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´evyprocess, 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 BM as a Gaussian process . .7 3.1 Elementary transformations . .8 3.2 Quadratic variation . .8 3.3 Paley-Wiener integrals . .8 3.4 Brownian bridges . 10 3.5 Fine structure of Brownian paths . 10 3.6 Generalizations . 10 3.6.1 Fractional BM . 10 3.6.2 L´evy's BM . 11 3.6.3 Brownian sheets . 11 3.7 References . 11 4 BM as a Markov process . 12 4.1 Markov processes and their semigroups . 12 4.2 The strong Markov property . 14 4.3 Generators . 15 4.4 Transformations . 16 0 J. Pitman and M. Yor/Guide to Brownian motion 1 4.4.1 Space transformations . 16 4.4.2 Bessel processes . 17 4.4.3 The Ornstein-Uhlenbeck process . 18 4.5 L´evyprocesses . 18 4.6 References for Markov processes . 20 5 BM as a martingale . 20 5.1 L´evy'scharacterization . 21 5.2 It^o'sformula . 22 5.3 Stochastic integration . 23 5.4 Construction of Markov processes . 26 5.4.1 Stochastic differential equations . 26 5.4.2 One-dimensional diffusions . 28 5.4.3 Martingale problems . 28 5.4.4 Dirichlet forms . 29 5.5 Brownian martingales . 29 5.5.1 Representation as stochastic integrals . 29 5.5.2 Wiener chaos decomposition . 30 5.6 Transformations of Brownian motion . 31 5.6.1 Change of probability . 32 5.6.2 Change of filtration . 33 5.6.3 Change of time . 35 5.6.4 Knight's theorem . 35 5.7 BM as a harness . 36 5.8 Gaussian semi-martingales . 36 5.9 Generalizations of martingale calculus . 37 5.10 References . 37 6 Brownian functionals . 38 6.1 Hitting times and extremes . 38 6.2 Occupation times and local times . 39 6.2.1 Reflecting Brownian motion . 41 6.2.2 The Ray-Knight theorems . 41 6.3 Additive functionals . 43 6.4 Quadratic functionals . 44 6.5 Exponential functionals . 45 7 Path decompositions and excursion theory . 45 7.1 Brownian bridge, meander and excursion . 45 7.2 The Brownian zero set . 48 7.3 L´evy-It^otheory of Brownian excursions . 48 8 Planar Brownian motion . 50 8.1 Conformal invariance . 50 8.2 Polarity of points, and windings . 50 8.3 Asymptotic laws . 51 8.4 Self-intersections . 52 8.5 Exponents of non-intersection . 54 9 Multidimensional BM . 55 9.1 Skew product representation . 55 J. Pitman and M. Yor/Guide to Brownian motion 2 9.2 Transience . 55 9.3 Other geometric aspects . 56 9.3.1 Self-intersections . 56 9.4 Multidimensional diffusions . 57 9.5 Matrix-valued diffusions . 57 9.6 Boundaries . 57 9.6.1 Absorbtion . 57 9.6.2 Reflection . 58 9.6.3 Other boundary conditions . 58 10 Brownian motion on manifolds . 58 10.1 Constructions . 58 10.2 Radial processes . 59 10.3 References . 59 11 Infinite dimensional diffusions . 60 11.1 Filtering theory . 60 11.2 Measure-valued diffusions and Brownian superprocesses . 60 11.3 Malliavin calculus . 61 12 Connections with analysis . 62 12.1 Partial differential equations . 62 12.1.1 Laplace's equation: harmonic functions . 62 12.1.2 The Dirichlet problem . 63 12.1.3 Parabolic equations . 64 12.1.4 The Neumann problem . 65 12.1.5 Non-linear problems . 65 12.2 Stochastic differential Equations . 66 12.2.1 Dynamic Equations . 67 12.3 Potential theory . 68 12.4 BM and harmonic functions . 68 12.5 Hypercontractive inequalities . 69 13 Connections with number theory . 69 14 Connections with enumerative combinatorics . 69 14.1 Brownian motion on fractals . 70 14.2 Analysis of fractals . 71 14.3 Free probability . 71 15 Applications . 71 15.1 Economics and finance . 71 15.2 Statistics . 72 15.3 Physics . 72 15.4 Fluid mechanics . 73 15.5 Control of diffusions . 73 References . 73 J. Pitman and M. Yor/Guide to Brownian motion 3 1. Introduction This is a guide to the mathematical theory of Brownian motion (BM) and re- lated stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical theory of partial differential equations associated with the Laplace and heat operators, and various general- izations 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 BM and related stochastic processes than can be found in these texts. The difficulty facing such a student is that there are now too many advanced texts on BM and related processes. Depending on what aspects or applications are of interest, one can choose from any of the following texts, each of which contains excellent treatments of many facets of the theory, but none of which can be regarded as a definitive or complete treatment of the subject. General texts on BM and related processes [139] D. Freedman. Brownian motion and diffusion (1983). [180] K. It^oand H. P. McKean, Jr. Diffusion processes and their sample paths (1965). [179] K. It^o. Lectures on stochastic processes (1984). [201] O. Kallenberg. Foundations of modern probability (2002). [211] I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus (1988). [206] G. Kallianpur and S. P. Gopinath. Stochastic analysis and diffusion pro- cesses (2014). [221] F. B. Knight. Essentials of Brownian motion and diffusion (1981). [313] P. M¨ortersand Y. Peres. Brownian motion (2010). [370] D. Revuz and M. Yor. Continuous martingales and Brownian motion (1999). [372] and [373] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales, Vols. I and II (1994). This list does not include more specialized research monographs on subjects closely related to BM such as stochastic analysis, stochastic differential geom- etry, and more general theory of Gaussian and Markov processes. Lists of such monographs classified by subject can be found in following sections. 1.1. History The physical phenomenon of Brownian motion was discovered by Robert Brown, a 19th century scientist who observed through a microscope the random swarm- ing motion of pollen grains in water, now understood to be due to molecular bombardment. The theory of Brownian motion was developed by Bachelier in J. Pitman and M. Yor/Guide to Brownian motion 4 his 1900 PhD Thesis [8], and independently by Einstein in his 1905 paper [113] which used Brownian motion to estimate Avogadro's number and the size of molecules. The modern mathematical treatment of Brownian motion (abbrevi- ated to BM), also called the Wiener process is due to Wiener in 1923.