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

Suppose you wanted to display the function $y=\sqrt{x}$ on a graphing calculator. Describe the process necessary to choose a proper window to display the graph.

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Key Concepts

1. The passage from discrete random walks to continuous stochastic processes, from the probability point of view and the partial differential equation point of view.

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Vocabulary

1. A diffusion process, or a diffusion for short, is a Markov process for which all sample functions are continuous. It is also a solution to a stochastic differential equation.

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Mathematical Ideas

The question is “How should we set up the limiting process so that we can make a continuous time limit of the discrete time random walk?” First we consider a discovery approach to this question by asking what should be the limiting process which assures us that we can visualize the limiting process. Next we take a probabilistic view using the Central Limit Theorem to justify the limiting process to pass from a discrete probability distribution to a probability density function. Finally, we consider the limiting process derived from passing from the difference equation from first-step analysis to a differential equation.

Visualizing Limits of Random Walks

The Random Walk

Consider a random walk starting at the origin. The $n$th step takes the walker to the position $T_n=Y_1+\cdots +Y_n$, the sum of n independent, identically distributed Bernoulli random variables $Y_i$ assuming the values $+1$, and $-1$ with probabilities $p$ and $q$ respectively. Then recall that the mean of a sum of random variables is the sum of the means:

and the variance of a sum of independent random variables is the sum of the variances:

Trying to use the mean to derive the limit

Now suppose we want to display a motion picture of the random walk moving left and right along the $x$-axis. This would be a motion picture of the “phase space” diagram of the random walk. Suppose we want the motion picture to display 1 million steps and be a reasonable length of time, say 1000 seconds, between 16 and 17 minutes. This fixes the time scale at a rate of one step per millisecond. What should be the window in the screen in order to get a good sense of the random walk? For this question, we use a fixed unit of measurement, say centimeters, for the width of the screen and the individual steps. Let $\delta$ be the length of the steps. To find the window to display the random walk on the axis, we then need to know the size of $\delta T_n$. Now

and

We want $n$ to be large (about 1 million) and to see the walk on the screen we want the expected end place to be comparable to the screen size, say 30 cm. That is,

so $\delta$ must be $3\cdot 10^{-5}\text{ cm}=0.0003\text{ mm}$ to get the end point on the screen. But then the movement of the walk measured by the standard deviation

will be so small as to be indistinguishable. We will not see any random variations!

Trying to use the variance to derive the limit

Let us turn the question around: We want to see the variations in many-step random walks, so the standard deviations must be a reasonable fraction $D$ of the screen size

This is possible if $\delta =D\cdot 3\times 10^{-2}\text{ cm }.$ We still want to be able to see the ending expected position which will be

To be consistent with the variance requirement this will only be possible if $\left(p-q\right)\approx 10^{-2}$. That is, $p-q$ must be comparable in magnitude to $\delta =3\times 10^{-2}$.

The limiting process

Now generalize these results to visualize longer and longer walks in a fixed amount or time. Since $\delta \to 0$ as $n\to \infty$, then likewise $\left(p-q\right)\to 0$, while $p+q=1$, so $p\to 1∕2$. The analytical formulation of the problem is as follows. Let $\delta$ be the size of the individual steps, let $r$ be the number of steps per unit time. We ask what happens to the random walk in the limit where $\delta \to 0$, $r\to \infty$, and $p\to 1∕2$ in such a manner that:

and

Probabilistic Solution of the Limit Question

In our accelerated random walk, consider the $n$th step at time $t=n∕r$ and consider the position on the line $x=k\cdot \delta$. Let

be the probability that the $n$th step is at position $k$. We are interested in the probability of finding the walk at given instant $t$ and in the neighborhood of a given point $x$, so we investigate the limit of $v_{k,n}$ as $n∕r\to t$, and $k\cdot \delta \to x$.

Remember that the random walk can only reach an even-numbered position after an even number of steps, and an odd-numbered position after an odd number of steps. Therefore in all cases $n+k$ is even and $\left(n+k\right)∕2$ is an integer. Likewise $n-k$ is even and $\left(n-k\right)∕2$ is and integer. We reach position $k$ at time step $n$ if the walker takes $\left(n+k\right)∕2$ steps to the right and $\left(n-k\right)∕2$ steps to the left. The mix of steps to the right and the left can be in any order. So the walk reaches position $k$ at step $n$ with binomial probability

From the Central Limit Theorem

Since $v_{k,n}$ is the probability of finding $T_n\cdot \delta$ between $k\cdot \delta$ and $\left(k+2\right)\cdot \delta$, and since this interval has length $2\cdot \delta$ we can say that the ratio $v_{k,n}∕\left(2\delta \right)$ measures locally the probability per unit length, that is the probability density. The last relation above implies that the ratio $v_{k,n}∕n$ tends to

It follows by the definition of integration as the sums of quantities representing densities times geometric lengths or areas, that sums of probabilities $v_{k,n}$ can be approximated by integrals and the result may be restated as

The integral on the right may be expressed in terms of the standard normal distribution function.

Note that we derived the limiting approximation of the binomial distribution

by applying the general form of the Central Limit Theorem. However, it is possible to derive this limit directly through careful analysis. The direct derivation is known as the DeMoivre-Laplace Theorem and it is the most basic form of the Central Limit Theorem.

Differential Equation Solution of the Limit Question

Another method is to start from the difference equations governing the random walk, and then pass to the differential equation in the limit. We can then obviously generalize the differential equations, and find out that the differential equations govern well-defined stochastic processes depending on continuous time. Since differential equations have a well-developed theory and many tools to manipulate, transform and solve them, this method turns out to be useful.

Consider the position of the walker in the random walk at the $n$th and $\left(n+1\right)$st trial. Through a first step analysis the probabilities $v_{k,n}$ satisfy the difference equations:

In the limit as $k\to \infty$ and $n\to \infty$, $v_{k,n}$ will be the sampling of the function $v\left(t,x\right)$ at time intervals $r$, so that $kr=t$, and space intervals so that $n\cdot \delta =x$. That is, the function $v\left(t,x\right)$ should be an approximate solution of the difference equation:

We assume $v\left(t,x\right)$ is a smooth function so that we can expand $v\left(t,x\right)$ in a Taylor series at any point. Using the first order approximation in the time variable on the left, and the second-order approximation on the right in the space variable, we get (after canceling the leading terms $v\left(t,x\right)$ )

In our passage to the limit, the omitted terms of higher order tend to zero, so may be neglected. The remaining coefficients are already accounted for in our limits and so the equation becomes:

This is a special diffusion equation, more specifically, a diffusion equation with convective or drift terms, also known as the Fokker-Planck equation for diffusion. It is a standard problem to solve the differential equation for $v\left(t,x\right)$ and therefore, we can find the probability of being at a certain position at a certain time. One can verify that

is a solution of the diffusion equation, so we reach the same probability distribution for $v\left(t,x\right)$.

The diffusion equation can be immediately generalized by permitting the coefficients $c$ and $D$ to depend on $x$, and $t$. Furthermore, the equation possesses obvious analogues in higher dimensions and all these generalization can be derived from general probabilistic postulates. We will ultimately describe stochastic processes related to these equations as diffusions.

Sources

This section is adapted from W. Feller, in Introduction to Probability Theory and Applications, Volume I, Chapter XIV, page 354.

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Problems to Work for Understanding

1. Consider a random walk with a step to right having probability $p$ and a step to the left having probability $q$. The step length is $\delta$. The walk is taking $r$ steps per minute. What is the rate of change of the expected final position and the rate of change of the variance? What must we require on the quantities $p$, $q$, $r$ and $delta$ in order to see the entire random walk with more and more steps at a fixed size in a fixed amount of time?
2. Verify the limit taking to show that
   $$v_{k,n}\sim \left(1∕\left(\sqrt{2\pi Dt}\right)\right)exp\left(-{\left[x-ct\right]}^2∕\left(2Dt\right)\right).$$

3. Show that
   $$v\left(t,x\right)=\left(1∕\left(\sqrt{2\pi Dt}\right)\right)exp\left(-{\left[x-ct\right]}^2∕\left(2Dt\right)\right)$$

   is a solution of

   $$\frac{\partial v\left(t,x\right)}{\partial t}=-c\frac{\partial v\left(t,x\right)}{\partial x}+\left(1∕2\right)D\frac{{\partial }^{2}v\left(t,x\right)}{\partial x^2}$$

   by substitution.

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Reading Suggestion:

References

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Outside Readings and Links:

1. Brownian Motion in Biology.. A simulation of a random walk of a sugar molecule in a cell.
2. Virtual Laboratories in Probability and Statistics.. Search the page for Random Walk experiment.

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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 1, 2010