Authors’ abstract: “Let \(\rho_t\) be the radial part of a Brownian motion in an \(n\)-dimensional Riemannian manifold \(M\) starting at \(x\) and let \(T= T_\varepsilon\) be the first time \(t\) when \(\rho_t= \varepsilon\). We show that \(E\rho^2_{t\wedge T}= nt- (1/6) S(x)t^2+ o(t^2)\), as \(t\downarrow 0\), where \(S(x)\) is the scalar curvature. The same formula holds for \(E\rho^2_t\) under some boundedness condition on \(M\)”.