Gaussian-type density bounds for solutions to multidimensional backward SDEs and application to gene expression

We obtain upper and lower Gaussian-type bounds on the density of the law of each component $Y^i_t$ of the solution $Y_t$ to a multidimensional backward SDE. Our approach is based on the Nourdin-Viens formula and the analysis of the associated semilinear parabolic PDE. Furthermore, we apply our results to stochastic gene expression; namely, we estimate densities of distributions of protein levels of separate genes in a gene regulatory network.