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