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.

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