Temporal Memory Kernels

📐 Definition

Given a kernel $K(t)$ defined for $t \ge 0$ , the memory operator acting on a time series $f(t)$ is

M[f](t) = \int_{0}^{t} K(t-\tau)\, f(\tau)\,d\tau

Domain and Codomain

Defined for $f \in L^1_{\text{loc}}([0,T])$ with kernels $K$ that are locally integrable (possibly weakly singular). Output shares the same temporal domain.

⚙️ Key Properties

Linearity and Convolution Form

$M$ is linear and is a (causal) convolution in time with kernel $K$ .

Laplace Transform

For suitable $f$ and $K$ ,

\mathcal{L}\{M[f]\}(s) = \hat{K}(s)\,\mathcal{L}\{f\}(s)

Kernel-Driven Behavior

The choice of kernel sets decay (exponential, stretched exponential, power law), regularity, and effective memory length.

🎯 Special Cases and Limits

Exponential kernels recover standard first-order relaxation.
Power-law kernels produce fractional integrals.
Compactly supported kernels localize memory.
As $K \to \delta$ , $M$ reduces to the identity operator (distributional sense).

Fractional integrals correspond to $K(t) \propto t^{\alpha-1}$ . Subordination integrals provide an alternative interpretation of nonlocal time effects via randomized operational time.

Usage in Oakfield

Oakfield implements temporal memory primarily through dedicated operator/integrator machinery:

Finite-memory fractional kernel: fractional_memory stores a rolling window of past states and applies history-weighted updates each step.
Subordination: the subordination integrator provides a nonlocal-in-time effect by mixing drift evaluations over operational time with a stretched-exponential weight.
Colored noise: stochastic_noise maintains internal state with a correlation timescale, providing another form of “memory” in stochastic forcing.

Historical Foundations

📜 Memory and Relaxation

Kernel-based memory operators unify classical relaxation laws (exponential kernels) with anomalous and long-memory behavior (power-law kernels).

🌍 Modern Perspective

Memory kernels provide a flexible, implementation-friendly interface for nonlocal-in-time dynamics, with Laplace-domain multipliers offering a complementary spectral view.

📚 References

Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity
Podlubny, Fractional Differential Equations