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Random Phenomena with Fractal-like Features

Patricia Alonso Ruiz

Communicated by Notices Associate Editor Scott Sheffield

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It is summer. My brother and I are walking along the seashore and arrive at a stony formation: The salty water that passes through the rocks from time to time creates tiny lagoons, where some kind of seaweed seems to find its ecosystem. We are captivated by these “microlakes” and my fascination grows when we discover how the seaweed seems to randomly climb and dry on the rock, creating a fractal pattern; see Figure 1. I draw my brother’s attention: “Look, this is a fractal!” Excitement rises, we take out our phones and capture yet another instance of random phenomena in nature exhibiting fractal features.

Figure 1.

Pointing to salty seaweed on a rock.

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An artist/mural painter himself, my brother is as curious an observer as I am and our discovery is going to initiate an interesting conversation. “What is, again, a fractal? How do you mathematicians know that something is fractal?”

1. What Makes It Fractal?

“How does a mathematician describe the notion of fractal?” My brother’s question puts me in an unusual situation because here we find a mathematical concept without a unique or generally established characterization. The term fractal was coined by Mandelbrot in Man75 in analogy to the Latin word fractus, which means broken. He would use it to describe objects whose geometry was too “broken” to fit into a traditional setting. His original mathematical definition, which he would later realize was too restrictive, was that a set is fractal when its Hausdorff dimension is strictly larger than its topological dimension. The topological dimension of a nonempty set is a nonnegative integer defined recursively: it is 0 if the set is either a singleton or totally disconnected, it is 1 if each point has arbitrarily small neighborhoods whose boundary has dimension 0, it is 2 if each point has neighborhoods with boundary of dimension 1 etc. For instance, the topological dimension of an interval is 1; see Figure 2. The Hausdorff dimension of an interval is also 1—Mandelbrot would thus say that is not fractal.

Figure 2.

Any point has an -neighborhood whose boundary, two totally disconnected points, has topological dimension 0.

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What about the Hausdorff dimension of a more intricate set? In contrast to the topological dimension, the Hausdorff dimension may be any nonnegative real number. Broadly speaking, this dimension captures the proportion between the diameter and the mass of an object, where the diameter is usually considered with respect to the Euclidean length and the mass refers to the Euclidean Hausdorff measure. The latter generalizes the concept of length (), area () and volume () to any and goes back to ideas of Carathéodory: Given a set and one first looks at all covers of by sets of diameter at most and tries to minimize the sum of the -power of the diameters of the cover, that is

Decreasing (the “degree of refinement” of the cover) reduces the possible coverings to take, and thus increases the value of . The -dimensional Hausdorff measure of is then defined as

and it may take any nonnegative value including 0 and infinity. The Hausdorff dimension of is now defined as the critical value where its -dimensional Hausdorff measure “jumps” from infinity to 0, i.e.,

In other words, the Hausdorff dimension determines which Hausdorff measure may be able to actually “see” the set: Looking at an interval with “2-dimensional” eyes we will see nothing (the area of an interval is 0!), while looking at it with “1-dimensional” eyes we will be able to measure its (finite, nonzero) length. To determine the Hausdorff dimension of a set is in general difficult. However, in the presence of certain similarity properties, exact computations are possible. Figure 3 shows a set created by the Swedish mathematician Helge von Koch in the early 1900s. This kind of set is called self-similar because it is made up of smaller copies of itself. In this case there are four copies, each scaled by a factor of .

Figure 3.

The von Koch curve is a self-similar set.

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Thus, the “Euclidean mass” of each copy scales by a factor of while its “Euclidean diameter” scales by a factor of . And since the Hausdorff dimension expresses the mass-to-diameter ratio, we expect to be the number such that . In general, a self-similar set consisting of scaled copies of itself with scaling factors respectively, has Hausdorff dimension , where solves the equation

see, e.g., Fal14, Theorem 9.3. For the von Koch curve, and , and indeed . As for the topological dimension of , it equals 1 in the same way as for the interval . The topological dimension of the von Koch curve is thus strictly smaller than the Hausdorff dimension! Mandelbrot would therefore say that the von Koch curve is indeed fractal .

A common rule of thumb to decide about the “fractality” of a set is to check whether its Hausdorff dimension is not an integer. This criterion ought to be taken with a grain of salt because it excludes objects one would consider fractal, for instance the paths of a Wiener process in three dimensions (more about this later!). My personal viewpoint follows K. Falconer in that the term “fractal” may be regarded similarly to how biologists may regard the term “life”: Most objects we would classify as fractal share certain properties, which we may call “fractal features,” and yet there will be interesting fractal objects lacking some of them. The properties listed by Falconer in Fal14 are:


having a fine structure, that is, detail at arbitrarily small scales.


being too irregular to be described in traditional geometrical language, both locally and globally.


being self-similar in some way, perhaps approximate or statistical.


having “fractal” (usually Hausdorff) dimension greater than the topological dimension.


being often defined in a simple way, perhaps recursively.

“Alright, to me the pattern in the rocks would classify as fractal.” My brother is checking items in the previous list. His curiosity does not stop here and he brings into the conversation another intrinsic aspect of nature: “I wonder whether the seaweed has deposited there somewhat randomly. Would that make the pattern a random fractal? How do you mathematicians describe such an object?”

2. Random Fractals

The mathematician in me feels compelled to first clarify that, technically speaking, the pattern we have found is a realization, that is, one of the many possible patterns created by the random process of seaweed deposition.

Introducing randomness in the construction of fractals allows us to reproduce irregular patterns closer to what we see in nature. In fact, many computer-drawn landscapes are constructed following some kind of random procedure. One example of a random analogue to the von Koch curve is shown in Figure 4.

Figure 4.

A random von Koch curve. Each generation consists of four random choices of the patterns above.

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A set like the random von Koch curve displayed is called statistically self-similar. Besides consisting of random variations of the whole, that randomness appears at all scales: each random variation consists again of random variations of the copy itself. Thus, enlarging a smaller part we shall see a set with the same distribution as the original one.

What about the Hausdorff dimension? Dealing with random objects, their properties are stated in probabilistic terms. In this case, with probability one, the Hausdorff dimension of a statistically self-similar fractal constructed analogously to the random von Koch curve from Figure 4 is the solution to

Here, are now random variables that determine the scale of the random copies the whole is made of; see, e.g., Fal14, Theorem 15.2. Indeed, my brother notices how 3 resembles 2!

As in the deterministic case, one can usually hope at least to obtain certain bounds for the Hausdorff dimension or the Hausdorff measure of a random fractal. Although the biological laws that govern a phenomenon like our concrete seaweed deposition may still be far too complex to fit a tractable mathematical model, there are random processes that do lead to random fractal sets. Significant examples are diffusion-limited aggregation (DLA) processes that model particle accumulation processes as seen in crystal growth or forked lightning, or random percolation models that simulate particle clustering.

My brother finds another rock that also displays a fractal pattern. “I see,” he says, “the fractal pattern is a product of the random process of seaweed deposition. That makes the process a random phenomenon with fractal-like features.” I couldn’t have said it any better . He is now scratching his forehead. “I wonder though: are there random phenomena one would consider fractal even without actually producing a fractal output?” Great question! This conversation is getting pretty interesting.

3. Fractals Behind Random Scenes

As my brother points out, our discussion has thus far focused on physical fractal objects produced by a random phenomenon. The world of fractals, however, expands rapidly the moment one starts searching (and finding!) the properties listed at the end of Section 1, not just in actual observables such as plants, snowflakes or metal particles, but also in relationships, which mathematicians encode in functions. Processes modeled by Brownian motion, also known as (the) Wiener process among mathematicians, constitute an important class of random phenomena that exhibit these kinds of fractal features.

The model became popular through the work of the botanist Robert Brown in 1827, who investigated it to describe the random movement of pollen particles suspended in water. The increasing interest by physicists in the model led to the work of Einstein in 1905, where he successfully applied Brownian motion to test the validity of the molecular-kinetic theory of heat. As he states in Ein56: “bodies of microscopically visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed in a microscope, on account of the molecular motions of heat.” The rigorous mathematical construction of Brownian motion as a random function was presented by Norbert Wiener in 1923, hence its namesake process. Nowadays Brownian motion is the basis to model not only the movement of particles suspended in a fluid, but also many other diverse natural phenomena. Changes in stock prices, medical imaging, topographical surfaces, and decision making algorithms are just a few of the fields where it plays a vital role.

Mathematically speaking, the standard 1-dimensional Wiener process is a random function of time. For each , is a random variable, and each element in the sample space defines a sample path . The graph of a possible outcome for a sample path is displayed in Figure 5.

Figure 5.

Graph of a simulated Brownian sample path .

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The main properties that characterize standard Brownian motion are:


with probability 1, the process starts at the origin, i.e., ;


with probability 1, the process has continuous sample paths, i.e., ;


the process has stationary increments, i.e., for any , is normally distributed with mean 0 and variance ;


the process has independent increments, i.e., for any , the increments are independent random variables.

A characteristic fractal feature of the Wiener process in appears in its space-time scaling, which we can analyze by looking at the graph from Figure 5,

Let us scale time by a factor and ask: How does space need to scale so that has the same statistical distribution as ? The definition of the process and the properties of the normal distribution imply that is normally distributed and the same happens with . We may thus write

where means that the left- and right-hand side are equal in distribution. So time and space scale differently! The mapping belongs to the class of affine transformations, which are linear transformations that contract with a different ratio in each direction. Because of 4, one says that the graph of is statistically self-affine.

What about its Hausdorff dimension? While there is no general method to compute it for an arbitrary statistically self-affine random set, one can show that the Hausdorff dimension of is 1.5 with probability 1. For Mandelbrot, a genuine fractal .

In the computation of the latter Hausdorff dimension, a fundamental property of the Wiener process plays a crucial role: sample paths are continuous and yet “rough.” This roughness is described by the fact that paths are almost surely Hölder continuous of order for any and they just fail to be so for . That means, for , there is a random constant for which

for any , and the latter statement fails when . In fact, one can show that

with probability 1 for . Loosely speaking, this means that a spatial increment of the Wiener process along a period of time is almost surely no larger than . Since is the standard deviation of such a spatial increment, the increments are often roughly of that size.

The Hölder regularity of the paths remains true in higher dimensions. For example, a sample path like the one displayed in Figure 6 also satisfies 5, cf. Fal14, Proposition 16.4.

Figure 6.

A sample path of 2-dimensional Brownian motion.

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These paths were Robert Brown’s initial object of analysis and they generally model the random trajectory of a heated particle. However, in contrast to the 1-dimensional case, the trail (and in fact any path with ) has almost surely Hausdorff dimension 2. This is an example of a fractal object whose Hausdorff dimension is in fact a whole natural number .

My brother has been listening to my talk with a mixture of amusement and awe. “You sounded really excited giving that explanation! May I try to put it in my own words?” Of course, I am all ears! He says: “The wind brings a grain of pollen to the surface of one of the minilakes we observe here between the rocks. The movement of that grain follows a random path which mathematicians describe as Brownian motion or Wiener process. And that movement is fractal because its space-time ratio is a fraction, namely .” Wow, he has really captured the essence of my talk!

“Now I wonder,” my brother’s observation and curiosity again at work: “Is the square root special? What if the pollen is suspended in a medium that is itself highly irregular, something fractal?” “Would the way the pollen moves tell anything about the geometric fractal features of the medium?”

4. Brownian Motion “Feels” Spatial Fractality

A natural yet challenging question! The mathematicians Kusuoka Kus87 and Goldstein Gol87 thought about such a situation in the 1980s and were able to give a rigorous construction of the analogue to Brownian motion on a fractal called the Sierpiński gasket (SG), approximated in Figure 7.

Figure 7.

Graphs approximating SG.

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Just as classical Brownian motion can be obtained as the scaled limit of random walks on lattices (the scaling ratio is again , think of time as number of steps), Kusuoka and Goldstein considered a sequence of random walks on the graph approximations of displayed in Figure 7. They found that the space-time ratio, instead of , is . Which means that, in the long run, the motion is on average slower than standard Brownian motion in .

This kind of behavior is called by physicists subdiffusive. Why would that be so? Intuitively, the physical structure of the Sierpiński gasket features many holes which the particle must “circumvent” to get from one place to another. The particle’s movement is thus constrained and slowed down by the intrinsic structure of the fractal medium in which it moves. If the medium is “wide enough” for the particle to move in it for a “long enough” period of time without reaching its boundary, the Hausdorff dimension of the particle’s trajectory (a sample path) will correspond to the inverse of the space-time scaling ratio. Remember how we had wondered that the Hausdorff dimension of a Brownian path was 2? Well, that is precisely the inverse of the ratio .

This intuitive idea is one of the reasons for physicists to often call the Hausdorff dimension of such a path the “dimension of the walk,” or walk dimension. Kusuoka and Goldstein’s work thus tell us that the walk dimension of the Sierpińksi gasket is

which means that Brownian motion on moves in time a distance comparable to .

“Wait a minute.” My brother has closed his eyes and I can feel he is trying to connect the dots. “Robert Brown’s grain of pollen moved in time a distance comparable to which corresponds to because the walk dimension is . Earlier on you mentioned that the space-time scaling was related to the (was it Hölder?) continuity of the paths.” Opening his eyes he points to expression 5, that is written on the sand. “Does the same happen with the Sierpiński gasket?”

Good question! The mathematicians Barlow and Perkins presented in BP88 a detailed analysis of Brownian motion on . In particular, they proved that for any there is a random variable such that

for any , and the latter fails when . One of the main ingredients to obtain 6 was to estimate the probability with which a particle performing Brownian motion on moves away from its starting point.

“By the way,” my brother’s observational skills are blossoming today, “I notice that we often speak of Hölder continuity as an indicator for fractal features. Are there other Hölder continuous functions associated with Brownian motion?”

His inquiry actually touches on an aspect that, in the case of diffusion processes on fractals, presents many challenges and open questions. When existent, one of the functions encoding Brownian motion is the heat kernel, which basically describes the probability that a particle performing Brownian motion moves from to in a period of time . In the case of there is an explicit expression for that function, known as the Gaussian heat kernel, which reads

where and . For example, the probability that a particle starting at a point lies in a region at time may be computed as

Here, denotes an operator called the heat semigroup. The existence of a neat explicit formula like 7 is nearly a miracle. In general, one can at most hope to understand the behavior of the heat kernel in terms of approximate relationships between the distance between two points and the time that is being considered. In the case of the Sierpiński gasket, Barlow and Perkins found that the heat kernel behaves up to (different) constants like

where is the Euclidean distance between and and denotes the Hausdorff dimension of SG, cf. BP88, Theorem 1.5. They were also able to show the existence of a constant such that

for any and distinct points in SG. In other words, the heat kernel on SG is Hölder continuous with exponent . Estimates like 9 and 10 are by now well-established results in many nonsmooth settings such as so-called fractional diffusion processes; see, e.g., Bar98, Definition 3.5. The latter include the Sierpiński gasket and the Sierpiński carpet; see Figure 8.

Figure 8.

The standard Sierpiński carpet (SC) and the gasket (SG).

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“Oh, the carpet is also an artful object!” My brother goes back and forth between the pictures of the carpet and the gasket. “They look somewhat related and yet different…. For instance, I could disconnect the gasket by removing just the three points connecting the three largest triangles. To disconnect the carpet one would need to remove much more than a few points…. Is the carpet more challenging when you study Hölder regularity of functions?”

Oh brother, sure it is! In fact, a major achievement of Barlow and Bass in BB92 was precisely to obtain the corresponding estimates 9 of the heat kernel for Brownian motion on SC. And challenges continue today! A few lines of computations, invoking also available estimates for the time-derivative , made it possible in ARBC21, Theorem 3.7 to derive from 10 the following Hölder regularity of the corresponding heat semigroup

for and bounded and measurable. The latter estimate has been named the weak Bakry-Émery curvature condition in ARBC21 because of its connection with the Bakry-Émery curvature dimension inequality that may be characterized as

for any compactly supported smooth function . In the case of a smooth Riemannian manifold, the parameter is the lower bound on the Ricci curvature of the manifold, cf. vRS05, Theorem 1.3.

In the context of fractals, the gradient operator appearing in 12 is less straightforward to conceptualize and to analyze.

A natural generalization of gradient in the metric measure space setting is the carré du champ operator . The name of the operator has its origins in the mathematical theory of electrostatics and means “square (norm) of the (electric) field.” In , and its connection with the (Brownian) heat semigroup is a fundamental result in the theory of Dirichlet forms, namely

cf. FOT11, Lemma 1.3.4 and Lemma 3.2.3. Due to its interpretation in Physics, is usually referred to as the “energy” of the system associated with . In this way, the carré du champ can be regarded as the Radon Nikodym derivative of the energy measure of with respect to the reference measure of the underlying space.

However, there are many fractals, including , for which the energy measure is singular with respect to the standard reference measure! Therefore, instead of the gradient, we decided to analyze in 11 the difference as a weak replacement of 12. Besides the reasonable desire to extend these to larger classes of fractal spaces, even in the case of the Sierpiński carpet it is still an open question whether estimates like 10 or 11 are optimal. In fact, we conjectured in ARBC21 that one may improve upon 11 in the Sierpiński carpet because it possibly satisfies the weak condition, that is

Here, denotes the topological Hausdorff dimension of the space, which roughly speaking corresponds to the largest possible Hausdorff dimension of the boundary of an open cover of the space.

While our discussion is warming my mathematical thoughts, it is actually getting colder on the shore and we decide to head back home. “One last question, though.” My brother is about to ask something almost philosophical: “Where does a conjecture come from? What brings you and your collaborators to ‘guess’ that 13 may be true?” His inquiry could fill another chapter of connections between probability, geometry, and analysis in the world of fractals—what an amazing observer an artist can be!

Walking back I try to convey a reason to conjecture the estimate 13. The Hölder regularity of the heat semigroup has a geometric implication that I personally find truly beautiful. In the 1990s, Ledoux discovered that the heat semigroup could be used to describe the perimeter of a ball by

cf. Led94. (By now, the time scaling is no surprise!) And what does 14 actually mean? Let us visualize it using Brownian motion. The expression inside the integral in 14 can be rewritten as

which represents the probability that by time a particle following the Brownian motion lies on the other side of the boundary than where it started. So the perimeter happens to be expressible in terms of Brownian motion crossing the boundary within a properly scaled short time period!

Figure 9.

Brownian motion crossing the boundary.

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The characterization 14 was later extended to any Borel set of a Riemannian manifold in MPPP07 and it partly motivated our project ARBC21 to search for a definition of perimeter for sets in a metric measure space by studying

with and Borel. In principle, this quantity may or not be finite depending on the value of . And where is the connection to the weak Bakry-Émery condition? If the space satisfies , then 15 is finite when the so-called upper -Minkowski content of is finite cf. ARBC21, Theorem 4.9 and Lemma 2.7. On the Sierpiński carpet it is possible to find open sets whose -Minkowski content is finite, whence

et voilà the guess .

My brother can feel my excitement telling this story when we arrive at his place. Opening the door he asks: “Could you make a picture of one of those open sets in the Sierpiński gasket?” I smile: Do you have a couple of colors to paint with here? My brother’s face brightens. Let the artistic evening begin!


I am grateful to my colleagues Roger Smith and Harold P. Boas, as well as to two anonymous referees. Their comments and feedback on an early draft version of this article greatly helped to improve it.


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Opening graphic is courtesy of Bruno Alonso Ruiz.

Figures 1–9 and photo of Patricia Alonso Ruiz are courtesy of Patricia Alonso Ruiz.