Introduction

One parameter semigroups provide a common language for evolution equations, Markov processes, diffusion, and resolvent methods. Standard analytic foundations appear in [Pazy1983], [EngelNagel2000]. Finite Markov chains and their coarse descriptions are treated in [KemenySnell1960], [Norris1997], [RogersPitman1981]. Conditional expectation and martingale difference decompositions are basic parts of probability theory [Williams1991]. Gaussian transfer and Hermite expansions provide a spectral model for the Ornstein Uhlenbeck semigroup [Janson1997]. Fractional powers of generators and subordinate semigroups belong to the functional calculus of positive and sectorial operators [Balakrishnan1960], [SchillingSongVondracek2012]. The Feynman Kac formula, projection methods, and phase type laws add three further views of transfer [Kac1949], [Feshbach1958], [Zwanzig1961], [Neuts1981].

These subjects share a simple operation. A state is propagated, resolved into declared components, compared across levels, and reconstructed. The same operation can be linear, as in a Markov semigroup, or normalized and nonlinear, as in Bayesian conditioning. Hidden components can be discarded from the state vector without disappearing from the dynamics. Their exact influence returns through a frequency dependent self energy or a time dependent memory kernel.

The first part of the article treats bounded semigroups on Banach spaces. A finite packet system resolves every operator into input and output blocks. The semigroup law becomes block convolution. Splitting the generator into its packet diagonal and transfer part gives a convergent Dyson expansion over directed packet paths. For a coarse map \(C\), the generator defect

produces the entire semigroup defect through a Duhamel integral. The same defect controls resolvents and holomorphic functions of the generators.

The next question concerns refinement towers. Exact intertwining gives a coordinatewise semigroup on the bounded inverse limit. Approximate refinement is more delicate. Even if every adjacent defect is small, reconstruction from a fine level can introduce a second error because a coarse state need not remain inside the reconstructed fine subspace. We separate these two effects. Summability of adjacent defects gives a uniform limit of reconstructed coarse evolutions. An asymptotic closure condition is then sufficient for the limiting family to satisfy the semigroup law.

The probabilistic part begins with finite Markov chains. Exact lumpability is the vanishing of a canonical obstruction on the kernel of aggregation. In coordinates, this is equality of the transition mass sent from any two states in the same source block into every target block. The usual finite horizon defect formula follows from telescoping. Conditional expectation is then treated as an orthogonal projector. A nested family of partitions produces mutually orthogonal martingale packets and an exact energy decomposition.

Bayesian updating requires more care because normalization depends on the current prior. Given a partition, every fine update has an exact two stage description. The block masses are updated by prior dependent average likelihoods, while the conditional distribution inside each block is updated by its centered likelihood variation. A fixed coarse Bayesian rule works for every prior precisely when the likelihood is constant inside each block. This gives a finite and exact form of Bayesian sufficiency. For a family of likelihoods with at least two common level sets, those level sets form the unique coarsest nontrivial fixed sufficient partition. Relative entropy then separates exactly into information retained by the coarse packet masses and information carried by the local conditional changes. The same part derives perturbation bounds, the replicator flow, a variance identity for accumulated evidence, and stability estimates for hidden Markov filtering.

For a positive operator \(L\), spectral packets define \(L^\alpha\) and \({\mathrm e}^{-tL^\alpha}\). Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound \(m\), the Balakrishnan formula gives the quantitative estimate

A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.

The final sections move from generators to source reconstruction and hidden variables. The one dimensional Dirichlet Green kernel is reproduced exactly by the scaled inverse of the second difference matrix at grid nodes. A discrete maximum principle gives a complete second order error estimate for smooth Poisson solutions. Finite state Feynman Kac transfer is proved in discrete and continuous time. Schur Feshbach elimination yields the exact projected resolvent, the retained memory equation, and the Laplace correspondence between memory and self energy. Every stable finite hidden system gives a polynomial exponential memory expansion. Primary blocks can be ranked optimally for the certified \(L^1\) bound, and a Volterra estimate carries kernel error to the retained solution. Survival, event density, moments, and competing risks are then read directly from a transient semigroup.

Several claims are intentionally absent. No arbitrary unbounded operator is asserted to generate a semigroup. The inverse limit theorem uses bounded generators or an explicit uniform continuity condition. Bayesian sufficiency is proved for a finite partition and a fixed likelihood. The fractional defect estimate uses a common spectral gap. The Poisson estimate concerns the stated one dimensional scheme. Memory compression is controlled for the declared finite or integrable kernels. General stochastic differential equations, posterior consistency, nonlinear partial differential equations, and optimal model reduction remain outside the article.

The same words have different meanings in the realizations below. The table records the object that is transferred and the condition that makes its coarse description exact.

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