The Probability Space of Brownian Motion
According to Einstein’s description, the Brownian motion can be defined by the following two properties: first, it has continuous trajectories (sample paths) and second, the increments of the paths in disjoint time intervals are independent zero mean Gaussian random variables with variance proportional to the duration of the time interval (it is assumed, for definiteness, that the possible trajectories of a Brownian particle start at the origin). These properties have far-reaching implications about the analytic properties of the Brownian trajectories. It can be shown, for example (see Theorem 2.4.1), that these trajectories are not differentiable at any point with probability 1 . That is, the velocity process of the Brownian motion cannot be defined as a real-valued function, although it can be defined as a distribution (generalized function) . Langevin’s construction does not resolve this difficulty, because it gives rise to a velocity process that is not differentiable so that the acceleration process, (t) in eq. (1.24), cannot be defined.
KeywordsBrownian Motion Probability Space Elementary Event Brownian Particle Roulette Wheel
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