##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics

## Magdalena Musat

#### UCSD

## The condenser problem

##### Abstract:

In classical potential theory on $R^n (n >= 3)$, the condenser theorem states the following. Let $U$ and $V$ be open subsets of $R^n$ with disjoint closures, where $U$ is relatively compact. There exists a Newtonian potential $p$ of a signed measure $mu = mu^+ - mu^-$ such that: \vskip .1in \noindent 1. $0 <= p <= 1$ \vskip .1 in \noindent 2. $p = 1 on U, p = 0 on V$ \vskip .1in \noindent 3. $mu^+$ is supported in the closure of $U$, $mu^-$ is supported in the closure of $V$ \vskip .1in \noindent The fundamental proof of this theorem was given by A. Beurling and J. Deny in the framework of Dirichlet spaces. Later K. L. Chung and R. K. Getoor studied the condenser problem in the probabilistic context of Markov processes. They showed that the condenser potential at a point x is simply the probability that Brownian motion starting at $x$ hits $U$ before it hits $V$. In this talk we discuss the condenser problem in the potential-theoretic framework of balayage spaces. We introduce the notion of a fine condenser potential, for which existence and uniqueness are proved for arbitrary superharmonic functions on sets $U, V$ which are not necessarily open. The probabilistic interpretation carries over to this setting by replacing the hitting times by the penetration times. These results are joint work with Jürgen Bliedtner.

Host:

### December 2, 2004

### 9:00 AM

### AP&M 6438

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