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We consider the stochastic equation $$X_t=x_0+{\int }_0^{t}b(u,X_u)\mathrm{d}{B}_u,t\ge 0,$$ where $B$ is a one-dimensional Brownian motion, $x_0\in ℝ$ is the initial value, and $b[0,\infty )\times ℝ\to ℝ$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$, 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 $b$ 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.