Backward heat equation ($u_t=-D{u}_{xx},D>0$)

A Fourier mode has multiplier $e^{D{k}^{2}t}$, amplifying arbitrarily small high-frequency data. For a fixed positive time $t_0$, initial modes $e^{-D{n}^2{t}_0/2}\sin (nx)$ tend to zero in $L^2$ while their values at $t_0$ grow without bound. This violates continuous dependence on initial data and gives an ill-posed problem in ordinary unweighted spaces. An analytic-data restriction or physical short-scale regularization changes that conclusion.

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 $f$.