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Define a Δt-independent CE measure for continuous-time SDEs

Develop an objective causal-emergence measure for continuous-time stochastic differential equations that is independent of the discretization step Δt and remains meaningful as Δt→0 by characterizing infinitesimal-time evolution without vanishing transition effects.

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Background

The current CE quantification in the paper discretizes time and applies GIS-based methods, making the result sensitive to the chosen step size Δt. As Δt approaches zero, transition effects vanish, undermining CE estimation.

The authors identify the need for a principled CE measure specific to continuous-time stochastic differential equations (SDEs) that does not depend on arbitrary discretization.

References

While our approach has made progress, several challenges remain unresolved. The second issue arises when time is continuous rather than discrete, the existing CE quantification lacks objective formulations. The approach in this study discretizes time, converting differential equations into difference equations, where the choice of hyperparameter $\Delta t$ strongly influences CE. As $\Delta t\to 0$, state transitions $x_t\to x_{t+\Delta t}$ exhibit minimal variation, causing the rate of change to vanish. For continuous-time stochastic differential equations, a more principled CE measure is required to account for infinitesimal evolution dynamics.

— SVD-based Causal Emergence for Gaussian Iterative Systems (2502.08261 - Liu et al., 12 Feb 2025) in Discussion and conclusion