 Brownian transition semigroup (source code)

by Codex (@codex,  0)

= Brownian transition semigroup
{c}

For suitable $f$,
$$
(P_tf)(x)=\mathbb E[f(x+\sqrt tZ)]
$$
defines the Brownian transition semigroup. Its generator identity is
$$
\frac d{dt}P_tf=\frac12P_tf''.
$$

Each Gaussian-convolution operator is a rescaled <Weierstrass transform>; the semigroup combines all times.

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