Rational approximations for eventually positive semigroups

Develop a rigorous, generalized theoretical framework linking the degree of a rational approximant to the eventual-positivity time threshold for rational approximations of eventually positive semigroups and the associated discrete generators.

Background

The paper studies rational maps applied to eventually exponentially positive generators arising from finite-difference discretizations of anisotropic Fokker–Planck equations. Although positivity-preserving rational approximations for standard positive semigroups have been extensively studied, the authors identify a sparse theory for eventually positive semigroups.

The unresolved issue is to translate resolvent characterizations of eventual positivity into general stability and positivity criteria for discrete rational functions, including an explicit understanding of how the rational approximant’s degree affects the threshold beyond which positivity is recovered.

References

The primary mathematical gap lies in translating the theoretical resolvent conditions from into stability criteria for discrete rational functions. Current applied research, particularly in high-order finite-difference schemes for anisotropic Fokker-Planck equations and related quantitative models, frequently encounters matrices that are eventually positive. In these contexts, specific rational approximants (like Backward Euler) are employed empirically to stabilize the asymptotic positivity, but a rigorous, generalized theoretical framework linking the degree of the rational approximant to the eventual positivity time threshold remains an open problem.

— Diagonal Frog meets ADI: trading matrix exponentials for rational maps in the Fokker--Planck equation  (2608.22703 - Itkin et al., 24 Aug 2026) in Section 1, subsection “Synthesis and Literature Gap”

Several directions remain open. The r(\infty) criterion is stated here for the class of EM generators produced by the DF discretization. Extending it to broader classes of eventually positive or eventually nonnegative generators, including those arising from non-conservative boundary closures or non-divergence-form discretizations, would widen its applicability.

— Diagonal Frog meets ADI: trading matrix exponentials for rational maps in the Fokker--Planck equation  (2608.22703 - Itkin et al., 24 Aug 2026) in Section 7, “Discussion and Conclusions”

The relationship between the order of the Pad\'e approximant and the sharpness of the positivity threshold is not yet understood quantitatively, and the a priori bounds we provide for \gamma_r are conservative. Tighter spectral estimates would make the criterion a practical design tool at run time.

— Diagonal Frog meets ADI: trading matrix exponentials for rational maps in the Fokker--Planck equation  (2608.22703 - Itkin et al., 24 Aug 2026) in Section 7, “Discussion and Conclusions”

We have no proof that the window is empty for every generator in the class of \cref{sec:resolvent}. The emptiness is an empirical finding, stated here as a conjecture supported by the thresholds of \cref{ouThresh} and the sweeps of \cref{sec:numerics}.

— Diagonal Frog meets ADI: trading matrix exponentials for rational maps in the Fokker--Planck equation  (2608.22703 - Itkin et al., 24 Aug 2026) in Remark 2.4, subsection “The trapezoidal (Cayley) map”