Heat balls and Fulks measures. (English) Zbl 0592.35057

Bauer, Heinz

Ann. Acad. Sci. Fenn., Ser. A I, Math. 10, 67-82 (1985).

On every heat sphere \(\Omega\) (z,c) with center \(z\in {\mathbb{R}}^{n+1}\) and radius \(c>0\) the measure \(\mu_{z,c}\) introduced by W. Fulks [Proc. Am. Math. Soc. 17, 6-11 (1966; Zbl 0152.105)] plays the same role for the solution of the heat equation as the surface measure on the Euclidean sphere does for the solution of the Laplace equation. It is shown that \(\mu_{z,c}\) is obtained by sweeping the Dirac measure at z on the complement of \(\Omega\) (z,c) or, in probabilistic terms, that \(\mu_{z,c}\) is the distribution of the first hit of the heat sphere \(\partial \Omega (z,c)\) by the space-time Brownian motion starting at z. This characterization of \(\mu_{z,c}\) implies that the hypertemperatures introduced by N. A. Watson [Proc. Lond. Math. Soc., III. Ser. 26, 385-417 (1973; Zbl 0253.35045); ibid. 33, 251-298 (1976; Zbl 0336.35046)] are the hyperharmonic functions of the harmonic space given by the heat equation.

Reviewer: W.Hansen

MSC:

35K05	Heat equation
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31D05	Axiomatic potential theory

Keywords:

heat equation; Laplace equation; hyperharmonic functions