The lifetime of conditioned Brownian motion in an angular domain (Q1898420)

We consider the integrability of the lifetime of conditioned Brownian motions in the angular domain \(A^d_m = \{(x^1, \ldots, x^d) \in R^d\), \(x^i > 0\), \(1 \leq i \leq m\leq d\} \) \((d \geq 2)\). Our main results are: If \(h(x) = x^1x^2 \cdots x^k\) \((1 \leq k \leq m)\), then \(E^h_x \tau < + \infty\) for \(1 \leq k \leq m - 3\); \(E^h_x \tau = + \infty\) for \(m - 2 \leq k \leq m\). If \(h(y) = \int_{\Pi^1_m} P_y \{B^j_\tau = z^j\), \(\tau_1 < \tau_j\), \(2 \leq j \leq m\}\mu_1 (dz) \in H_0\), \(y \in A^d_m\) where \(\Pi^1_m = \{z \in \partial A^d_m, z' = 0\}\), \(d \geq 2\), then (i) \(E^h_x \tau \equiv \infty\) as \(m = 1\), \(d = 2\); (ii) \(E^h_x \tau < + \infty \Leftrightarrow \int_{\Pi^1_m} {z^2 z^3 \cdots z^m \over (1 + |z |)^{d + 2m - 4}} \mu_1(dz) < + \infty\) otherwise.