Functional integration: Difference between revisions

Revision as of 10:54, 23 October 2019

Revision as of 17:45, 29 October 2019

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In the [[Wiener process|Wiener integral]], a probability is assigned to a class of [[Brownian motion]] paths. The class consists of the paths ''w'' that are known to go through a small region of space at a given time. The passage through different regions of space is assumed independent of each other, and the distance between any two points of the Brownian path is assumed to be [[Normal distribution|Gaussian-distributed]] with a [[variance]] that depends on the time ''t'' and on a diffusion constant ''D'':

:<math>\Pr\big(w(s + t), t \,\big|\,mid w(s), s\big) = \frac{1}{\sqrt{2\pi D t}} \exp\left(-\frac{\|w(s+t) - w(s)\|^2}{2Dt}\right).</math>

The probability for the class of paths can be found by multiplying the probabilities of starting in one region and then being at the next. The Wiener measure can be developed by considering the limit of many small regions.