Vol. 59, No. 1, 1975

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ISSN: 0030-8730
On Gross differentiation on Banach spaces

Hui-Hsiung Kuo

Vol. 59 (1975), No. 1, 135–145
Abstract

Let pt(x,) denote the Wiener measure in an abstract Wiener space (H,B) with variance parameter t > 0 and mean x in B. It is shown that if f L2pt(x,),t > 0 and x are fixed, then the function ptf defined by ptf(x + h) = Bf(y)pt(x + h,dy) for h in H is infinitely Gross differentiable at x. The first two derivatives are given by (Dptf(x),h) = t1 Bf(y)(h,y x)pt(x,dy) and (D2ptf(x)k,h) = t1 Bf(y){t1(h,y x)(k,y x) (h,k)}pt(x,dy), where h and k are in H. Moreover, D2ptf(x) is a Hilbert-Schmidt operator and D2ptf(x)2 √2-t1{ B|f(y)|2pt(x,dy)}12. An application to Uhlenbeck-Ornstein process is also given.

Mathematical Subject Classification 2000
Primary: 28A40
Secondary: 60J65, 58D15
Milestones
Received: 7 March 1975
Published: 1 July 1975
Authors
Hui-Hsiung Kuo