Sur une conjecture de M. Kac. (On a conjecture of M. Kac)
We consider the following heat conduction problem. Let K be a compact set in Euclidean space \({\mathbb{R}}^ 3\). Suppose that K is held at the temperature 1, while the surrounding medium is at the temperature 0 at time 0. Following \textit{F. Spitzer} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 3, 110-121 (1964; Zbl 0126.335)] we investigate the asymptotic behaviour of the integral \(E_ K(t)\) which represents the total energy flow in time t from the set K to the sourrounding medium \({\mathbb{R}}^ 3- K\). An asymptotic expansion is given for \(E_ K(t)\) which refines a theorem due to Spitzer. This expansion also verifies and improves a formal calculation of Kac. Similar results are proved in higher dimensions. Up to the constant m(K), the quantity \(E_ K(t)\) can be interpreted as the expected value of the volume of the Wiener sausage associated with K and a d-dimensional Brownian motion. This point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion.
scientific article; zbMATH DE number 4069969