The Degree Theorem on Wiener Space

Abstract

Let Ψ be a C ¹ map from ℝⁿ to ℝⁿ and let D be a bounded domain in ℝⁿ. The topological degree of Ψ at a point p in ℝⁿ is defined as follows. Let Ψ ⁻¹{ p } ⋂ D denote the intersection of D with the inverse image of p under: Ψ

$${\psi ^{ - 1}}\left\{ p \right\} \cap D = \left\{ {x \in D:\psi \left( x \right) = p} \right\}$$

and let J _Ψ (x) denote the Jacobian of Ψ at x, $x,{J_\psi }\left( x \right) = \det {\left( {{\partial _{\psi i}}\left( x \right)/{\partial _{xi}}} \right)_{n \times n.}}$