Vol. 0 (0000) A guide to Brownian motion and related stochastic processes Jim Pitman and Marc Yor Dept. Statistics, University of California, 367 Evans Hall # 3860, Berkeley, CA 94720-3860, USA e-mail:
[email protected] Abstract: This is a guide to the mathematical theory of Brownian mo- tion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical the- ory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof. As a typical reader, we have in mind a student, familiar with the basic concepts of probability based on measure theory, at the level of the graduate texts of Billingsley [43] and Durrett [106], and who wants a broader perspective on the theory of Brow- nian motion and related stochastic processes than can be found in these texts. Keywords and phrases: Markov process, random walk, martingale, Gaus- sian process, L´evy process, diffusion. AMS 2000 subject classifications: Primary 60J65. Contents 1 Introduction................................. 3 1.1 History ................................ 3 1.2 Definitions............................... 4 2 BM as a limit of random walks . 5 3 BMasaGaussianprocess......................... 7 3.1 Elementarytransformations . 8 3.2 Quadratic variation . 8 3.3 Paley-Wiener integrals . 8 3.4 Brownianbridges........................... 10 3.5 FinestructureofBrownianpaths . 10 arXiv:1802.09679v1 [math.PR] 27 Feb 2018 3.6 Generalizations . 10 3.6.1 FractionalBM ........................ 10 3.6.2 L´evy’s BM . 11 3.6.3 Browniansheets ....................... 11 3.7 References............................... 11 4 BMasaMarkovprocess.......................... 12 4.1 Markovprocessesandtheirsemigroups. 12 4.2 ThestrongMarkovproperty. 14 4.3 Generators .............................