Heat flow, Brownian motion and Newtonian capacity (Q876125)

The author studies the asymptotic behaviour as \(t\to\infty\) of the unique weak solution of \(\partial_{t}u=\Delta u\) on \((\mathbb R^{m}\setminus K)\times(0,\infty)\), with \(u(x,0)=0\) on \(\mathbb R^{m}\setminus K\) and \(u(x,t)=1\) on \(\partial K\times(0,\infty)\). Here \(K\) is a compact non-polar set in \(\mathbb R^{m}\), \(m\geq 3\). It is known that, for \(t\to\infty\), \[ u(x,t)=h_{K}(x)-\left(\frac{m}{2}-1\right)^{-1}(4\pi)^{m/2}C(K) (1-h_{K}(x))t^{(2-m)/2}+o(t^{(2-m)/2}), \] where \(h_{K}\) is harmonic on \(\mathbb R^{m}\setminus K\), equals 1 on the regular points of \(K\) and which vanishes at infinity [see \textit{S. C. Port} and \textit{C. J. Stone}, Brownian motion and classical potential theory. Probability and mathematical Statistics. A Series of Monographs and Textbooks. New York et al: Academic Press (1978; Zbl 0413.60067)]. The present paper concerns the analysis of the remainder estimate \(o(t^{(2-m)/2})\), and it is shown that it can be improved to \(O(t^{-m/2})\), the proofs being different for \(m\geq 5\), \(m=4\) and \(m=3\).