Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 30 / 6 / b / Solution

Apply Q5(b) in each independent coordinate, with $\mu =\lambda$, to change from the Ornstein-Uhlenbeck path law to d-dimensional Wiener measure ${\mathbb{Q}}^x$. Its density isMultiplication by the positive potential factor cancels the second term exactly. ThereforeThe integral in its exponent is an Itô integral under Wiener measure, not an ordinary pathwise line integral. The Itô formula for $|\omega {|}^2$ givesHence an equivalent endpoint-only Wiener integral is