Before revision:  We constructed quantile diffusion processes by transforming each marginal of a given univariate diffusion process under a composite map consisting of a distribution function and quantile function, which in turn produces the marginals of the resulting quantile process.
After revision:  We constructed quantile diffusion processes by transforming each marginal of a given univariate diffusion process under a composite map consisting of a distribution and a quantile function. Such maps, analogous to rank transmutation maps, produce the marginals of the resulting quantile process.