A variational principle for stationary distributions of driven Markov processes
In plain words
In equilibrium, the Boltzmann distribution gives the probability of every state from its energy alone. For small driven systems that hop between states, no equivalent rule is known, and proposed rules such as minimum entropy production work only for weak driving.
Precise statement
For finite-state continuous-time Markov jump processes whose rates obey local detailed balance, $k(x\to y)/k(y\to x) = \operatorname{exp}(s(x,y))$ with s the entropy flux to the environment, determine whether a variational functional built from entropy production and dynamical activity (frenesy, the time-symmetric part of the path weight) has an extremum that gives the stationary distribution to all orders in the driving, as minimum entropy production does to linear order. The Landauer blowtorch example shows stationary occupations depend on kinetic details beyond the entropy flux. Answer: yes with the functional and its domain, or a no-go theorem.
What would settle it
A theorem identifying such a functional for a stated class of jump processes, or a no-go theorem with explicit counterexamples.