Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States

Abstract

We develop a Banach-valued lateral Stieltjes calculus adapted to linear evolution equations on a Gelfand triple \[ V\hookrightarrow H\simeq H^*\hookrightarrow V^*. \] The evolution problem is formulated with the posterior state, \[ D_g u(t)+A(t) S_g u(t)=f(t),\qquad u(a)=u_0. \] This choice is decisive at atoms of the Stieltjes measure: if $\delta=\Delta_g(r)>0$, the equation yields the implicit variational resolvent \[ (I+\delta A(r))u(r^+)=u(r)+\delta f(r). \] We prove a Banach-valued lateral product formula, a weak--strong reconstruction theorem, and a Stieltjes analogue of the Lions--Magenes energy identity. In the separable Hilbert setting considered here, the stable projection chain used in the proof is shown to exist automatically by a spectral-band construction. The latter contains the exact positive jump contribution \[ \frac12\sum_r\|u(r^+)-u(r)\|_H^2. \] Existence is constructed independently by Galerkin approximation and by a Rothe scheme adapted to the Stieltjes mass and its atoms. The Rothe construction does not use spatial projections. We also explain why the available Aubin--Lions theorem for anterior Stieltjes--Bochner evolution graphs is not automatically applicable to the posterior energy space and, in the present linear coercive setting, is not needed. A heat equation with Stieltjes time is included as a model application. The claims are positioned relative to earlier Stieltjes parabolic equations, general measure-driven evolution, and operator-valued-measure parabolic systems; no priority is claimed for measure-driven parabolic theory or Galerkin approximation in general.