality: for some constant C
Modelling Transport in Disordered Media
This type of inequality extended
arises when the Laplacian
has a spectral
the diagonal,
we must examine
the most likely paths.
Brownian
ramified
motion.
then the greatest
the short
fractals,
paths,
where Tf = inf{t
the embedded
which correspond
is that
the following
If we think of the transition
contribution
density
to the sum will come from
whereNn,n+mis
the number
in F,. This inequality, of Wzp, contains enough following
branching
process
encodes
the small time behaviour to branching
processes
super-branching
> Tf_, : X, E Fk \ {XT;_,))
make the ith step on level n. Then,
path
with few offspring.
and T!:, = 0, that
the super-branching
of the
In fact, all
Let IV? = T,” - TiT1,
holds.
is, WF is the time required
inequality
path in F,+,
of steps in the shortest
the whole path
of pt(z, y), we need to know
inequality
to
is
between
two adjacent
points
coupled with a weak uniform estimate on the tail of the distribution information to get an exponential estimate on the crossing time, of the
form P” (WT < t) < c2 exp (--~a (E” (Wl”) t)-‘)
cr,cz
are constants.
the mean
crossing
self-similar
fractal.
The off-diagonal radius
term and we use a density away from
For a short time the most likely path will be close to the shortest
In order to examine
we need to observe
where
can be
the points.
For finitely about
the off-diagonal of the transition
the short paths in the fractal.
x, y as a sum over all paths,
between
gap, and this approach
to the case where d, 2 2.
As yet there are no analytic techniques for calculating probabilistic argument. In order to estimate the behaviour between
139
p is determined
time,
as @ = log(Nn,,+,)/log(E(Wy+n)/N,,,+,)
upper
bound
d(z, y)/2 about
neighbourhood
The exponent
follows from a probabilistic
x and split the probability
of y into two parts,
of moving
by conditioning
t < 1,
,
‘~xEF,
by the length
(10)
of shortest = d,/(d,
argument.
We construct
from the neighbourhood
on whether
path
and
- d,), for a a ball B of of x to the
or not the process is in B at time
t/2. As the process is reversible, we can deal with each part in the same way. We can, without loss of generality, consider the process conditioned on not being in B at time t/2. Now, further decompose the transition probability, time t/2. The first half of the sample
this time conditioning on the position of the process at path can be dealt with using (lo), as we know the process
must have crossed a set of size at least d(x,y)/2. The second half of the sample controlled using (9). Combining these, we obtain the upper bound of (1) and (7). There
are three
steps to obtaining
a lower estimate
for the transition
density.
path
can be
The first is to
get an on-diagonal estimate. As we have an upper bound on the transition density, the process cannot exit a ball about x too rapidly, and hence, from the upper bound itself or (lo), see [4j, we can show that there is a constant cd such that pt(x,x) 2 c4t-ds/2. The next step is to determine a ‘near’ diagonal estimate. In other words, what is the size of ball for which the on-diagonal estimate is good? This involves an estimate on the Holder continuity of the heat kernel. In the finitely-ramified case, we control functions in the domain of the Dirichlet form in terms of the effective resistance metric T(Z, y), see [8]
If(x) - f(Y)12 I +7 Y)E(f>fL
Vx,y~
F,
f E3.
As pt(x, .) is in the domain F, we have immediately that its Holder order is d,./2. This sort of estimate is available only when d, < 2, and we have an effective resistance metric. Other
B. HAMBLY
140
AND 0. D. JONES
techniques are required if d, > 2. Combining this with the on diagonal estimate gives the result that there exists a constant cs such that for all 2, y E F and 0 < t < 1, if d(x, y) < c&dw.
pt(z, y) > yJ2,
(11)
The final step in the procedure is to use a chaining argument. We consider the shortest path between the points x and y and determine the scale at which we should view the path according to the allowed time to move between the points, then cover the path with balls of the appropriate size to apply our near diagonal estimate. We let 5 = d(x, ~)~w/t, and for the short time behaviour we examine < large. In order, to apply (11) we must choose a path {zi}~=r in the fractal such that d(xi, xi+l) = d(x, y)/N
5 cg(t/N)l/dW,
then for r = d(x, y)/N,
we have
Using our previous estimates, and the fact that the natural measure ~1is the Hausdorff measure, we have pt(x,y)
> (:
($)-dS”)N
((cs
(;)“‘JdJN-l
2 c&%:.
The choice of N 5 c&l/(dw-l) required to fit the criteria gives the lower bound of (1). The lower bound of (7) comes from taking into account the scaling of the shortest path.
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in Disordered Media
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19. M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, Poteatiat Analysis 1, l-35 (1992). 20. M.T. Barlow and B.M. Hambly, Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets, Preprint (1996). 21. B. Hambly, Brownian motion on a random recursive Sierpinski gasket, Ann. Probab. 25, 1059-1102 (1997). 22. O.D. Jones, Transition probabilities for the simple random walk on the Sierpinski graph, Stoc. Proc. Appl. 61, 45-69 (1996). 23. T. Lindstrom, Brownian motion on nested fractals, Number 420, In Memoirs of the AMS, American Mathematical Society, Providence, RI, (1990). 24. J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math. 6, 259290 (1989). 25. J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Tmns. AMS 335 (2), 721-755 (1993). 26. M.T. Barlow and RF. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. Inst. Henri Poincark 25, 225-257 (1989). 27. M.T. Barlow, RF. Bass and J.D. Sherwood, Resistance and spectral dimension of Sierpinski carpets, J. Phys. A 23, L253-258 (1990). 28. S. Kusuoka and X.Y. Zhou, Dirichlet forms on fractals: Poincare constant and resistance, Prob. 7% Rel. Fields 93, 169-196 (1992). 29. S.O.G. Nyberg, Brownian motion on simple fractal spaces, Stochastics 55, 21-45 (1995). 30. H. Ossda, Self-similar diffusions on a class of infinitely ramified fractals, J. Math. Sot. Japan 47 (4), 591616 (1995). 31. C. Sabot, Existence and unicite de la diffusion sur un ensemble fractal, C.R. Acad. Sci. Paris. SCrie 1321, 1053-1059 (1995). 32. S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. RIMS 25, 659-680 (1989). 33. M. Fukushima, Dirichlet forms, diffusion processes and spectral dimensions for nested fractals, In Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, In Memory of R. Hoegh-Krohn, Volume 1, (Edited by Albeverio, Fenstad, Holden and Lindstrem), pp. 151-161, Cambridge Univ. Press, (1992). 34. P.G. Doyle and J.L. Snell, Random Walks and Electric Networks, Math. Assoc. Amer., (1984). (1989). 35. A. Telcs, Random walks on graphs, electric networks and fractals, Prob. Th. Ret. Fields 82,435-449 36. M.T. Barlow, Harmonic analysis on fractal spaces, Se’minaire Bourbaki 5, 345-368 (1992). 37. T. Kumagai, Construction and some properties of a class of nonsymmetric diffusion processes on the Sierpinski gasket, In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on Fractals (Edited by K.D. Elworthy and N. Ikeda), Pitman, Montreal, (1993). 38. J.E. Hutchinson, Fractals and self similarity, Indinana Univ. Math. J. 30, 713-747 (1981). 39. K. Hattori, T. Hattori and H. Watanabe, Gaussian field theories on general networks and the spectral dimensions, Prog. Th. Phys. Suppl. 92, 108-143 (1987). 40. M.T. Barlow, Random walks, electrical resistance and nested fractals, In Asymptotic Problems in Probability Theory: Stochastic Models and Diffusions on FVactals, (Edited by K.D. Elworthy and N. Ikeda), Pitman, Montreal (1993). 41. J. Kigami, Effective resistances for harmonic structures on p.c.f. self-similar sets, Math. Proc. Camb. Phil. Sot. 115, 291-303 (1994). 42. M.T. Barlow, K. Hattori, T. Hattori and H. Watanabe, Restoration of isotropy on fractals, Phys. Rev. Letters 75, 3042-3045 (1995). 43. T. Kumagai, Rotation invariance of a class of self-similar diffusion processes on the Sierpinski gasket, In Proc. Hayashibaren Forum 1992, “New Bases for Engineering Science”, (Edited by Y. Takahashi), Plenum, (1992). 44. K. Hattori, T. Hattori and H. Watanabe, Asymptotically one-dimensional diffusions on the Sierpinski gasket and the abc-gaskets, Prob. Th. Ret. Fields 100, 85-116 (1994). one-dimensional diffusions on scale-irregular gaskets, Technical Report UTUP45. T. Hattori, Asymptotically 105, Faculty of Engineering, Utsunomiya University, Ishii-cho, Utsunomiya 321, Japan (1994). 46. T. Hattori and H. Watanabe, On a limit theorem for nonstationary branching processes, In Seminar on Stochastic Processes, 1992, (Edited by E. Cinlar), pp. 173-187, Birkhauser, Boston, MA, (1993). 47. O.D. Jones, On the convergence of multi-type branching processes with varying environments, Technical Report 444/95, Prob. and Stat. Section, School of Math. and Stat., University of Sheffield, (1995). 48. B. Hambly and T. Kumagai, Heat kernel estimates and homogenization for asymptotically lower dimensional processes on some nested fractals, Preprint (1995). 49. T. Kumagai and S. Kusuoka, Homogenization on nested fractals, Prob. Th. Ret. Fields 104, 375-398 (1996). 50. O.D. Jones, Random walks on pre-fractals and branching processes. Ph.D. Thesis, Statistical Laboratory, University of Cambridge (1995). 51. B. Hambly, On the limiting distribution of a supercritical branching process in a random environment, J. Appt. Prob. 29, 499-518 (1992). 52. N.H. Bingham, On the limit of a supercritical branching process, J. Appl. Prob. 25A, 215-228 (1988). 53. O.D. Jones, Bounds on the limiting distribution of a branching process with varying environment, Bull. Austral. Math. Sot. 52 (3) (1995). 54. B. Hambly, On constant tail behaviour for the limiting random variable in a supercritical J. Appt. Prob. 32, 267-273 (1995).
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PERGAMON
Mathematical
and Computer
Modelling 31 (2000)
129-142 www.elsevier.nl/locate/mcm
Modelling Transport in Disordered Media via Diffusion on Fkactals B. HAMBLY Department of Mathematics and Statistics The University of Edinburgh, Edinburgh, U.K.
0. D. JONES Department of Mathematics, The University of Queensland Brisbane, Australia Abstract-When
viewed at an appropriate scale, a disordered medium can behave ss if it is strictly less than three-dimensional. As fractals typically have noninteger dimensions, they are natural models for disordered media, and diffusion on fractals can be used to model transport in disordered media. In particular, such diffusion processes can be used to obtain bounds on the fundemantal solution to the heat equation on a fractd. In this paper, we review the work in this area and describe how bounds on branching processes lead to bounds on heat kernels. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Disordered
media, Transport,
Fractals,
Diffusion, Heat kernels.
1. INTRODUCTION The term ‘disordered medium’ has no strict mathematical definition, but is used to describe ‘real’ structures with highly irregular geometry. In this review, the disordered media we consider are objects existing but not dense in Rd, which exhibit a similar structure at many different scales of observation,
a property termed statistical self-similarity.
Numerous constructs have
been put forward as models for disordered media. In particular, random walk paths, diffusion limited aggregation (and other growth models), and critical percolation have all received intense scrutiny. These are all examples of random fractals, which exhibit statistical self-similarity. That is, when averaged over the whole space, properties such as density scale at a fixed rate, strictly less than that of the Euclidean space they reside in. The rigorous study of diffusion on random fractals such as these is extremely difficult. However, our understanding of diffusion on random fractals is aided by our understanding of the (presumably) analogous behaviour of diffusion on regular geometric fractals, which are in general more tractable. In this paper, we aim to provide a brief survey of the current mathematical literature in this area. The study of diffusion on a fractal F, from the probabilistic point of view, is the construction of a continuous time and space stochastic process on F. The relationship with the applied mathematicians view of diffusion, as the study of the heat equation on F, is given by the following correspondence, even when there is no differentiable structure on the set. The Laplacian A on F is defined by the fact that it is the generator of the canonical diffusion process or ‘Brownian motion’ X on F. Thus, the transition density of X is the heat kernel of A, that is the fundamental
08&j-7177/CKl/$- see front matter PII: SO895-7177(00)00080-7
@ 2000 Elsevier Science Ltd. All rights reserved.
Typeset
by -G&‘&F
B. HAMBLYAND 0. D. JONES
130
solution to the heat equation -= on F. a’ Au dt ’ The first rigorous construction of diffusion on a fractal was of Brownian motion on the Sierpinski gasket [l-3]. Let
7(i,g)}
vo= { (O,O),(LO)
and
and recursively define (VI, El), (Vi, Ez), (VS, Ez), . . . by Vn+l = V, u [(2n,~) + v,J u [(2n-1,2n-1di) E n+l = En U [(2n,0) + En] U [(2,-‘,29
+ vn]
and
+ En] .
Let V = V, U [-VW] and E = E, u [-Em] and put Go = (V, E). We call Go the infinite Sierpinski graph or the pre-Sierpinski gasket. Let G, = 2-mGo and define the Sierpinski gasket G = U,“=, 2-mV.
Figure 1 gives a picture of Gc.
Figure 1. The Sierpinski gasket and Sierpinski carpet.
Let X,, be the simple nearest-neighbour random walk on G,, then as n + 00, Xn(t) := Xn(59) converges weakly to a continuous strong Markov process X on G. The time scaling is determined by requiring the expected crossing time EzEGo inf{t : Xn(t) E Go \ {cc}} to remain constant as n ---t 00. X is locally invariant w.r.t. the local isometries of G, and is accordingly termed Brownian motion on G. The X,
can in fact be nested so that for any m < n, X,
is precisely
the process obtained by observing X, only when it moves from one point in G, to another, i.e., the G,-decimation of X,. This can be used to establish the a.s. convergence of _%,. The analysis of Barlow and Perkins [3] included (among many other things) the following uniform bounds on the transition density pt (z, y) of X. There exist positive constants cl, c2, c3, c4 such that for all z, y E G and t > 0, crt-d./sexp
i-c2
(z-~]dwJ1”dw-l’)
ivt
(z,y)
< c3tKdJ2 exp
{ _c4 (
,5
y&n)
‘i’d*u-l)} ,
(‘)
Modelling Transport in Disordered Media where d, = 2 log 3/ log 5 and d, = log 5/ log 2. These scaling given the following
formal
interpretations,
eigenvalue Note that that
of the Laplacian
we write
N to mean
these exponents
(and one other)
can be gasket:
of G in a Euclidean ball radius r scales as m(r) N rdf); from the origin in time t scales as E/X(t)/ N tlldw);
(the integrated
density
of states
N(r)
= #{X
: X is an
A, X 5 r} N rdsi2).
bounded
are related
exponents
which can be made precise for the Sierpinski
df: fractal dimension (the mass m(r) d,: walk dimension (the displacement d,: spectral or fracton dimension
131
above and below by constants.
It is generally
believed
by the expression ,& = F
and this has been established The physics literature fractals
in a number
of situazons
(see [9-161 and the references
has concentrated
largely
12 - yI = T of pt(z, y). For finitely
ramified
for P(r, t), that
and lower bounds
therein)
on diffusion
P(r, t), the average
on the ‘propogator’
analogous
upper
[3-81. contained
fractals,
bounds
on geometric
(in some sense)
over
of the form (1) immediately
is, there
exist constants
such that
give for all
r,t > 0, ,st-ds/2 exp(
Physicists have estimated arguments); the so called diffusion
5 P(r,t)
--G ($)l”dw-l’}
equation
5 qd-“l2ug(
-Cg ($)“‘dw-l’}.
(3)
P(r, t) using a variety of methods: renormalization groups continuous time random walk formalism and generalizations
using fractional
derivatives.
These have lead to refinements
(scaling of the
of (3) of the form
P(r, t) = t-d*‘2S” exp {-co<‘}, for various asymptotic 0, then
LYand p, where E = r/tlidw. form of P(r, t) has p = d,/(d,-1)
appears
as a first-order
and p = d,
do better.
mathematics
literature.
One factor behind constancy behaviour is, small oscillations
correction
As yet, rigorous
Numerical simulations suggest that as < t 00 the and cy = 0. The 5” factor, with (I( = (d, -df)/?/2 < for smaller
asymptotics
<. Contrastingly, of analogous
as < + 0, values of a = 0
form have not appeared
in the
the lack of precise asymptotics in the mathematics literature is the near of the crossing time W = inf{t : X(t) E Go \ X(0) ( X(0) E Go}. That are present
in the distribution
of W [17,18]., These make exact
asymptotics
for W and thus X difficult. As yet the physics literature seems unaware of this behaviour though it is possible that it does not cross over from geometric to random fractals.
of W,
Another integrated
in the
important factor behind the problem is the fact that there can be oscillation density of states, N(X). This means that for the Sierpinski gasket, by [19],
The existence of this gap corresponds to the existence of localized eigenfunctions for the Laplacian on the gasket. For spatially homogeneous random fractals there is even worse behaviour than in the geometric case. Indeed, for the random fractals studied in [7,20], there exists dependent on the convergence rate of the environment sequence, such that ’ < ‘iyizf
N(X)
f(A)-lAd,/B
’
lim sup x+m
a function
f,
N(X) f(X)XdJ2
<
O”.
However, for random recursive fractals [21], there may be an ergodic effect which leads to more precise asymptotics. Of course, unlike fractals, disordered media are not statistically self-similar at all scales of measurement. Certainly not at the atomic scale. It is generally assumed in the physics literature that this is not a problem. This assumption has been justified to some extent in [22], where it is shown that for t 2 ~(a: - yl, the transition probabilities of X0 are of the same form (1) as the density of X.
B. HAMBLY AND 0.
132
D. JONES
2. BROWNIAN MOTION ON SELF-SIMILAR FRACTALS The construction
of Brownian
gasket to ‘nested fractals’ work of Fitzsimmons of the Laplacian geometric nested
fractals
constructing fractals.
referred
it is possible but
Brownian
the construction
gives approximations ramified).
on both nested
Other
of the Sierpinski
gasket
of Brownian
self-similar
Zhou
the class of
fractals,
and include
of a ‘harmonic
by Barlow
structure’
motion.
Brownian
and Bass [4,26,27], which
[28] have provided
and a similar
a method
of
class of infinitely-ramified
than infinitely-ramified
walks (see below).
subset can be isolated to look at a diffusion
constructions
and
fractals
Prior to the
to the construction
and hence, a Brownian
are more tractable of random
approach
the existence
carpet
and Kusuoka
of ‘nested’ sequences
one for which any bounded this means it is sufficient
a Laplacian,
on the Sierpinski
fractals
[23] from the Sierpinski
sets’, which are essentially
By assuming
to construct
motion
Finitely-ramified
a parallel
as finitely-ramified
fractals.
infinitely-ramified,
by Lindstrerm
et al. [6] to ‘affine nested fractals’.
‘p.c.f. self-similar
to by physicists
has also been constructed
is self-similar
was extended
[24,25] developed
He defined
and affine nested
on the fractal, motion
et al. [6], Kigami
on fractals.
fractals
motion
and then by Fitzsimmons
ones, as they allow
A finitely-ramified
fractal
is
by removing a finite set of points. In practice, on the fractal at such cut points. Figure 1
(finitely motion
ramified) on fractals
carpet
(infinitely
which are not usually
and Sierpinski
embedded
in Rd are given in [29,30]. All of these fractals
can be defined
as the closure
of the limiting
vertex
graphs, each a successively finer approximation to the fractal. A diffusion be constructed as the limit of a sequence of time-scaled nearest-neighbour approximating
graphs.
two-dimensional Brownian
subsets
motions
(Note that
on these sets to construct
and Zhou [28] showed that Justification
which self-similar
motion
the class of nested
on a fractal fractals
motion
on the limit.
established
However,
Kusuoka
walks in this case.) It is unclear
The uniqueness
by Sabot
of
of reflecting
is still based on local invariance
all posses to some degree.
which is not self-similar.
was recently
and a sequence
and random
of
on the fractal can then random walks on the
and Bass [26] used a sequence carpet,
motion
to use graphs
Brownian
fractals
of Barlow
the Sierpinski Brownian
it is also possible
for calling these processes
local isometries, Brownian
the construction
of R2 to approximate
set of a sequence
under
how to define
of Brownian
motion
on
[31].
The original constructions of Barlow and Perkins [3] and Lindstrom [23] use estimates of crossing times to prove the time-scaled sequence of random walks converges, and that the limit is a continuous strong Markov process. However, more successful has been the use of Dirichlet forms, developed largely in Japan [8,24,25,32,33], which give the Laplacian directly, rather than as the generator of a Brownian motion. The link between random walks and Dirichlet forms is provided by electric networks [34-361. This approach does however place some restrictions on the sort of random walks you can use. In particular, diffusion on a fractal it is still necessary Dirichlet
forms allow us to use a wealth
they must be symmetric. To construct nonsymmetric to use a sequence of random walks [37]. Also, while of analytic
techniques,
at present
we still need to know
something about crossing times to provide off diagonal bounds for the transition probabilities of the limit process. It is in the study of crossing times of diffusion on finitely-ramified self-similar fractals that branching processes appear. Define self-similar fractals in Rd as follows. For i = 1, . . . , N let & be a ‘similitude’, defined by &(z) = ai + (X - ai)/bi for ai E Rd and bi E (1,oo). For A c Rd let v(A) = @, q&(A). Hutchinson [38] proved cp has a unique fixed point F, which we call a self-similar fractal. Let W’s be the set of fixed points of the &. A point z E Ws is an essential fixed point if there exist 1 L i, j I N, and y # z E Wo such that &(lc) = +j((y). Let VObe the set of essential fixed points. Given some regularity conditions on F, in particular that it is connected and that there is no overlap of components (though their edges can meet, this is called the open-set condition), we can reconstruct F from its essential fixed points.
Modelling Transport in Disordered Media
Let h(l)...i(k)
=
h(l)
o . ” o &(k)
where 1 5 i(j)
133
5 n for all j, and define
N
v, =
U
h(l)...i(k)(h)-
i(l),...,i(k)=l
Vk then F = V,. we form graphs Fk = (vk, Ek) Clearly, VOC VI C . . . . If we put V, = UF=“=, as follows. Let EO be the edge set of the complete directed graph on VOand define for Ic > 0 E,, =
$i(q...i(k)(Eo).
U i(l),...,i(k)=l
For 1 5 i(l), . . . ,i(k) 5 IV, call C$i(l)...i(k)(F) a /c-complex. F is finitely-ramified if you can only get from one k-complex to another only via a point in vk. Specifically, for distinct sequences i(l), . . . , i(k) and j(l), . . . ,.?@), h(q...i(k)(F) f’ &(l)...j(k)(F) = h(l)...i(k)(h) f- @j(l)...j(k)(h). This property (called the nesting property) enables the definition of a nested sequence of random walks. The connected, open-set and nesting properties are essentially those required to specify p.c.f. self-similar sets, that is, finitely-ramified self-similar fractals. will assume F is such a fractal for the remainder of this section. Let X,
be a nearest-neighbour
random walk (n.n.r.w.)
inf{lc > 0 : Xn(k) E Vm} and T,,,(i + 1) = inf{lc > T,,,(i) the F, decimation of X, is given by Xm+(i) = X,(&,,(i)). will be a n.n.r.w. on F,.
If F is infinitely-ramified,
to another not directly joined by an edge in
on F,.
Unless stated othewise, we For m 5 n define T,,,(O)
=
: X,(k) E V, \ X,(2”,,,(i))}, then If F is finitely-ramified, then X,,,
then X,,, may jump from one point in V, A sequence (Xn)rzo such that for all m 5 n,
E,.
x,
% x,,, is called a nested sequence. Clearly, this is only possible for finitely-ramified selfsimilar fractals. For infinitely-ramified fractals a different approach is needed (see below). The first problem in constructing
Brownian motion on F is to find such a sequence. The correct time
scaling X, is then obtained by requiring the expected FO crossing times of Xn(t) := xcn(Xlzt) to remain constant. Clearly, Brownian motion will have to be symmetric and the restriction to any n-complex must be a scaled version of the process on F. These conditions enable us to specify that all the X,
be defined in terms of probabilities
p(x, y) = p(y,x)
for each (2,~)
E EO and
weights r(i) for 1 5 i 5 N. For any (z, y) E E, there is a unique sequence i(l), . . . ,i(n) that (z, y) = &(r~...~(~)(a, b) for (a, b) E Eo. Define the X, transition probabilities by
Pn(?Y)
m
such
da7b) 71
kElr(i(k))~
We need to know if this sequence of n.n.r.w. is nested.
For any given set of weights r(i),
the
Fe-decimation of Xi defines a map D on the set of EO transition probabilities. The fixed points of this map correspond to nested X,, sequences. Such sequences are sometimes called ‘decimation invariant’. The existence of such fixed points is proved for various classes of fractal by Fitzsimmons et al. [6] and Lindstrmm [23]. Hattori et al. [39] have shown that for some self-similar finitely-ramified fractals, no nondegenerate fixed point exists. As yet no one has characterized those fractals for which it does exist. The uniqueness of such fixed points is also still an open question for general PCF sets [31,40]. Clearly, we can look for other nested sequences of random walks. Generally, for this to be tractable some restriction needs to be made on the random walks, e.g., that they be spatially homogeneous. Such nested sequences have been constructed, and lead to diffusions which clearly are not Brownian motion. We will look at these in Section 3. The method used by Kusuoka and Zhou [28] to deal with infinitely-ramified fractals uses a different sequence of approximating graphs. Let z be an element of the interior of the convex hull
B. HAMBLY AND 0. D. JONES
134
I
I l I
I
Figure 2. Using essential fixed points, and alternatively using a point in the convex hull, to form approximating (level 2) graphs for the Sierpinski carpet.
of h, then put FL = (VI, EL> for k 2 0, where V,l = U~l~,...,i~k~=l h(l)...i(k)({~}) and if x and y are in adjacent k-complexes and the dimension of their common boundary do and df , for some do. For finitely-ramified
fractals
and we must take de = 0. For infinitely-ramified Figure 2 gives both type of graph approximation,
the boundary
(GY)
E EL
is between
can only be a single
point
fractals more complex boundaries are possible. Fz and Fl for the Sierpinski carpet.
Unfortunately, V,l $?!Vj,, for any k, nor is there any clear embedding of VL in Vi+I, so the idea of nested random walks cannot be applied. Nevertheless, it is still possible to consider the crossing times
of a sequence
of n.n.r.w.
X,
on FA and if these converge
(after appropriate
scaling),
we can hope to find a limit process. The potential theory approach also works, though analogous problems, and is the approach used by Kusuoka and Zhou [28]. For the infinitely-ramified fractals of [26,28], the uniqueness of the limiting process an open
problem.
the approximating transition density behaviour. 2.1.
sequence estimates
techniques
.An electric
Networks network
of its edges.
only give weak convergence
along
suffers is still
a subsequence
of
of processes. This has not been a problem for the computation of for the carpet [4], as all subsequential limit points have the same
In [28], the existence
Electric
sistance)
The current
then
of a self-similar
and
Dirichlet
on graph
diffusion
process on the carpet
was established.
Forms
F, is defined
in terms
Let a,(~, y) be the conductivity
C, a,(~, y). Th e D iric hl e t form associated
of the conductivities
(reciprocal
of re-
of edge (z:,y) E En and let b,(x)
with the network
=
(F,, a,) is defined for f, g : F, -+ R
by Wf,g)
= &(x7 Z,Y
Y)(f(X) - f(Y))MX)
&“( f, f) is interpreted as the energy dissipated when a potential The discrete Laplacian An on F, is defined by A”f(z)
= c
a&,
- S(Y))* f is maintained
on the network.
Y) f(Y;)$(x).
Y w h ere the inner product (., .) is w.r.t. b,, that is, and E” can be recovered from each other. A” is also the (f, 9) = C, f(x)g(x)bn(x). So, An generator of the n.n.r.w. X,, on F, defined by An and E” satisfy
&“(f,g)
P(&(i
= -(Anf,g),
4x7 Y) + 1) = y I &z(i) = x) = Pnb, Y) = b,(z).
Suppose now we have a finitely-ramified self-similar set F with approximating graphs F, as described above. Given conductivities a, for each F,, let (Xn)rzo be the corresponding sequence
Modelling Transport in Disordered Media of n.n.r.w.
The Laplacian
X m,n, is Am provided potentials
only at points
Equivalently, Kigami
corresponding
the electrical in V,,
the effective resistance
is equivalent
time-scaled
random
of Dirichlet
forms is monotone
of the Dirichlet
forms is implicit
Dirichlet
form,
to the convergence
of their Dirichlet
under the compatibility
forms is that
F. The effective
holds for graphs points, d;:
that
a, required
resistance
metric
to be the effective
dimension
process
on the fractal
them
for the
networks.
is a symmetric
regular Markov
of the Dirichlet form for analysing A, the
to the Euclidean
r(. , .) is defined
between
Scaling
of a limiting
Knowledge techniques
alternative
(r-metric)
resistance
but will not define a metric fructal
The sequence
to get compatible
the existence
the corresponding
gives rise to a natural
is, if d, > 2. Formally.
r-metric
be zero.
and hence, the limit
for which the limit is finite.
once we establish
know that
corresponding Laplace operator. The electric network approach two points
forms.
condition,
process, a fact which requires some effort to prove otherwise. of the limit process also allows access to a range of analytic
between
points
given by
we immediately
a fractal
at other
Finding a sequence of compatible random walks, and the convergence of
of nested
in the choice of conductances
of Dirichlet
of
when we specify V,
‘compatible’.
F is defined to the those f E C(F)
The advantage
is, the generator
in V, is the same for each network.
any two points
a sequence
increasing
forms is simply
where the domain Dirichlet
between
walks is equivalent
that
to (F,, a,)
let the net flow of current
of networks
to finding
of X,,
(F,, a,) is equivalent
and then
[8] calls such sequences
networks
to the F,-decimation
network
135
by setting
[41]. Note that
if the associated
metric
on
the distance this definition
process
does not hit
we can consider (the mass ,mr(r)
of F in a r-metric
ball radius
T scales
as
mT(7’) N r”?); d’,:
r-metric
walk dimension
(the displacement
The spectral dimension d, is unaffected sets, and conjectured to hold more widely
E r(X(O), X(t))
N tlld:u).
by a change in metric. It is true for p.c.f. self-similar [41], that d, = 2d’f/(d; + 1) and thus, if (2) also holds,
that d& = d; + 1.
So, we effectively d,:
resistance
(6)
lose a parameter
here, but ga.in instead
dimension
effective
(the
resistance
between
two points
x and
y scales
as
T(X, y) nJ ]Z - yldV,). Clearly,
d> = drldr
and dG = d,/d,.
Applying
this to (6) we obtain
the Einstein
relation
d, = df + d,
3. GENERALISATIONS The construction
methods
of Brownian
construct diffusion-which may be Brownian self-similar, and to construct on self-similar
motion
on self-similar
fractals
have been extended
to
motion-on geometric fractals which are not exactly fractals diffusions which certainly are not Brownian
To date, these extensions have been restricted to finitely-ramified fractals and have motion. hinged on the production of more general nested sequences of random walks, or equivalently, compatible sequences of Dirichlet forms. (Though recently Barlow et al. [42] have announced results in this area for the Sierpinski carpet.) In constructing Brownian motion on finitely-ramified self-similar fractals we look for fixed points of the map D on the space {p(a,b) = (a, b) E Ee, xca,b) p(a, b) = 1, p(a, b) = p(b, a)}, determined by decimating the random walks X, defined by equation (5). This leads to a ‘decimation invariant’ nested sequence. By relaxing the condition p(a, b) = p(b, a), Kumagai [37,43]
B. HAMBLY AND 0. D. JONES
136
obtained a nested sequence on the Sierpinski gasket which gives in the limit a diffusion which is nonsymmetric construction Hattori
but invariant under rotation of the fractal.
There is no analogous Dirichlet form
for this diffusion. et ~2. [44,45] obtained
nested sequences
by looking at trajectories
of the inverse
They have done this for the Sierpinski gasket, producing diffusion which has lo-
map D-l.
cal behaviour which is horizontal movement asymptotically, a&-gaskets,
for which no nondegenerate
and for a class of fractals
called
fixed point exists for the map D, for certain values of
a, b, c. Hattori et al. use new results on branching processes to show their nested random walks converge [45,46] ( see also [47]).
Recently
however, Hambly and Kumagai
[48] have used the
electric network framework to reproduce and extend these results and provide diagonal bounds for the associated
transition
densities.
Off diagonal bounds have yet to be produced for these
diffusions. It is also possible to consider (forward) trajectories
of the map D. These arise when considering
infinite fractals, which can be constructed by applying the inverse similitudes $il
to the fractal F.
We can specify the behaviour of an anisotropic random walk at a given (microscopic)
scale, and
then ask about its large scale (macroscopic) behaviour. It has been shown for a variety of fractals that as the scale increases you get increasingly isotropic behaviour [42,48,49]. This corresponds to the convergence of the forward trajectories
of D to a nondegenerate fixed point, representing
Brownian motion. This is a form of homogenization for operators on fractals which is induced purely by the geometry of the fractal, and not through some ergodicity of an underlying random operator. In the class of geometric fractals there are some which are not exactly self-similar, the random recursive Sierpinski gasket of [21], or the scale-irregular
gaskets of [7,20,45].
such as These
are produced by allowing some variation in the family of maps used in the construction of the fractal, and as such can be regarded as a step closer to disordered media than exactly self-similar fractals.
These examples are still finitely-ramified
however, and it is still possible to construct
nested sequences of random walks for them. For the scale-irregular gaskets the variation in the maps is spatially homogeneous and governed by an environment sequence. For random recursive fractals, which have no spatial homogeneity, we can still use the nested sequence of random walks. However, some care must be taken when choosing the ‘levels’ of approximation. This can be achived by working in the effective resistance metric,
as the probabilities
for the Brownian motion to exit a region are determined
by the
resistance. The effective resistance metric is then the appropriate metric in which to get transition density estimates. 4.
CROSSING
Let F be a finitely-ramified
TIMES
AND
BRANCHING
self-similar fractal and (F,)~&
PROCESSES
a sequence of approximating graphs
as described in Section 2. If (Xn)F& is a nested sequence of random walks on F, then the implicit branching process is constructed using the jumps of these random walks. The original ancestor is a single jump made by Xo, the first generation are the corresponding (via the decimation process) set of steps made by Xi, and so on. So, the children of any individual in the nth generation are the jumps made by Xn+ir which, when decimated, produce the given X, jump. The growth rate of the branching process immediately gives the time scaling for the random walks, and the normed limit of the branching process is the crossing time of the limiting diffusion. In general, the branching process will be multitype. Set the type of each jump to be the type of the edge (2, y) along which it travels. However, as #En -+ 00 we need some way of rationalizing the number of types before the branching process can be useful. In the case of Brownian motion, when conditoned on staying within a given n-complex, the motion of X,, depends only on {~(a, b) : (a, b) c Eo}. S0, it is sufficient to restrict ourselves to a single set of types {t(a,b) : (a, b) E Eo}, and set the type of an edge (z, y) E En to t(a, b), where
Modelling
(GY)
Transport
in Disordered
dql)..+)(a, b) forSOme 1 I i(l), . . . , i(n)
=
(a, b) E I&}
are a fixed point
Media
137
< N. In this case, the probabilities
of D, and so we get a classical
multitype
However, nested sequences corresponding to trajectories of D-l with varying environment, as the underlying transition probabilities, the number
of jumps,
the construction process
change at each level [45-47,501.
of Brownian
in a random
The crossing
environment
times
sequence
same as the time scaling on the normed
limits
process is just
of branching
of bounding
a large deviation
problem
produce branching processes and thus, the distribution of
the nested
random
X are given
the expected
used for the random
The problem
Similarly,
on a homogeneous
process.
fractal,
sequence
produced
in
gives rise to a branching
[7,51].
of the limit
The norming
process.
motion
:
{p(a,b)
branching
processes
the normed [17,18].
walks.
by the normed
limit
size of the branching
of the branching
process,
This has lead to a number
and is the
of recent
results
[17,18,47,51-541.
limit of a (supercritical)
In particular,
branching
the event of minimal
process
growth
is essentially
is of central
im-
portance. Determining the minimal growth rate is nontrivial for multitype processes [5,6], and when constructing Brownian motion on a finitely-ramified self-similar fractal, is equivalent to determining a shortest path metric on the fractal [6]. The shortest path metric is called chemical distance
in the physics
ch,(z, y) be the length
literature,
and, if it exists,
of the shortest
path
is determined
5 and y in F,.
between
grows like 1” for some 1 > 1, then we define the chemical d,:
chemical dimension
For the Sierpinski Lindsrtom
d?: ch-metric VP(r) Again, When
y)/P
we have 1 = 1.
and carpet
If the length
of this path
d, by
N ]z - y]“’ =: ch(x, y)).
[23]. As with the effective resistance
For an example metric,
with
we can redefine
1 > 1 see the the fractal
and
using this metric fractal dimension walk dimension
d, is unaffected
the relation
(the mass me(r)
(the displacement
by a change
of F in a ch-metric
ball radius
r scales as
in metric.
Ech(X(O),
X(t))
N t”“:‘).
We have d? = df/d,
and d”, = d,/d,,
so they
df and d, do.
(2) whenever
ch(. , .) exists, off diagonal
shown to hold for the Sierpinski carpet
ch,(x,
dimension
For 2, y E V,, let
N rd?);
d&: ch-metric satisfy
gasket
snowflake
walk dimensions
(lim,,,
as follows.
bounds
gasket,
for the transition
nested
fractals,
density
affine nested
of the form below have been fractals,
and the Sierpinski
[3-6,361
It is worth
noting,
that
bounds
of this form (using d; rather
t,han d,)
are not currently
seen in
the physics literature. When fractals are not exactly self-similar such tight uniform bounds on the heat kernel are generally not possible. In the case of scale-irregular gaskets, we can obtain estimates which are best possible for the diffusion. They are much as (7), but are modified by the introduction (if there is any) of the of a correlation factor [20], which depends on the speed of convergence environment sequence to its ergodic limit. For random recursive fractals [2], there are as yet no best possible bounds on the transition density, as the relationship between the short paths in the fractal and the effective resistance metric is still not well understood. However, it is clear that there will be oscillation in the bounds. It is tempting to think that by scaling differently the time taken by jumps of different type, you might obtain a different limit. From our understanding of the limiting mechanism for branching
B. HAMBLY AND 0. D. JONES
138 processes,
we can see that
of a branching for different
process
this is pointless.
is deterministic,
types of jump
the heat
kernel,
transition
with respect in Section
motion.
density
ESTIMATION
solution
to reflection
motion
in the perpendicular
of the limit
process.
of another
[47].
that
as its generator.
This means
on F, is the transition
self-similar
bisectors
compact
of the essential
fractal,
that
density
of the ideas used in the estimation
on F, an exactly
limit
FOR FRACTALS
to the heat equation
We give here an outline
for Brownian
less than
F has the Laplacian
Xt on a fractal
the fundamental
of the Brownian
types in the normed
So, scaling time differently
time scaling
rate of one type is strictly
5. HEAT KERNEL motion
between
would lead only to a deterministic
This is true even when the growth
The Brownian
The distribution
only the total size is random.
of the
symmetric
hxed points,
as defined
2.
The following
heuristic
argument
shows how the functional
form of the transition
density
arises
from the natural scaling of the process. Prom the exact self-similarity, and the definition of d,, for certain lengths 1. That we should have locally the relationship Xt = Z-IX ldwt in distribution, is P”(Xt where
BT(z)
Laplacian
E B,(z))
is the ball of radius
A on F is a compact
T about operator:
sition, with which we can construct Hausdorff measure ~1on the fractal. should
= Pll(Xld”,t 2.
E B&r)),
The transition
by Mercer’s
(8)
density,
Theorem,
pt(z,g),
A admits
will exist as the
a spectral
decompo-
the density. Note that this is a density with respect to the Now, use (8) with a change of variable, to show that ~~(2, y)
satisfy ?‘t(& Y) = ~%ldwt(&
Hence, we can deduce
that
the functional
&,
for some continuous
function
y)
VX,~EF,
l?/),
form of the density
N
t-d./2g
and determine
should
be
d(x;yJdw )
( )
g. Here, we have assumed
In order to prove this estimate
O
that
the function
d, = 2df/d,,
as in (2).
g, we begin by establishing
a bound
on pt(x, z) which is uniform in 2. Initial techniques [3-51, relied on estimating the Green density and using Tauberian theorems to turn this into estimates on the heat kernel. This use of this idea is restricted
to the case where d, < 2, when the Green
density
is bounded.
Subsequent techniques have used analytic results, which relate the time decay of the heat kernel to various inequalities for functions in F, the domain of the Dirichlet form. The Nash inequality is said to hold if there exist constants A and u such that
Ilf II;+4’”5 A (WJ) + Ilfll;) Ilfll:‘“7 This is equivalent to the following that for 0 < t < 1,
on-diagonal
heat kernel bound:
sup pt(x, y) L Cp@. WIEF
V’fEF. there exists a constant
cl such
(9)
In [6] the Nash inequality is obtained directly in the case where v = d, < 2, using the scaling in the fractal. In [48], ideas in [28] are used to get such an on-diagonal bound from a Poincare inequality: for some constant C
Modelling Transport in Disordered Media
This type of inequality extended
arises when the Laplacian
has a spectral
the diagonal,
we must examine
the most likely paths.
Brownian
ramified
motion.
then the greatest
the short
fractals,
paths,
where Tf = inf{t
the embedded
which correspond
is that
the following
If we think of the transition
contribution
density
to the sum will come from
whereNn,n+mis
the number
in F,. This inequality, of Wzp, contains enough following
branching
process
encodes
the small time behaviour to branching
processes
super-branching
> Tf_, : X, E Fk \ {XT;_,))
make the ith step on level n. Then,
path
with few offspring.
and T!:, = 0, that
the super-branching
of the
In fact, all
Let IV? = T,” - TiT1,
holds.
is, WF is the time required
inequality
path in F,+,
of steps in the shortest
the whole path
of pt(z, y), we need to know
inequality
to
is
between
two adjacent
points
coupled with a weak uniform estimate on the tail of the distribution information to get an exponential estimate on the crossing time, of the
form P” (WT < t) < c2 exp (--~a (E” (Wl”) t)-‘)
cr,cz
are constants.
the mean
crossing
self-similar
fractal.
The off-diagonal radius
term and we use a density away from
For a short time the most likely path will be close to the shortest
In order to examine
we need to observe
where
can be
the points.
For finitely about
the off-diagonal of the transition
the short paths in the fractal.
x, y as a sum over all paths,
between
gap, and this approach
to the case where d, 2 2.
As yet there are no analytic techniques for calculating probabilistic argument. In order to estimate the behaviour between
139
p is determined
time,
as @ = log(Nn,,+,)/log(E(Wy+n)/N,,,+,)
upper
bound
d(z, y)/2 about
neighbourhood
The exponent
follows from a probabilistic
x and split the probability
of y into two parts,
of moving
by conditioning
t < 1,
,
‘~xEF,
by the length
(10)
of shortest = d,/(d,
argument.
We construct
from the neighbourhood
on whether
path
and
- d,), for a a ball B of of x to the
or not the process is in B at time
t/2. As the process is reversible, we can deal with each part in the same way. We can, without loss of generality, consider the process conditioned on not being in B at time t/2. Now, further decompose the transition probability, time t/2. The first half of the sample
this time conditioning on the position of the process at path can be dealt with using (lo), as we know the process
must have crossed a set of size at least d(x,y)/2. The second half of the sample controlled using (9). Combining these, we obtain the upper bound of (1) and (7). There
are three
steps to obtaining
a lower estimate
for the transition
density.
path
can be
The first is to
get an on-diagonal estimate. As we have an upper bound on the transition density, the process cannot exit a ball about x too rapidly, and hence, from the upper bound itself or (lo), see [4j, we can show that there is a constant cd such that pt(x,x) 2 c4t-ds/2. The next step is to determine a ‘near’ diagonal estimate. In other words, what is the size of ball for which the on-diagonal estimate is good? This involves an estimate on the Holder continuity of the heat kernel. In the finitely-ramified case, we control functions in the domain of the Dirichlet form in terms of the effective resistance metric T(Z, y), see [8]
If(x) - f(Y)12 I +7 Y)E(f>fL
Vx,y~
F,
f E3.
As pt(x, .) is in the domain F, we have immediately that its Holder order is d,./2. This sort of estimate is available only when d, < 2, and we have an effective resistance metric. Other
B. HAMBLY
140
AND 0. D. JONES
techniques are required if d, > 2. Combining this with the on diagonal estimate gives the result that there exists a constant cs such that for all 2, y E F and 0 < t < 1, if d(x, y) < c&dw.
pt(z, y) > yJ2,
(11)
The final step in the procedure is to use a chaining argument. We consider the shortest path between the points x and y and determine the scale at which we should view the path according to the allowed time to move between the points, then cover the path with balls of the appropriate size to apply our near diagonal estimate. We let 5 = d(x, ~)~w/t, and for the short time behaviour we examine < large. In order, to apply (11) we must choose a path {zi}~=r in the fractal such that d(xi, xi+l) = d(x, y)/N
5 cg(t/N)l/dW,
then for r = d(x, y)/N,
we have
Using our previous estimates, and the fact that the natural measure ~1is the Hausdorff measure, we have pt(x,y)
> (:
($)-dS”)N
((cs
(;)“‘JdJN-l
2 c&%:.
The choice of N 5 c&l/(dw-l) required to fit the criteria gives the lower bound of (1). The lower bound of (7) comes from taking into account the scaling of the shortest path.
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