Holomorphic Processes in Banach Spaces and Banach Algebras

The main result is that if F is an analytic multifunction and B t is a complex Brownian motion, then F( B t ) 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(B t) is a local conformal martingale, φ subharmonic ⇒ φ(B t) 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.