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  April 1, 2002

Mathematician Uses Probability
to Describe Random Processes
By Brent Evans

In a still room, cigarette smoke hangs in slowly drifting serpentine clouds. The distinctive shape of a wisp of smoke is due to tiny, random motions of gas particles. And the math behind such random phenomena is a topic Richard Bass is studying.

Bass, a professor of mathematics, uses math to describe random events, which by definition have no pattern and are therefore unpredictable.

One type of random motion, that exhibited by smoke particles, is called Brownian motion. Brownian motion occurs when a small particle is bombarded from all directions, such as when a gas molecule repeatedly bumps into its neighbors. A Brownian particle jitters continually.

Bass has made several discoveries about Brownian motion. For one, he showed how a randomly moving particle behaves when it collides with a boundary: no matter how the boundary is shaped, the particle tends to move perpendicular to the boundary immediately after the collision.

This is because the boundary, or wall, gives the particle a kick in a definite direction, instead of just contributing to the jittering motion.

Bass's work uses probability to describe random processes. The real world can often be described this way, especially in finance, he says: "Many theorems in probability have applications to finance."

The stock market exhibits Brownian motion in some cases, he says. It certainly does respond predictably to stimuli such as changing interest rates.

Over a short period of time, however, the market's ups and downs are unpredictable, and often do not have discernable causes.

Some of Bass's work can be applied toward understanding general features of the market. He has not, however, discovered a means to make detailed predictions about the stock market. "You can't predict anything for sure," he says.

Just as for particles, there are boundaries for the Brownian motion of the stock market, Bass says such as the current psychological barrier of 10,000 points for the Dow-Jones average, and its former barrier of 4,000 points. Although this is not a hard boundary like a wall, he says his theory may give insight into the market's fluctuations near these boundaries.

In another problem involving random processes, Bass disproved a longstanding idea about how heat is distributed in a room, postulated in 1985 by mathematician Isaac Chavel. Imagine being in a closed room and turning a space heater on and then immediately off. Then, measure the temperature at one spot in the room as a function of time. Chavel postulated that, for a smaller room, the temperature will always be higher.

Bass disproved this idea by coming up with a counterexample. Using a heat model that incorporated random heat distribution, he discovered some shapes of rooms for which the bigger room would sometimes be warmer.

Bass and Chris Burdzy, a professor of mathematics at the University of Washington, disproved Chavel's claim in a matter of a month, after others had spent 10 years trying to prove the wrong thing: that Chavel was correct. "We were the ones to solve the problem," he says, "because everyone else was trying to prove it."

Burdzy says Bass's work has placed him at the forefront of the field of probability, citing Bass's invited talk at the 1994 International Congress of Mathematicians. Only 10 probabilists are selected every four years to be among the mathematicians who deliver these addresses.

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