## Brownian motion constructions

Interpolation (Levy-Ciesielski construction)

Cameron-Martin space eigenfunctions

White noise

Limit of Random walk

Hyperreals construction

see “A Non-standard Construction of Multi-Dimensional Brownian Motions and Option Pricing””.

Heat equation

The $u(x,t)=E_{x}\left[g(B_{t})\chi\{t\leq \tau_{\Omega}\}\right]$ solves the heat equation with initial data $g$ in domain $\Omega$. So conversely by starting with the heat semigroup, we construct the Markov semigroup.

This entry was posted in Notes. Bookmark the permalink.

This site uses Akismet to reduce spam. Learn how your comment data is processed.