= Martingale problem for Brownian motion
{title2=$Lf=\tfrac12\Delta f$}
A continuous adapted process started at a fixed point is <Brownian motion> exactly when $f(X_t)-f(X_0)-\frac12\int_0^t\Delta f(X_s)ds$ is a <martingale> for every $f\in C_c^\infty(\mathbb R^d)$. Cutoff coordinate and coordinate-product tests give continuous <local martingales> with <quadratic covariations> $[X^i,X^j]_t=\delta_{ij}t$. The <Itô formula> makes $\exp(iu\cdot(X_t-X_0)+|u|^2t/2)$ a local martingale, whose deterministic modulus bound on bounded horizons makes it a true martingale. Its conditional expectation identifies every increment as centered Gaussian with covariance $(t-s)I$, independent of the past. This proves the converse via the <Lévy characterization of multidimensional Brownian motion>. The forward implication is the <Itô formula>.
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