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We consider the stochastic equation
where is a one-dimensional Brownian motion, is the initial value, and is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients , beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on ensuring the existence as well as the uniqueness in law of the solution.
We give a complete analytical characterization of the functions transforming reflected Brownian motions to local Dirichlet processes.
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