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A Remark on Stochastic Path Integrals

Naturevolume 187page968 (1960) | Download Citation

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Abstract

VARIOUS stochastic functional of the type: where X(u) is a stochastic process, have been studied in recent years (see, for example, Kac1). It seems worth noting that for X(u) Markovian, there is often no particular difficulty in setting up characteristic function equations, of the type derived in my book2 on stochastic processes (see especially the end of chapter 3), applicable not only to (1) but also to rather more general types of integral. Thus for: provided we may set up an operator equation of the form : where CE {exp(iθS(t) + iϕ X(t)}. A particular case given on page 86 of my book is vX(uu. As further examples, consider v ≡ [ΔX(u)]2 for (i) the normal linear Markov process (which includes ‘Brownian motion’ as a special case) and (ii) the simple birth-and-death process, for which S(t) records the total number of transitions. For (i), S(t) then increases regularly with t (as is well known) ; for (ii), C is readily evaluated, either by (3) or even more easily from the corresponding ‘backward’ equation. A further amplification of these points will be given elsewhere.

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References

  1. 1

    Kac, M., “Probability and Related Topics in Physical Sciences” (New York, 1959).

  2. 2

    Bartlett, M. S., “An Introduction to Stochastic Processes” (Cambridge, 1955).

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  1. The University, Manchester, 13

    • M. S. BARTLETT

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https://doi.org/10.1038/187968b0

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