Characterize functions satisfying Li's strong-completeness criterion

Determine for which functions \(\rho:[0,\infty)\to\mathbb{R}\) Li's general strong-completeness condition based on moments of the Jacobian of the stochastic flow is satisfied for the stochastic differential equation \(dX(t)=-X(t)\,dt+\rho(|X(t)|)\begin{pmatrix}0&1\\-1&0\end{pmatrix}X(t)\,dt+dW(t)\).

Background

The paper studies strong completeness of the stochastic flow generated by a planar stochastic differential equation with linear dissipation, additive two-dimensional Brownian noise, and a radius-dependent rotational drift $\rho(|X|)$. Strong completeness requires almost-sure global existence simultaneously for all initial conditions and is substantially stronger than weak completeness, which holds for the considered equation for every admissible function $\rho$.

Li’s cited criterion provides a general sufficient condition for strong completeness in terms of moments of the stochastic-flow Jacobian. The paper explicitly leaves unresolved the task of identifying the functions $\rho$ for which that criterion applies. This question is distinct from the paper’s own sharp strong-completeness results under growth conditions on $\rho$ or $\rho'$, because it concerns the applicability of Li’s broader Jacobian-moment criterion.