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Extra info for PROB - Dynamical Theories of Brownian Motion
Sample text
Let us make this notion precise. Let I be an interval that is open on the right, let x be an -valued stochastic process indexed by I, and let Pt for t in I be an increasing family of σ- algebras such that each x(t) is Pt -measurable.
See the following and its bibliography: THE WIENER INTEGRAL 35 [16]. Irving Segal, Algebraic integration theory, Bulletin American Math. Soc. 71 (1965), 419–489. For discussions of Wiener’s work see the special commemorative issue: [17]. Bulletin American Math. Soc. 72 (1966), No. 1 Part 2. We are assuming a knowledge of the Wiener process. For an exposition of the simplest facts, see Appendix A of: [18]. Edward Nelson, Feynman integrals and the Schr¨ odinger equation, Journal of Mathematical Physics 5 (1964), 332–343.
E. Uhlenbeck, On the theory of Brownian motion II, Reviews of Modern Physics 17 (1945), 323–342. The first mathematically rigorous treatment, and additionally the source of great conceptual and computational simplifications, was: [24]. J. L. Doob, The Brownian movement and stochastic equations, Annals of Mathematics 43 (1942), 351–369. All four of these articles are reprinted in the Dover paperback “Selected Papers on Noise and Stochastic Processes”, edited by Nelson Wax. [24]. E. Nelson, Regular probability measures on function space, Annals of Mathematics 69 (1959), 630–643.