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Basic Properties of Brownian Motion

Stat205B: Probability Theory (Spring 2003)Lecture: 15 Basic Properties of Brownian MotionLecturer: James W. PitmanScribe: Rui this lecture, we discuss some Basic Properties of Brownian Motion , including various transformations, thetransition semigroup and its Motion lies in the intersection of several important classes of processes. It is a gaussian Markovprocess, it has continuous paths, it is a process with stationary independent increments (a L evy process),and it is a martingale. Several characterizations are known based on these consider also the following variation of Brownian Motion :Example a Brownian Motion (Bt, t 0) starting from 0.

is a Gaussian processes, i.e. all its FDDs (finite dimensional distributions) are multivariate normal. Note that X is a Markov process, with stationary independent increments, with x the initial state, δ the drift parameter, σ2 the variance parameter. These three parameters determine all the FDDs of (X t,t ≥ 0), which

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