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A Guide to Brownian Motion and Related Stochastic Processes

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A Guide to Brownian Motion and Related Stochastic Processes 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 .............................. 15 4.4 Transformations ........................... 16 0 J. Pitman and M. Yor/Guide to Brownian motion 1 4.4.1 Spacetransformations . 16 4.4.2 Besselprocesses.. .. .. .. .. .. .. .. .. .. 17 4.4.3 The Ornstein-Uhlenbeck process . 18 4.5 L´evyprocesses ............................ 18 4.6 ReferencesforMarkovprocesses . 20 5 BMasamartingale ............................ 20 5.1 L´evy’s characterization . 21 5.2 Itˆo’sformula ............................. 22 5.3 Stochastic integration . 23 5.4 ConstructionofMarkovprocesses. 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 Changeoftime ........................ 35 5.6.4 Knight’s theorem . 35 5.7 BMasaharness ........................... 36 5.8 Gaussian semi-martingales . 36 5.9 Generalizations of martingale calculus . 37 5.10References............................... 37 6 Brownianfunctionals............................ 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 TheRay-Knighttheorems . 41 6.3 Additive functionals . 43 6.4 Quadratic functionals . 44 6.5 Exponential functionals . 45 7 Pathdecompositionsandexcursiontheory. 45 7.1 Brownianbridge,meanderandexcursion. 45 7.2 TheBrownianzeroset ........................ 48 7.3 L´evy-ItˆotheoryofBrownianexcursions . 48 8 PlanarBrownianmotion.......................... 50 8.1 Conformalinvariance. .. .. .. .. .. .. .. .. .. .. 50 8.2 Polarity of points, and windings . 50 8.3 Asymptoticlaws ........................... 51 8.4 Self-intersections . 52 8.5 Exponentsofnon-intersection . 54 9 Multidimensional BM . 55 9.1 Skewproductrepresentation. 55 J. Pitman and M. Yor/Guide to Brownian motion 2 9.2 Transience............................... 55 9.3 Othergeometricaspects . 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 10.1Constructions............................. 58 10.2Radialprocesses ........................... 59 10.3References............................... 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 TheNeumannproblem . 65 12.1.5 Non-linear problems . 65 12.2 Stochastic differential Equations . 66 12.2.1 Dynamic Equations . 67 12.3Potentialtheory ........................... 68 12.4 BMandharmonicfunctions . 68 12.5 Hypercontractive inequalities . 69 13 Connectionswithnumbertheory . 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 15Applications................................. 71 15.1 Economicsandfinance . 71 15.2Statistics ............................... 72 15.3Physics ................................ 72 15.4 Fluid mechanics . 73 15.5 Controlofdiffusions . .. .. .. .. .. .. .. .. .. .. 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. Itˆoand H. P. McKean, Jr. Diffusion processes and their sample paths (1965). [179] K. Itˆo. 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¨orters and 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 [436]. Wiener proved that there exists a version of BM with continuous paths. L´evy made major contributions to the theory of Brownian paths, especially regarding the structure of their level sets, their occupation densities, and other fine fea- tures of their oscillations such as laws of the iterated logarithm. Note that BM is a Gaussian process, a Markov process, and a martingale. Hence its importance in the theory of stochastic process. It serves as a basic building block for many more complicated processes. For further history of Brownian motion and related processes we cite Meyer [307], Kahane [197], [199] and Yor [455]. 1.2. Definitions This section records the basic definition of a Brownian motion B, along with some common variations in terminology which we use for some purposes. The basic definition of B, as a random continuous function with a particular family of finite-dimensional distributions, is motivated in Section 2 by the appearance of this process as a limit in distribution of rescaled random walk paths. Let (Ω, , P) be a probability space. A stochastic process (B(t,ω),t 0,ω Ω) is a BrownianF motion if ≥ ∈ (i) For fixed each t, the random variable Bt = B(t, ) has Gaussian distribu- tion with
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