Ornstein-Uhlenbeck Path Integral and Its Application

Abstract

Introducing a path integral for the Ornstein-Uhlenbeck process distorted by a potential V(x), we find out the T → ∞ limit of the probability distributions of X[ω] := 1/T ^ν ∫ ^T₀ V(ω(t)) dt for Ornstein-Uhlenbeck process ω(t), with appropriate values of the exponent ν that depend on V. The results are compared with those for the Wiener process.

It is a great pleasure to dedicate this article to Prof. Takeyuki Hida at the occasion of his seventieth birthday. We are deeply grateful to him for his constant leadership for many years in the mathematical physics circle, in particular in the study of path integrals and white noise, not only in Japan but also globally. I wish him many happy returns.