Holomorphic Processes in Banach Spaces and Banach Algebras

The main result is that if F is an analytic multifunction and Bt is a complex Brownian motion, then F(Bt) is a subholomorphic process. It has previously been shown that such processes enjoy many interesting sample-path properties. As special cases of the theorem above, we recover f holomorphic => f(Bt) is a local conformal martingale,[phi] subharmonic => [phi](Bt) is a local submartingale. We also prove a stochastic form of Radó's theorem, and a holomorphic selection theorem for convex-valued subholomorphic processes of a nature quite different from the usual type of measurable selection theorem.