Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1031where L is the Lagrangian, given byL(˜z(τ), ˙˜z(τ); τ) = 12σ 2 [ ˙˜z(τ) − A ] 2,and the integral is performed (with functional measure D[·]) over thepaths ˜z(·) belonging to C, i.e., all the continuous functions with constrains˜z(t ′ ) ≡ z ′ , ˜z(t ′′ ) ≡ z ′′ . As carefully discussed in [Bennati et. al. (1999)],a path integral is well defined only if both a continuous formal expressionand a discretization rule are given. As done in many applications, the Itôprescription is adopted here (see subsection 6.2.5 above).A first, naïve evaluation of the transition probability (6.45) can be performedvia Monte Carlo simulation, by writing (6.45) as∫ +∞−∞p(z ′′ , t ′′ |z ′ , t ′ ) =∫ +∞ n∏{1· · · dg i √2πσ2 ∆t exp − 1}2σ 2 ∆t [z′′ − (z n + A∆t)] 2 , (6.46)−∞iin terms of the variables g i defined by the relationdg k ={dz√ k2πσ2 ∆t exp − 1}2σ 2 ∆t [z k − (z k−1 + A∆t)] 2 , (6.47)and extracting each g i from a gaussian distribution of mean z k−1 + A∆tand variance σ 2 ∆t. However, as we will see, this method requires a largenumber of calls to get a good precision. This is due to the fact that eachg i is related to the previous g i−1 , so that this implementation of the pathintegral approach can be seen to be equivalent to a naïve MC simulation ofrandom walks, with no variance reduction.By means of appropriate manipulations [Schulman (1981)] of the integrandentering (6.45), it is possible, as shown in the following, to get a pathintegral expression which will contain a factorized integral with a constantkernel and a consequent variance reduction. If we define z ′′ = z n+1 andy k = z k − kA∆t, k = 1, . . . , n, we can express the transition probabilitydistribution as∫ +∞ ∫ {+∞1· · · dy 1 · · · dy n √(2πσ2 ∆t) ·exp − 1n+1∑n+1 2σ 2 [y k − y k−1 ]},2∆t−∞−∞(6.48)in order to get rid of the contribution of the drift parameter. Now let usk=1