 Space-time harmonic functions along Brownian motion

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 $t\ge 0$. Necessity follows because a zero finite-variation integral has continuous integrand $f_t(B_t,t)+\frac{1}{2}{f}_{xx}(B_t,t)$ 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 $f$.