Space-time Hermite polynomial
= Space-time Hermite polynomial
{title2=$H_n(x,t)$}
= Hermite polynomial martingale
{c}
{synonym}
The space-time <Probabilists' Hermite polynomial> $H_n(x,t)=t^{n/2}\mathrm{He}_n(x/\sqrt t)$ extends polynomially to $t=0$ and solves $\partial_tH_n+\frac12\partial_x^2H_n=0$. Therefore $H_n(B_t,t)$ is a <martingale> for a standard <Brownian motion>, by the <Itô formula> and Gaussian <moments>.