Accessibility of infinite dimensional Brownian motion to holomorphically exceptional set (Q1917582)

The first author [J. Math. Kyoto Univ. 34, No. 4, 849-857 (1994)] suggested the notion of holomorphically exceptional sets of the complex Wiener space. In particular, he pointed out the following remarkable relation between holomorphically exceptional sets and the standard Brownian motion \((Z_t)_{t \geq 0}\) on the complex Wiener space: \(Z_t\) does not hit a holomorphically exceptional set until time 1 almost surely. In any finite-dimensional space, if the Brownian motion does not hit a certain set until time 1 almost surely, neither does it after time 1. So one may guess that the infinite-dimensional Brownian motion never hits a holomorphically exceptional set after time 1, either. But we show in the present paper that the above guess is false. That is, we construct a holomorphically exceptional set which the Brownian motion \((Z_t)_{t \geq 0}\) hits after a certain time \(t_0 > 1\) almost surely. The reason why such an example can exist lies essentially in the fact that the distributions of \((Z_t)_{t \geq 0}\) at different times are mutually singular.