Kolmogorov backward and forward equations

For a nonexplosive Markov diffusion and a terminal observable $g$, define $u(t,x)$ as the expected value of $g(X_T)$ when the process starts from $x$ at time $t$. If this function is once differentiable in time and twice in space with the needed integrability, applying Itô's formula along the process shows that its drift must vanish. It therefore solves the backward equation with terminal, rather than initial, data. Time-homogeneous elapsed-time notation reverses the time derivative and yields the semigroup equation instead.

A clean sufficient classical setting is smooth bounded coefficients with bounded derivatives and a uniformly positive definite covariance, together with smooth compactly supported terminal data. More general settings exist, but global Lipschitz existence of an SDE alone does not automatically supply all these PDE derivatives. Unbounded observables can also be treated when suitable moment and growth controls are separately established. An explicit polynomial example below verifies those controls through a Gaussian solution rather than invoking a bounded-data theorem outside its assumptions.

Let $\mu_t$ be the distribution of the process. The martingale test identity implies that its pairing with a compactly supported smooth test changes by the pairing with $L$ applied to that test. This is a distributional forward equation for a family of measures. It remains meaningful when the law has atoms. For example, a zero-diffusion equation with a deterministic initial state keeps a point mass, so there is no ordinary Lebesgue density to differentiate.

When a sufficiently smooth density exists and integration by parts has no remaining boundary terms, the adjoint operator gives the Fokker–Planck equation. In one dimension the drift becomes a negative first derivative of the drift-density product, and the diffusion becomes one half of a second derivative of the variance-density product. Coefficients cannot be pulled outside these derivatives when they depend on position. On a bounded domain, killing, reflection, or another boundary rule also changes the corresponding flux or boundary condition.

For constant drift and positive constant volatility, transition laws from a fixed point are Gaussian at every positive elapsed time. Their variance is volatility squared times elapsed time, not half that quantity. The half appears in the differential operator. At zero elapsed time the law is a point mass, so its density formula is singular there. The correct initial condition is convergence of measures to the point mass, or convergence of expectations of bounded continuous tests.

The backward and forward equations are linked by duality: evolving a test and pairing with an initial law equals pairing that test with the evolved law. This equality explains their different variables and signs without choosing one as more physical. A PDE calculation should identify which variable is being differentiated, whether time is elapsed or terminal, and whether the claimed solution is classical or distributional. These three labels prevent most apparent contradictions between correct versions of the same transition formula.