For \(0< \alpha \leq 1\), the authors consider the parabolic operator \(L ^{(\alpha)}=D_t +(-\Delta)^\alpha\). Their central result is the following characterization of a strip, in which \(W^{(\alpha)}\) denotes the fundamental solution for \(L^{(\alpha)}\): If \(D\) is an open subset of \(\mathbb R^{n+1}\) such that \((\mathbb R^n \times ]-\infty,\tau[)\setminus D \neq\emptyset\) for any \(\tau \in \mathbb R\), and \[ \iint_D W^{(\alpha)}(x-y,t-s) dx dt= \int_{\mathbb R^n}W^{(\alpha)}(x-y,-s) dx \] whenever \((y,s)\in\mathbb R^{n+1}\setminus D\), then \(D=\mathbb R^n\times ]a,a+1[\) for some \(a\in[-1,0[\). This generalizes a result proved for the heat equation by the reviewer [N. Z. J. Math. 25, 243-248 (1996; Zbl 0868.31002)], and the new proof is a substantial improvement. Much of the paper is devoted to proving general results about \(L^{(\alpha)}\)-harmonic measures in order to establish the converse of the quoted theorem.