  In this paper, we develop a general 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 \& Perkins [BP09] in the standard diffusion case, the proposed methodology
allows one to deal with processes whose probability laws are singular with
respect to the Lebesgue measure. To illustrate our methodology, we prove weak
existence and uniqueness in two examples : a diffusion process with
coefficients depending on its running symmetric 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.
