Brownian Motion and Martingale Theory
The applications of the Itô integral in Sections VIII.12 to VIII.14 exhibit aspects of the intimate relation between Brownian motion and martingale theory. In the following we shall go from simple examples of this relation to an analysis by means of martingale theory of the composition of the basic functions of the potential theory for Laplace’s equation [the heat equation] with Brownian motion [space-time Brownian motion]. This will be effected by a direct method without the use of the Itô integral, but there will be a slight repetition of some of the most elementary topics in Chapter VIII.
KeywordsBrownian Motion Transition Density Harmonic Measure Sample Function Nonempty Open Subset
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