$u(t,B_t)$ is a martingale if $u(t,x)$ is polynomial in each variables and satisfies the heat equation I want to show that for $u(t,x)$ which is a polynomial in $t$ and $x$ such that $$\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}$$
we have $u(t,B_t)$ is a martingale where $B_t$ stands for the standard Brownian motion.
Durrett's "Probability theory and examples" shows that $ E_x u(t, B_t) $ is constant in $t$ and concludes right away that $u(t,B_t)$ is a martingale. How is this possible?
Any help is appreciated.

A mean-constant process with markov property is a martingale.
