Subordination Integrals

📐 Definition

Given a Markov semigroup $S(\tau)$ and a nonnegative density $\eta(\tau,t)$ with $\int_{0}^{\infty} \eta(\tau,t)\,d\tau = 1$, the subordinated operator is

$$T(t) = \int_{0}^{\infty} \eta(\tau,t)\,S(\tau)\,d\tau$$

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representing evolution under a randomized operational time $\tau$.

Domain and Codomain

Acts on the same function space as $S(\tau)$ (e.g., $L^2$ or probability densities). Requires $\eta(\cdot,t)$ integrable for each $t \ge 0$ and $S(\tau)$ strongly continuous.

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⚙️ Key Properties

Positivity and Linearity

If $S(\tau)$ preserves positivity (and mass, in probabilistic settings), then $T(t)$ inherits these properties because it is an average over $\tau$.

Laplace-Domain Representation

When $\eta$ corresponds to a subordinator with Bernstein function $\psi$, Laplace transforms encode the time change. In many common cases, the operational-time mixture corresponds to replacing $s$ by $\psi(s)$ in transform formulas.

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🎯 Special Cases and Limits

• $\eta(\tau,t)=\delta(\tau-t)$ recovers the original semigroup $S(t)$.
• Inverse-stable subordination yields time-fractional diffusion and Mittag–Leffler-type relaxation.

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🔗 Related Functions

Temporal memory kernels encode similar nonlocal time effects via convolution; fractional integrals arise from inverse-stable subordinators.

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Usage in Oakfield

Oakfield exposes subordination directly as an integrator:

• subordination integrator approximates subordinated evolution by quadrature over an operational-time parameter, repeatedly evaluating the context drift in a side-effect-free drift mode.
• This integrator is used from Lua via sim.sim_create_context_integrator(ctx, "subordination", {...}) and supports complex primary fields.

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Historical Foundations

📜 Semigroup Subordination

Bochner’s subordination framework links stochastic time changes and semigroup theory, providing a principled route from classical diffusion to anomalous transport models.

🌍 Modern Perspective

Subordination remains a standard mechanism for constructing non-Markovian effective dynamics from Markovian building blocks.

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📚 References

• Schilling, Song, Vondraček, Bernstein Functions
• Meerschaert & Sikorskii, Stochastic Models for Fractional Calculus