Nonlinear diffusions and stable-like processes with coefficients depending on the median or VaR

The paper is devoted to the well-posedness for nonlinear McKean-Vlasov type diffusions with coefficients depending on the median or, more generally, on the α-quantile of the underlying distribution. The median is not a continuous function on the space of probability measures equipped with the weak convergence. This is one reason why well-posedness of the SDE considered in the paper does not follow by standard arguments