Evaluation Formulas for Conditional Function Space Integrals I

Abstract

In this article, we establish several explicit conditional function space integration formulas for functionals defined on a very general function space C _a,b [0,T]. The formulas we obtain are rather simple and don't involve function space integrals. In particular we obtain a formula for the conditional function space integral of each of the functionals exp{∫₀ ^T x(t)db(t)}, exp{−[∫₀ ^T x(t) db(t)]²}, and exp{− ∫₀ ^T x² (t) db(t)} which arise naturally in quantum mechanics.

Keywords:

• Conditional Cameron-Martin translation theorem
• Conditional function space integral
• Function space integral
• Generalized Brownian motion process

AMS Mathematics Subject Classification:

• 28C20
• 60J65