When does the infinitesimal generator completely and uniquely encode its associated stochastic process?
For an Ito diffusion specifically, it would seem to do so, except maybe up to a rotation of the diffusion matrix coefficient.
2026-04-23 16:01:46.1776960106
Does infinitesimal generator fully encode SDE
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To summarize the comments, for an Ito diffusion specifically, if $dX_t$ has bounded, Lipschitz continuous coefficients, then it is a Feller process, which implies that its infinitesimal generator fully encodes it.
Anyone who wants to comment on more general cases are welcome to do so! :)