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Sur une conjecture de M. Kac. (On a conjecture of M. Kac). (French) Zbl 0655.60067

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 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.

MSC:

60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60H25 Random operators and equations (aspects of stochastic analysis)
60J55 Local time and additive functionals
60J65 Brownian motion

Citations:

Zbl 0126.335
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References:

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