Nonlinear Perron–Frobenius conjecture for statistical Hamiltonians

Let $H_{\boldsymbol{\gamma}}({\boldsymbol{p}},{\boldsymbol{z}})$ be a biased Hamiltonian describing a nonlinear Markov process, and consider its stationary Hamilton–Jacobi equation at energy $E$. Assume that the absolute maximum of $H_{\min}({\boldsymbol{z}})$ is non-degenerate, with $H_0^\star(\boldsymbol{\gamma})>H_1^\star(\boldsymbol{\gamma})$, and write

$$H_0^\star(\boldsymbol{\gamma})=\max_{{\boldsymbol{z}}}\min_{{\boldsymbol{p}}}H_{\boldsymbol{\gamma}}({\boldsymbol{p}},{\boldsymbol{z}}).$$

Nonlinear Perron–Frobenius conjecture. There exists a value $E^\star(\boldsymbol{\gamma})$ such that: for $E>E^\star(\boldsymbol{\gamma})$, all orbits tend towards the system boundaries in forward and backward time and contain no critical manifold; for $E<E^\star(\boldsymbol{\gamma})$, the Hamilton–Jacobi equation has no global solution and the reduced action along every bounded orbit is non-negative; and for $E=E^\star(\boldsymbol{\gamma})$, the equation has at least two global solutions up to an additive constant, exactly one globally stable solution $W_{\mathrm{s}}({\boldsymbol{z}},\boldsymbol{\gamma})$ and exactly one globally unstable solution $W_{\mathrm{u}}({\boldsymbol{z}},\boldsymbol{\gamma})$, which coincide on each critical manifold. The dominant fixed point $({\boldsymbol{p}}_0^\star,{\boldsymbol{z}}_0^\star)$ lies on both solutions, and

$$E^\star(\boldsymbol{\gamma})=\max_{{\boldsymbol{z}}}\min_{{\boldsymbol{p}}}H_{\boldsymbol{\gamma}}({\boldsymbol{p}},{\boldsymbol{z}})=H_0^\star(\boldsymbol{\gamma}).$$

This is proposed as a nonlinear analogue of the Perron–Frobenius theorem, replacing positivity and uniqueness properties of dominant eigenvectors by global stable and unstable Hamilton–Jacobi solutions. The conjecture is used to identify the dominant long-time trajectories and remains unproved in the stated generality.