The investigator will study reinforced random walk. If a fair coin is flipped repeatedly and X(n) represents the number of heads in the first n flips minus the number of tails in the first n flips, the sequence of integers X(1), X(2),... is called fair random walk. Reinforced random walk is constructed in a similar manner except that if X(n) is the maximum price seen so far then the coin which determines X(n+1)-X(n) has probability p<1/2 of coming up heads, while if X(n) is a record minimum the coin has probability 1-p, which of course exceeds 1/2, of coming up heads. In words, this walk is fair random walk with partial reflection back to previously assumed values at records. The fact that the history {X(1),X(2),..X(n)} of the process is needed to determine the probability of the next jump means that reinforced random walk is not a Markov process, and renders the vast theory of such processes inapplicable. The investigator has shown that X(n) divided by the square root of n converges in distribution as n goes to infinity. He hopes to refine this result, for example by identifying the limiting distributions, and to prove its analog for the related processes which have probability greater than one half of setting new records, if they are at a record. Oddly, this situation will require very different methods. In another direction, the investigator will use Brownian motion to study positive solutions of the Dirichlet heat equation in planar domains, more specifically to find geometrical properties of the shape of the domain which determine or ensure that the ratio of any two solutions at any positive time is bounded throughout the domain, above and below, by positive constants, a property which is called intrinsic ultracontractivity of the Dirichlet heat semigroup. The investigator will study reinforced random walk. Suppose the price of a stock always changes by one dollar. Track the price after the first price change, the second, the third, and so on. Fair random wal k models this under the assumption that each change is equally likely to be plus one dollar or minus one, regardless of all earlier changes. Reinforced random walk behaves the same way except when the current price is a record high, in which case the next change is more likely to be down, or when the current price is a record low, in which case the next change is more likely to be up. That is, the walk shies away from setting records. It is a primitive model of a buy-low-sell-high market. It is less tractable than fair random walk because the past price fluctuations are used to determine the probabilities of the next step being plus one or minus one. The investigator has recently shown how the average net change in price behaves after a large number of changes, namely he has shown that it is on the order of the square root of the number of changes. He will try to refine this result and to prove a companion theorem for the related class of processes which like to set records. In another direction, probabilistic methods will be used to study how a flat metal plate cools, if it looses heat only out the edges, which are held at absolute zero, more specifically how the shape of the plate influences this cooling.
