The Langevin equation describes a particle buffeted by thermal noise and friction. The standard version is Markovian — the future depends only on the present position and velocity, not on the history. Adding memory — friction that depends on the particle's past trajectory — makes the equation non-Markovian. Adding relativity — a velocity-dependent mass that prevents exceeding the speed of light — makes it nonlinear. Combining both memory and relativity produces a non-Markovian, nonlinear stochastic differential equation that resists standard analysis.
The paper shows that when the memory kernel is a sum of exponentials, the non-Markovian system is exactly equivalent to a Markovian system with additional auxiliary variables. Each exponential in the kernel contributes one auxiliary degree of freedom. The reformulation preserves all statistical properties while restoring the Markov property — the price is working in a higher-dimensional state space.
The Markovian reformulation permits rigorous analysis. The system is well-posed, polynomially ergodic, and converges algebraically to its unique Gibbs distribution. In the small-noise limit, it recovers relativistic underdamped Langevin dynamics. In the Newtonian limit (c → ∞), it recovers the classical generalized Langevin equation. The two limits — low noise and non-relativistic — commute.
The structural point: memory and relativity are independent complications, and their combination is not more than the sum of the parts. Memory is eliminated by adding dimensions; relativity is handled by the nonlinear mass-velocity relation; and the two operations don't interfere. The system's complexity is additive, not multiplicative.