Brownian transition density
= Brownian transition density
{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 $\mathbb P_x(X_t\in A)=\int_Ap_t(x,y)\,dy$.