Abstract

A new class of Gaussian (‘‘Bernstein’’) diffusion processes associated with semiclassical quantum dynamics is introduced and studied according to a new Euclidean strategy for the quantization of systems of particles in a potential (‘‘Euclidean quantum mechanics’’). All the measures known by other means, up to now, are included as particular cases of the present framework. It is shown that the theory may be regarded as a well‐defined probabilistic version of Feynman’s space‐time approach distinct, in particular, from the one due to Kac, and with much closer relations with classical Lagrangian dynamics. Many explicit examples are considered.

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