On nonincrease of Brownian motion. (English) Zbl 0719.60086

The author’s opening paragraph states the contents of this note: D. Aldous pointed out in his recent book [Probability approximations via the Poisson clumping heuristic. New York etc.: Springer-Verlag (1989; Zbl 0679.60013), K7, K13] that none of the published proofs of the nonincrease of Brownian motion is totally satisfactory. The original proof of A. Dvoretzky, P. Erdős and S. Kakutani [Proc. 4th Berkeley Symp. Math. Stat. Probab. 2, 103–116 (1961; Zbl 0111.15002)] is hard; those of O. Adelman [Isr. J. Math. 50, 189–192 (1985; Zbl 0573.60030)], I. Karatzas and S. E. Shreve [Brownian motion and stochastic calculus. New York etc.: Springer-Verlag (1988; Zbl 0638.60065)] or F. B. Knight [Essentials of Brownian motion and diffusion. Providence, R.I.: AMS (1981; Zbl 0458.60002)] do not shed much light on the difference between points of increase and local maxima. Aldous’ own argument is only a “Poisson clumping heuristic”.
This note is an attempt at the impossible, i.e., a proof which is at the same time rigorous, short, simple, elementary, and last but not least, elucidates the difference between the points of increase and local maxima. I believe that the proof is new, although its ideas may be traced back to Adelman (loc. cit.), Aldous (loc. cit.) and B. Davis [Z. Wahrscheinlichkeitstheor. Verw. Geb. 64, 359–367 (1983; Zbl 0506.60078)].


60J65 Brownian motion
60G17 Sample path properties
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