Let and be independent Wiener processes, and consider the task of estimating a diffusion solving the stochastic DEdxt=f(xt)dt+dwton the basis of noisy observationsdefined bydyt=xtdt+dbt. This problem is governed by the filtering equationfor the unnormalized conditional densitywithA*the forwarded operatorTheorem: ifthen the fundamental solution of the filtering equation can be written explicity in terms of a small number of statistics satisfying a matrixvector equation. The Lie algebraic interpretation of this result is studied and described. Extensions to many dimensions and applications to optimal stochastic control readily follow.