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Brownian transition semigroup

{c}
{title2=$p_t(x,y)$}

The transition density of standard Brownian motion in $\mathbb R^d$ is the <heat kernel>
$$
p_t(x,y)=\frac{1}{(2\pi t)^{d/2}}
\exp\left(-\frac{|y-x|^2}{2t}\right),
\qquad t>0.
$$
Thus $\mathbb P_x(X_t\in A)=\int_Ap_t(x,y)\,dy$.

Brownian transition density

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 5 / b / Solution

Brownian transition density ( $p_{t} (x, y)$ )

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Brownian transition density (pt​(x,y))

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