Measure derivative and its applications to $sigma$-multifractals

J. Korean Math. Soc. 36 (1999), No. I, pp. 229-241

MEASURE DERIVATIVE AND ITS APPLICATIONS TO IT-MULTIFRACTALS

TAE SIK KIM, TAE HOON AHN AND GWANG IL ^KIM

ABSTRACT. The fractal space is often associated with natural phe- nomena with many length scales and the functions defined on this space are usually not differentiable. First we define a O'-multifractal from O'-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the

multifractal. We show that the non-differentiable functiori on the O'-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary dif- ferential equations and diffusion processes on O'-multifractal spaces.

1. Introduction

Recently there have been many attempts to investigate the fractal sets and apply them to the multiple length-scale phenomena of the na- ture [5]. The archetype property of the fractal sets is identified as the self similarity. The simplest sets with these self-similar properties can be obtained by using the iterated function systems [2]. The iterated function systems have been widely studied since they can be applied to the various branches of science, for example, the image and signal processings [3] and the crystal lattice dynamics [4]. Now the math- ematical research on the anomalies of fractals includes the studies of quantum mechanics with fractal support [9] and the SchrOdinger equa- tion in complex fractal graphs [10]. There have been continuous efforts to apply the fractal geometry to understand diffusion in fractal media by means of fractal operations [6].

In the studies of fractal phenomena, it became increasingly evident that the mathematical calculus, which is not appropriate on fractals,

Received August 24, 1998.

1991 Mathematics Subject Classification: 28A80, 58G32, 60J18, 60J60.

Key words and phrases: O'-multifractal, measure derivative, diffusion process,

Brownian motion.

should be extended to the fractal space. For example, Kigami at- tempted to develop calculus on the self-similar fractals based on the harmonic measure [8] and Giona generalized the ordinary differential equation to the fractal support with the help of corresponding integral equations [7].

In this paper we first define the u-multifractals with the help of u-iterated function systems with probability and introduce a measure derivative on these sets. We show that the non-differentiable function on u-multifractal becomes differentiable with respect to the measure derivative. As an illustration, we apply the measure derivative method to some examples of ordinary differential equations and diffusion pro- cesses on u-multifractal processes.

2. u-iterated function systems and u-multifractals

Let X =

Xi be the union of non-overlapping metric space Xi'S with metric d. And suppose that for each Xi, there exists a finite number

of contraction maps

= (w}, w; ,... ,W?i) on Xi, w{ :

Xi

Xi. Then there exists the invariant set F

called the self-similar set, such that

F i = Uj;;l w {(Fi).

This self-similar set is a typical type of fractal sets and in some cases is simply called the fractal. In order to treat a more general form of these similar sets, we define the u-similar set, F =

and consider the measure structure on this set. For each i, let the probability vector Pi = (pLp~,··· ,p~J, in which 0 < p} < 1 and Ej~l p} = 1, be associated with the map

Then (Wi, Pi) is said to be the iterated junction system with probability (IFSP) [2]. We also call {w = (Wi),P = (Pi)) the u-iterated function system with probability (u-IFSP).

Let M(X

be the space of all probability measures defined on the u-algebra of the Borel sets of Xi. Then the Markov operator 1i :

M(X

M(X

defined by

ni

Tdm(A)] = LP;m

^(wj)-l(A),

^M(X),

j=l

has a unique invariant meaure J..Li with support on F

such that

TdJ..Li] = J..Li·

Measure derivative and its applications to u-multifractals 231

And define the measure J1. = E~l J.Li on the attractor of IT-IFSP (w, P) by

00

J.L(E) = L ^J.Li(X

^nE).

i=l

Then this measure has support on the IT-similar set, :F =

We can see that :F has a multi-fractal structure. So from now on, we call this set, :F, the IT-multifractal with invariant measure J1. or simply the J.L-multifractal.

3. Measure derivative on IT-multifractals

Let:F be a J1.-multifractal, that is, IT-multifractal with invariant mea- sure J1. as in Section 2. And let >.. be a J.L-absolutely continuous mea- sure on :F, that is, >.. < < J.L. Then by the Radon-Nikodym theorem [10], there exists a J1.-measurable function f on:F, denoted formally by

~~

f, such that

>"(E)

L f(w}dJ1.(w).

From now on, we focus on the J.L- multifractal :F in

DEFINITION 1. Let:F be a IT-multifractal, >.. be the measure defined above and let F be a function defined

Then for x

:F, the measure derivative of F(x), dA1

F(x) == DAF(x), is defined by

d>..~x]F(X) ⁼ l~[F(x ⁺ h) - F(x - h)]j>"[(x - h, x + h)], when the limit exists.

REMARK

1. When>.. is Borel measure and Z = {... ,-1,0,1"" }, the measure derivative of the continuous function F at a.e. x can be defined by using the dyadic intervals, that is,

d>..~x]F(X) ⁼ !~[F(k ^{+ 2-}

^{(i + 1))}

- F(k + 2-

i)j>"[(k + 2-

i, k + 2-

(i + 1))],

where (k + 2-

i, k + 2-

(i + 1)) is a dyadic interval of length 2-

containing x, k E Z, i = 0,1" .. ,2

1.

Let Xi = [ai, bi] be the closed interval and {wn~;l be given such that WHai) = ai, w~i(bd = bi and

dist{ w{ (Xi), W~(Xi)} ~ max{ diam[w{ (Xi)], diam[w~(Xi)]}

for each neighboring interval w{ (Xi) and W~(Xi). Then F is a general- ized Cantor set as a JL-multifractaL

THEOREM

1. Let F be a generalized Cantor set defined as above and F be a continuous function defined on

such that

F(x) = F(W{l

w{2

w{k(bi»

. . . l If If

for each x

(wr owr

owik(bi), w/ ow/

ow/'(ai» where

. . . If If If

wi

owi2

owik(Xi) ^and w/ ow/

.ow/(Xi) ^{are the}

basic intervals. Then d

dp.[x]F(x)

= lim F(W{l

w{2

W{k(~i». - F(u:{l

w{2

w{k(ai» ,

k-+=

PIlPI2 ... PIk

for almost all x

F such that x

w{2

w{k (ai),

w{2

w{k(bi)J.

Proof. For each x

F

there exists a sequence (jl,h,"'), ik =

1,2"" ,ni, such that x E

OW{2

OW{k(Xi). Then for each b _{asIC In} . . t _er val wi

^Wi

^Wi

ik(X) - i - wi [ir

wi h

^Wi ... jk() ai, wi

W{2 0...ow{k(bi )], ^{let hk} = W{lOw{2

ow{k(bi)-W{lOW{2

ow{k(ai)' Then hk '\t 0 as k

00 and by the invariance of p.,

JL(x - h, x + h)

= JL([w{l

w{2

w{k (ai),

w{2

w{k (b i )])

= pflpf2 ... ptk.

Then as in Remark 1, d

dp.[x]F(x)

F(W~l

W~2

w~k(b·» - F(W:!l

w!2

w~k(a·»

li

=k-+~

pflpf2 ... pfk '0

Measure derivative and its applications to cr-multifractals 233

PROPOSITION 1. For D>..-difIerentiable functions, F and G on :F, and two real numbers a and b, we have

(a) d>..1x] [aF(x) + bG(x)] = a d>..1x)F(x) + bdMx]G(x), (b) d>..1x] [F(x)G(x)] = [d>..1x)F(x)]G(x) + F(x)[d>..1x]G(x)].

In [6], Giona defined the derivative V/L on IFSP with invariant mea- sure {L as the inverse operator of integration, that is, when

I/LF(x) = lX ^{F(w)d{L = G(x)}

he defined I;lG(X) = V/LG(x) = F(x). And he used this derivative to transform a differential equation to the integral equation. In follow- ings, we show that the measure derivative coincides with the Giona's derivative in this sense. Therefore, we can efficiently transform the integral equation into the differential equation on fractals and investi- gate the diffusion process on fractals through the differential equation under the measure derivative.

PROPOSITION 2. Let J.L and A be defined as above and define

vex) = lX j(w)d{L(w), and [v(x)]n = vn(x). Then

(a) D>..[vn(x)] = V>.. [vn(x)].

(b) Let F(x) be an infinitely D>..-difIerentiable at x = a. Then F(x) can be extended to the following form of Taylor series on a neigh- borhood of x in which the series converges:

(1)

00

F(x) = L an [v(x) - v(a)t

n=O

where an = Din) F[v(a)JjnL

Proof (a) d>..1xj v(x) = limh----toJ:~:dA(w)jA[(x-h, ^x+h)]

^1.

Then d d

dA[X] vn(x) = n[dA[X] v(x)]v n

-

(x) = nvn-1(x),

by Proposition 1. Now put VO(x) = 1, VI(X) = vex) and define vn(x) by

vn(x)

lX d)"(xn) lxn d).,(Xn-I)" ·lx2 !(xl)dj.L(XI)

= l ^x d)"(xn) lxn d).,(Xn-I)" .lx2 ^d).,(xr).

Since

Vn(X) l

l

^vn-l(w)

- , - = vn(x) = vn-l(w)d)"(w) = ( _ 1)1 d)"(w),

n.

n .

we have

Vn(X) = lX nvn-l(w)d)"(w).

Therefore V>.[lIn(x)] = I;l[vr..(x)] = nvn-l(x) and (2)

(b) A series expansion can be done similarly to the usual calculus. 0

4. Applications

It can be shown that when F(x) defined on

is differentiable in . usual sense, it is also D>.-differenti.able since

can be regarded as the u-multifractal with Lebesgue measure. But the converse does not hold in general. For example, the Cantor ternary function F (x) is not differentiable on the Cantor set in usual sense. However, for the s- dimensional Hausdorff measure, )." with s

log 2/ log 3, we have for every x E C,

Thus the measure derivative is well-defined and can be seen as a gen-

eralized form of the derivative on fractals, which can be applicable to

the motion on the u-multifractal :F.

Measure derivative and its applications to cr-multifractals 235

4.1 Differential equation on u-multifractals with respect to the measure derivative

Let x

1R, y and H are n-dimensional vector functions defined on 1R and 1R

respectively. Then the solution of the Cauchy problem

(3) dy

dx = H[y(x)], y(O) = yo,

has a unique solution in a neighborhood of x = 0 when H is Lipschitz continuous. Note that the above differential equation in ⁽¹⁾ can be recast in the form of a nonlinear Volterra-type integral equation

y(x) = Yo + i

H[y(~)Jd{,

which is particularly useful in functional analysis.

From this point, we can extend the initial value problem involving ordinary differential equations on !R to one in u-multifractal. That is, supposing that y and H are real valued functions defined on !R, the differential equation in (3) is extended to the u-multifractal :F as follows:

(4) or

d'x[x] dy

H[y(x)], ^{y(O) = yo,}

y = Yo + l

H[y(w)]d'x(w),

where the integral on the right hand side also exists under the condition of Lipschitz continuity of H.

EXAMPLE

1. Let Xi = ri, i+1], w}(x) = ~x+~, W[(x) = ~x+~+~

and p; ⁼ p; ^{= 1/2 for all i E Z. Then :F = !R and J-L} is the Lebesgue measure. Thus for

= J-L, vex) = 'x([O, x]) = x and d'x(x) = dx. Thus in this case, the equations (3) and (4) coincide.

Since the solution of the usual Cauchy problem in 1R has the form

of

y(x) = L ^anx

n=O

we can see that the solution of a similar Cauchy problem in the a- multifractal F also has the form of

00

y(x) = L ^{an [v(x)t,}

n=O

from Proposition 3.

EXAMPLE

2. As in

we define the exponential type function on the a-multifractal F by

(5)

00

exp[v(x)] = ~)v(x)t ^/nL

n=O

Then it can be easily shown that y = Yo exp[kv(x)] is the solution.of the differential equation

d.A[x] dy = ky(x), y(O) = ^Yo,

on the a-multifractal:F with respect to the measure.

In the following figures, we show the graphs of two functions on the subset [0, 2] of a-multifractal generated by the IFSP in Example 1 where probabilities, for example, are chosen to be PB ⁼ 1/5 and

pi

^3/4.

vex)

...

...

... ...

Fig 1. y = vex)

exp[v(x)]

I .

a.~

Fig 2. y = exp[v(x)]

Measure derivative and its applications to er-multifractals 237

4.2 Multifractal diffusion process on the u-multifractals Let :F be the u-multifractal with invariant measure J.L and let A and v be defined as in Section 2. For the random variable X

of:F defined on the probability space n with probability measure P, define XZI(t) by [XZI(t)](w) = v[Xt(w)] for each wEn.

DEFINITION 2. The process {XZI(t) : t 2: O} is the multifractal dif- fusion process on F with drift 8 if

(i) it has a stationary independent increment, (ii) it has a distribution of

(XZI(t) = XZI(O) + 8t, ([XZI(t) - X ZI(0)]2) ⁼ t and ([XZI(t) - XZI(O)]k)

o(t) for k > 2.

NOTE. When XZI(t) is normally distributed with XZI(O) = 0, it can be seen that XZI(t) = B(t) + 8t for the standards Brownian motion process {B(t)}

However, in this paper XZI(t) does not need to be the Brownian motion process.

Let XZI(t) be distributed with the density function p[v(x) , t] on v(F).

Since P[XZI(t)

B] = P[X(t)

v~l(B)] a.e.[A], we can define the probability density function p(x, t) on F such that

1 p(x, t)dA(X) = ( p[v(x), t]dv(x).

ZI-1(B) lB

From the independent and stationary increment, the conditional probability density of XZI satisfies

p[v(x), t + s; v(y), s] == ^P[XZI(t + s) = v(x); XZI(s) = v(y)]

= P[v(x), t; v(y), 0] == p[v(x), t; v(y)]

which corresponds to the conditional density function p(x, t;y).

Now we will show that the above definition is reasonable in that

the conditional density function satisfies two types of the diffusion

equation, Kormogrov's backward and forward diffusion equations (also

called Fokker-Planck equations).

THEOREM

2. The conditional density function p(x, t; y) defined above satisfies the Kormogrov's backward diffusion equation, that is,

Proof. From P(X(t)

A;y) = fAP(x,t;y)d>"(x) = fAf:FP(xh,h;y) p(x, t - h; xh)d>"(Xh)d>"(x) = fA E[P(x, t - h; xh)]d>"(x) , we have

p(x, t; y) = E[P(x, t - h; X(h»] a.e.[>..].

Recall that (Xv(h) - v(y) = 8h, ([Xv(h) - v(y)]2) = hand ([Xv(h) - v(y)]k)

o(h) for k > 2 from the definition. We expand the right hand side of the above equation in Taylor series about (x,t;y) as in Proposition 3(1) and use the change of variables so that

p(x, t; y)

= E[P(x, t - h; X(h))]

8 8

= Efp(x, t; y) + (-h) atP(x, t; y) + (X(h) - y) 8>"[x]P(x, t; y)

h2 8

(X(h) - y)2 82

+"2 at2P(x,t;y) + 2 8>,,[xj2P(x,t;y) + ... ]

8 8

= p(x, t; y) - h atP(x, t; y) + E[{Xv(h) - v(y)}] 8>"[x]P(x, t; y)

h2 8

E[{(Xv(h) - v(y)}2] 82

+ "2 at2P(x, t; y) + 2 8>..[xj2P(x, t; y) + ...

8 8

= p(x, t; y) - hatP(x, t; y) + 8h 8>" [x] p(x, t; y) h 8

+ 2" 8>..[xj2P(x, t; y) + o(h).

Rearranging terms and then taking the limit h

0, we have

1 8

8 8

2 8>..[xj2P(x, t; y) + 8 8>"[x]P(x, t; y)

atP(x, t; y). 0

Measure derivative and its applications to u-multifractals 239

3. The conditional density function p{x, t; y) defined above satisfies the Kormogrov's forward diffusion equation, that is,

Proof. The proof is similar to that of Theorem 1 except that we

work with conditioning on X (t - h). 0

Ai:, an example of the diffusion process on multifractals, consider a random walk on the O'-multifractal.

EXAMPLE

3. Let Z be the set of integers. And let Xi = ri, i+ 1] for i

Z, w}{x) = r}x+{1-rl)i and w;{x) = r;x+{1-r;)(1+i) with 0 <

pI, p; < 1, 0 < rt, r; and rt + r; ::; 1 for all i

Z. For this IT-IFS,

multifractal set :F c

is a countable union of Cantor type sets with an independent geometric structure. Now we define some Brownian type motion on this fractal with respect to its invariant measure. Suppose that at every time step of

a particle standing at Xi goes to either xi+! with probability p, 0 < p < 1, if lI{Xi+l) - lI{Xi) =

or Xi+!

with probability 1- p if lI{xd -lI{Xi+l) =

Suppose that v{xo) = 0 for the initial state Xo of the process." Let

When X{t) denotes the position during time t, we have

in which [tj

denotes the largest integer not greater than tj

Since for each i,

we have

(Xv{t)) =

- 1)

and

Put D.v vzs:t, ^p = 1/2(1 + 8vzs:t) and let D.t

O. From the central limit theorem the distribution of Xv(t) converges to a normal distribution such that

Thus we define A-Brownian-type motion {Xv(t) : t 2:: O} on the u- multifractal :F so that Xv(t) is normally distributed with a mean Xv(O) + 8t and a variance t with respect to the measure A, that is,

p(x,y; t) = 1/V21rtexp[-{v(x) - v(y) - 8t}2/2t],

where exp[v(x)] is defined as in (5). Clearly the above process {Xv(t) : t 2:: O} is the multifractal diffusion process from Definition 2 anti so p(x, t;y) satisfies the diffusion equations, which can also be shown by direct differentiation using (2).

ACKNOWLEDGEMENT. This research has been supported through the Postdoctoral program of the Korean Research Foundation. We also acknowledge support from the Korea Science & Engineering Founda- tion through the Global Analysis Research Center at the Seoul National University. And we would like to thank Prof. Seunghwan Kim. for his helpful advice and hearty encouragement during the discussions.

References

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Sons, NY, 1974.

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[5] M. Fleischmann, D. J. Tildesley and R. C. Ball, Fractals in the Natural Sci- ences, Proceeding of the Royal Society of London, Princeton University Press, Princeton, 1989.

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Roman, Fractional diffusion equation on fractals; one dimensional case and asymptotic behaviour, J. Phys. A 25 (1992), 2093-2106.

[7] M. Giona, Fractal Calculus on [0, 1], Chaos, Solitons & Fractals 5 (1995),

987-1000.

Measure derivative and its applications to u-multifractals 241

[8] J. Kigami, Harmonic calculus on P.C.F. self-similar sets, Trans. Am. Math.

Soc. 335 (1993),721-755.

[9] M. S. El Naschie, Quantum mechanics and the possibility of a Cantorian space- time, Chaos, Solitons & Fractals 1 (1991),485-487.

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Tae Sik Kim

Department of Mathematics

Nonlinear & Complex Systems Lab.

Pohang University of Science & Technology Pohang 790-784, Korea

Tae Roon Ahn

The Nonlinear Centre

Department of Applied Mathematics and Theoretical Physics University of Cambridge

Silver Street CB3 9EW ENGLAND

Gwang Il Kim

Department of Mathematics

Kyeongsang National University

Chinju 660-701, Korea