measurevalued

Measure-valued objects are mathematical objects whose values are measures on a given base space. A measure-valued stochastic process is a process (X_t) that takes values in the space M(S) of measures on a measurable space S, typically with finite total mass or as finite measures or probability measures.

The state space is usually the space M(S) or M+(S) with topologies such as weak convergence or the vague topology, turning it into a Polish space in common cases. Convergence and continuity of paths are defined in terms of test functions f, by the scalar process ⟨X_t, f⟩ = ∫ f dX_t.

Prominent examples include Dawson–Watanabe (or superprocesses), which arise as scaling limits of branching particle systems and satisfy measure-valued branching equations; the Fleming–Viot process, a measure-valued diffusion used in population genetics; and various interacting particle systems that converge to measure-valued limits.

Measure-valued processes are often constructed via martingale problems, stochastic partial differential equations, or particle representations. They often admit duality relations with function-valued processes or genealogical representations.

Applications span population genetics, ecology, and Bayesian nonparametrics (random measures and processes on measures, such as the Dirichlet process). The study blends probability, analysis, and functional analysis, with a rich theory of existence, uniqueness, and long-term behavior.