Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

A. Stochastic calculus in outlinecontinuous but not differentiable. The probability distribution P governing W spreadsout with increasing time, until as t →∞, P → 0 everywhere.increment as:dW (t) =W (t + dt) − W (t),We define the Wiener(A.3)which has the following expectation valuesE[dW ] = 0 (A.4a)E[dW (t) 2 ] = dt (A.4b)E[dW (t) 2+n ] = 0 n>2 (A.4c)E[dW (t)dW (t ′ )] = 0. (A.4d)Any process with white-noise correlations:〈ξ(t)ξ(t ′ ) 〉 = δ(t − t ′ ), (A.5)can be written in terms of the Wiener process asξ(t) =dW (t). (A.6)dtA.3 Ito and Stratonovich stochastic calculiAn Ito stochastic equation of the formdx i (t) =A i (x,t)dt + ∑ jB ij (x,t)dW j(A.7)is equivalent to the Stratonovich equation⎛⎞dx i (t) = ⎝A i (x,t) − 1 ∑B kj (x,t) ∂ B ij (x,t) ⎠ dt + ∑ 2∂x kjkjB ij (x,t)dW j .(A.8)In other words, Eq. (A.7) when interpreted according to the rules of Ito calculus givesthe same solution as Eq. (A.8) when interpreted according to the rules of Stratonovichcalculus.The two versions of stochastic calculus arise from different ways of defining the integralof a stochastic variable. The Ito integral evaluates the terms in the Riemann sum at thebeginning of the time interval, and thus corresponds to an explicit integration procedure,170