Singular HJB equations with applications to KPZ on the real line

This paper is devoted to studying Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which are not well-defined in the classical sense and understood by using paracontrolled distribution method introduced (Gubinelli et al. Forum Math Pi 3(6):1, 2015). By a new characterization of weighted Hölder spaces Zvonkin’s transformation we prove some priori estimates, therefore establish global well-posedness for singular HJB equations. As applications, obtain polynomial KPZ type on real line, as well modified Cole–Hopf applicable.