Viscosity Characterization of the Explosion Time Distribution for Diffusions

We show that the tail distribution $U$ of the explosion time for a multidimensional diffusion (and more generally, a suitable function $\mathscr{U}$ of the Feynman-Kac type involving the explosion time) is a viscosity solution of an associated parabolic partial differential equation (PDE), provided that the dispersion and drift coefficients of the diffusion are continuous. This generalizes a result of Karatzas and Ruf (2013), who characterize $U$ as a classical solution of a Cauchy problem for the PDE in the one-dimensional case, under the stronger condition of local H\"older continuity on the coefficients. Furthermore, we show that $\mathscr{U}$ is dominated by any nonnegative classical supersolution of this Cauchy problem, and that $\mathscr{U}$ is the smallest lower-semicontinuous viscosity supersolution of that PDE with an appropriate boundary condition, provided it is a classical solution. We also consider another notion of weak solvability, that of the distributional (sub/super)solution, and show that $\mathscr{U}$ is no greater than any nonnegative distributional supersolution of the relevant PDE. Finally, we establish the joint continuity of $\mathscr{U}$ in the one-dimensional case.