Necessary and sufficient conditions are given for transforming (constructively) a one-dimensional diffusion process described by a Kolmogorov equation into the Wiener process. These conditions are shown to be equivalent to invariance of a parabolic partial differential equation under a six-parameter Lie group of point transformations. Moreover these conditions correspond to a significant generalization of Cherkasov’s result; i.e., a much wider class of Kolmogorov equations can be derived from the Wiener process than was previously realized.