= Space-time harmonic functions along Brownian motion
{title2=$f_t+\tfrac12f_{xx}=0$}
For $f\in C^2(\mathbb R^2)$ and standard <Brownian motion> $B$, the drift in <Itô formula> for $f(B_t,t)$ vanishes exactly when the displayed equation holds for every spatial point and every $t\geq0$. Necessity follows because a zero finite-variation integral has continuous integrand $f_t(B_t,t)+\tfrac12f_{xx}(B_t,t)$ equal to zero, and at every positive time the <normal distribution> of $B_t$ has full support. Continuity extends the equation to time zero. There is no condition at negative times except compatibility with the global smoothness of $f$.
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