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We propose a new classification scheme for diffusion processes for which thebackward Kolmogorov equation is solvable in analytically closed form byreduction to hypergeometric equations of the Gaussian or confluent type. Theconstruction makes use of transformations of diffusion processes to eliminatethe drift which combine a measure change given by Doob's h-transform and adiffeomorphism. Such transformations have the important property of preservinganalytic solvability of the process: the transition probability density for thedriftless process can be expressed through the transition probability densityof original process. We also make use of tools from the theory of ordinarydifferential equations such as Liouville transformations, canonical forms andBose invariants. Beside recognizing all analytically solvable diffusion processknown in the previous literature fall into this scheme and we also discoverrich new families of analytically solvable processes.

Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions