If \(K\subset \mathbb R^{m}\) is a compact subset, we set \(E_{K}(t) = \int_{\mathbb R^{m}\setminus K} u(x,t)\,\roman dx\), where \(u\) is the (unique) weak solution to the heat equation \(\partial_{t}u = \Delta u\) in \((\mathbb R^{m}\setminus K)\times \mathbb R_+\) with initial condition \(u(\cdot,0) = 0\) and boundary condition \(u=1\) on \(\partial K\times \mathbb R_+\). Let \(K_1\), \(K_2\) be disjoint compact sets in \(\mathbb R^{m}\), the heat exchange \(H_{K_1,K_2}: \mathbb R_+\to \mathbb R\) is defined by \(H_{K_1,K_2}(t) = E_{K_1}(t) + E_{K_2} - E_{K_1\cup K_2}(t)\). It is known that \(H_{ K_1,K_2} = O(t^{(N+1)/2})\) as \(t\to 0\) for every \(N\in\mathbb N\), provided the boundaries \(\partial K_1\) and \(\partial K_2\) are \(C^\infty\)-smooth. In the paper under review, this result is generalized and refined, the proofs being based on probabilistic methods. If \(m=2\) it is assumed that the Lebesgue measure of \(K_{i}\cap B(x,\varepsilon)\) is strictly positive for all \(x\in K_{i}\), \(\varepsilon>0\), \(i=1,2\), where \(B(x,\varepsilon)\) is the closed ball centered at \(x\) with radius \(\varepsilon\). If \(m \geq 3\) it is supposed that the Newtonian capacity of \(K_{i}\cap B(x,\varepsilon)\) is strictly positive for all \(x\in K_{i}\) and \(\varepsilon>0\). It is proved that \(t\mapsto H_{K_1,K_2}(t)\) is strictly increasing and \(\lim_{t\to 0} t\log H_{K_1,K_2} (t) = - d^2/4\), where \(d = \text{dist} (\partial K_1,\partial K_2)\).