A fundamental rule of probability theory just broke for systems that are not linear.
April 29, 2026
Original Paper
Non-uniqueness of nonlinear Markov processes in the sense of McKean associated with parabolic PDEs
arXiv · 2604.25851
The Takeaway
Classical Markov processes are defined by the rule that knowing the current state tells you everything about the future. This paper proves that for nonlinear processes, the distribution of states at a single point in time is not enough to identify the process. Multiple different systems can appear identical at any one moment but behave entirely differently over time. This discovery contradicts a century of assumptions in probability and statistics. It means that monitoring the status of complex, nonlinear systems requires more historical data than previously thought.
From the abstract
We derive a general scheme to construct infinitely many probabilistic counterparts for solutions to nonlinear PDEs by recasting the latter as different nonlinear Fokker--Planck equations and by constructing, for each of these equations, a solution to the associated McKean--Vlasov SDE with one-dimensional time marginal densities given by the PDE solution. We utilize this scheme to prove that nonlinear Markov processes in the sense of McKean as introduced by Rehmeier--Röckner (J.\,Theor.\,Probab.