, Volume 78, Issue 3, pp 389-402

Sur une conjecture de M. Kac

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Summary

We consider the following heat conduction problem. Let K be a compact set in Euclidean space ℝ3. Suppose that K is held at the temperature 1, while the surrounding medium is at the temperature 0 at time 0. Following Spitzer 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 surrounding medium ℝ3K. 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.