Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

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Note: To read these pages properly, you will need the latest version of the Mozilla Firefox browser, with the STIX fonts installed. In a few sections, you will also need the latest Java plug-in, and JavaScript must be enabled. If you use a browser other than Firefox, you should be able to access the pages and run the applets. However, mathematical expressions will probably not display correctly. Firefox is currently the only browser that supports all of the open standards.

Rating

Mathematically Mature: may contain mathematics beyond calculus with proofs.

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Question of the Day

Provide an example of a continuous function which is not differentiable at some point. Why does the function fail to have a derivative at that point? What are the possible reasons that a derivative could fail to exist at some point?

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Key Concepts

1. With probability $1$ a Brownian Motion path is continuous but nowhere differentiable.

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Vocabulary

1. In probability theory, the term almost surely is used to indicate an event which occurs with probability 1. In infinite sample spaces, it is possible to have meaningful events with probability zero. So to say an event occurs “almost surely” is not an empty phrase. Events occurring with probability zero are sometimes called negligible events.

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Mathematical Ideas

Properties of the Path of Brownian Motion

To state this as a theorem may seem strange in view of property 4 of the definition of Brownian motion. Property 4 requires that Brownian motion is continuous. However, some authors weaken property 4 in the definition to only require that Brownian motion be continuous at $t=0$. Then this theorem shows that the weaker definition implies the stronger definition used in this text. This theorem is difficult to prove, and well beyond the scope of this course. In fact, even the statement above is imprecise. Specifically, there is an explicit representation of the defining properties of Brownian Motion as a function in which (with probability $1$) $W\left(t,\omega \right)$ is a continuous function of $t$. We need the continuity for much of what we do later, and so this theorem is stated here again as a fact without proof.

This property is even deeper and requires more machinery to prove than does the continuity theorem, so we will not prove it here. Rather, we use this fact as another piece of evidence of the strangeness of Brownian Motion.

In spite of one’s intuition from calculus, Theorem 2 shows that continuous, nowhere differentiable functions are actually common. Indeed, continuous, nowhere differentiable functions are useful for stochastic processes. One can imagine non-differentiability by considering the function $f\left(t\right)=\left|t\right|$ which is continuous but not differentiable at $t=0$. Because of the corner at $t=0$, the left and right limits of the difference quotient exist but are not equal. Even more to the point, the function $t^{2∕3}$ is continuous but not differentiable at $t=0$ because of a sharp “cusp” there. The left and right limits of the difference quotient do not exist (more precisely, they approach $\pm \infty$) at $x=0$. One can imagine Brownian Motion as being spiky with tiny cusps and corners at every point. This becomes somewhat easier to imagine by thinking of the limiting approximation of Brownian Motion by coin-flipping fortunes. The re-scaled coin-flipping fortune graphs look spiky with corners everywhere. The approximating graphs suggest why the theorem is true, although this is not sufficient for the proof.

From Theorem 4 and the continuity we can deduce that for arbitrarily large $t_1$, there is a $t_2>t_1$ such that $W\left(t_2\right)=0$. That is, Brownian Motion paths cross the time-axis at some time greater than any arbitrarily large value of $t$.

From Theorem 4 and the inversion $tW\left(1∕t\right)$ also being a standard Brownian motion, we deduce that $0$ is an accumulation point of the zeros of $W\left(t\right)$. That is, Standard Brownian Motion crosses the time axis arbitrarily near $0$.

This means that the graph of a Brownian Motion path is “fuzzier” or “thicker” than the graph of, for example, a continuously differentiable function which would have Hausdorff dimension $1$.

Sources

This section is adapted from: Notes on Brownian Motion. by Yuval Peres, University of California Berkeley, Department of Statistics.

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Problems to Work for Understanding

1. Provide a more complete heuristic argument based on Theorem 4 that almost surely there is a sequence $t_n$ with $\underset{t\to \infty }{lim}{t}_n=\infty$ such that $W\left(t\right)=0$
2. Provide a heuristic argument based on Theorem 5 and the shifting property that the zero set of Brownian Motion
   $$\left\{t\in \left[0,\infty \right):W\left(t\right)=0\right\}$$

   has no isolated points.

3. Looking in more advanced references, find another property of Brownian Motion which illustrates strange path properties.

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Reading Suggestion:

References

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Outside Readings and Links:

1. Notes on Brownian Motion. Yuval Peres, University of California Berkeley, Department of Statistics

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Information on this website is subject to change without notice.

Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1

Email to Steve Dunbar, sdunbar1 at unl dot edu

Last modified: Processed from LATEX source on November 12, 2010