Criterion ensuring integral separation for all parameter values

Establish conditions on the continuous matrix function A(z,s) that guarantee, for every parameter z, that the random dynamical system generated by the linear differential equation dy/dt = A(z,\omega_t)y is integrally separated.

Background

The numerical approximation theorem for the derivative of the leading Lyapunov exponent relies essentially on an integral-separation assumption for the random dynamical system generated by the parameter-dependent linear differential equation. Although the set of integrally separated systems is known to be open and dense in the space of bounded linear random differential equations under suitable conditions, the paper does not provide a directly verifiable condition on the coefficient function A(z,s) ensuring that the system is integrally separated for every parameter value z.

Resolving this problem would make the numerical derivative formula applicable under assumptions stated directly in terms of the model’s coefficient function, rather than requiring integral separation as an external hypothesis.

References

Even if \citet{CongSon2016} established that the set $\Rrond$ of integrally separated systems is open and dense in the set of bounded linear random differential equations, when the dynamical system $(\Omega,\Frond,\PP, \theta_t)$ is not of circle type (see Remark 1, section 3 of \citet{CongSon2016}), we are unable to provide an assumption on the matrix function $A(z,s)$ that ensures that for all $z$, $M(\omega)=A(z,\omega_0) $ is in $\Rrond$.

— Differentiability of the Leading Lyapunov Exponent of a linear differential equation with random coefficients Application to the Calculation of the Selection Gradient in Random Environments  (2609.02711 - Carmona, 2 Sep 2026) in Section 1, Introduction