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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.

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

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