A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients

We study measure-valued solutions of the inhomogeneous continuity equation partial derivative rho t + div (v(rho t)) = g(rho t), where the coefficients v and g are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinger-Kantorovich energy is finite. This principle gives a decomposition of the solution into curves t bar right arrow h(t)delta(gamma(t)) that satisfy the characteristic system (gamma)over dot(t) = v(t, gamma(t)), (h)over dot(t) = g(t,gamma(t))(h)over dot(t) in an appropriate sense. In particular, it provides a generalization of existing superposition principles to the low-regularity case of g where characteristics are not unique with respect to h. Two applications of this principle are presented. First, uniqueness of minimal total-variation solutions for the inhomogeneous continuity equation is obtained if characteristics are unique up to their possible vanishing time. Second, the extremal points of dynamic Hellinger-Kantorovich-type regularizers are characterized. Such regularizers arise, for example, in the context of dynamic inverse problems and dynamic optimal transport.