Sur une conjecture de M. Kac
- Cite this article as:
- Le Gall, J.F. Probab. Th. Rel. Fields (1988) 78: 389. doi:10.1007/BF00334202
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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 EK(t) which represents the total energy flow in time t from the set K to the surrounding medium ℝ3−K. An asymptotic expansion is given for EK(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 EK(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.