# On uniqueness of solutions to viscous HJB equations with a subquadratic nonlinearity in the gradient

## TL;DR

This paper proves the uniqueness of positive solutions to certain viscous Hamilton-Jacobi-Bellman equations with subquadratic gradient nonlinearities under mild growth conditions on the potential function, using a novel measure-based approach.

## Contribution

It introduces a new method involving infinite dimensional linear programming for measures, establishing uniqueness without strong regularity assumptions.

## Key findings

- Unique positive solution for a broad class of coercive functions
- Method applicable to larger Hamiltonian classes studied by Ichihara
- Verification of optimality in ergodic control problems without parabolic equations

## Abstract

Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form $-\Delta u(x) + \frac{1}{\gamma} |D{u}(x)|^\gamma = f(x) - \lambda$, with $f$ a coercive function and $\lambda$ a constant, in the subquadratic case, that is, $\gamma\in(1,2)$, appears to be an open problem. Barles and Meireles [Comm. Partial Differential Equations 41 (2016)] show uniqueness in the case that $f(x) \approx |x|^\beta$ and $|D f(x)|\lessapprox |x|^{(\beta-1)_+}$ for some $\beta>0$, essentially matching earlier results of Ichihara, who considered more general Hamiltonians but with better regularity for $f$. Without enforcing this assumption, to our knowledge, there are no results on uniqueness in the literature. In this short article, we show that the equation has a unique positive solution for any locally Lipschitz continuous, coercive $f$ which satisfies $|D f(x)| \le \kappa\bigl(1 +|f(x)|^{2-\frac{1}{\gamma}}\bigr)$ for some positive constant $\kappa$. Since $2-\frac{1}{\gamma}>1$, this assumption imposes very mild restrictions on the growth of the potential $f$. We also show that this solution fully characterizes optimality for the associated ergodic problem. Our method involves the study of an infinite dimensional linear program for elliptic equations for measures, and is very different from earlier approaches. It also applies to the larger class of Hamiltonians studied by Ichihara, and we show that it is well suited to provide verification of optimality results for the associated ergodic control problems, even in a pathwise sense, and without resorting to the parabolic problem.