  We study the well-posedness of an infinite-dimensional Hamilton-Jacobi
equation posed on the set of non-negative measures and with a monotonic
non-linearity. Our results will be used in a companion work to propose a
conjecture and prove partial results concerning the asymptotic mutual
information in the assortative stochastic block model in the sparse regime. The
equation we consider is naturally stated in terms of the Gateaux derivative of
the solution, unlike previous works in which the derivative is usually of
transport type. We introduce an approximating family of finite-dimensional
Hamilton-Jacobi equations, and use the monotonicity of the non-linearity to
show that no boundary condition needs to be prescribed to establish
well-posedness. The solution to the infinite-dimensional Hamilton-Jacobi
equation is then defined as the limit of these approximating solutions. In the
special setting of a convex non-linearity, we also provide a Hopf-Lax
variational representation of the solution.
