Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 2 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 2 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 29K / a / Solution

{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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 Incoming links: Brownian transition semigroup