In this article, we develop a methodology to prove weak uniqueness for stochastic differential equations with coefficients depending on some path-functionals of the process. As an extension of the technique developed by Bass and Perkins (2009) in the standard diffusion case, the proposed methodology allows one to deal with process whose probability laws is singular with respect to the Lebesgue measure. To illustrate our methodology, we prove weak existence and uniqueness in the two following examples, a diffusion process with coefficients depending on its running local time and a diffusion process with coefficients depending on its running maximum. In each example, we also prove the existence of the associated transition density and establish some Gaussian upper-estimates.
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