Long-Time Average of Field Measured by an Ornstein–Uhlenbeck Wanderer

Abstract

For a given field V ( x ) on R d , we shall find (i) the exponent ν such that \(X_T[\omega] = T^{-\nu}\int_0^{T} V(\omega(t))\,\mathrm{d}t\) converges as T →∞ to have a non-trivial probability distribution, and (ii) the distribution itself, where ω( t ) denotes the d -dimensional Ornstein–Uhlenbeck process starting from ω(0) = x at t = 0. The exponent and the probability distribution are different depending on whether or not a weighted average \(\int_{\mathbb{R}^d}V(x) \exp[-\beta x^2/2D]\,\mathrm{d}^{d} x\) of V vanishes, where D and β are constants of the Ornstein–Uhlenbeck process. In the non-vanishing case, they are ν=1 and a Dirac delta distribution, while in the vanishing case, ν=1/2 and the Gaussian distribution. The results have applications to the physics of chemoreception in immune systems.