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A Langevin equation with a friction memory kernel, rather than only instantaneous friction. The equation couples the velocity to its earlier history through a causal convolution. Equilibrium noise and friction are related by a generalized fluctuation-dissipation relation. The name extends the Langevin equation associated with Paul Langevin.

Paul Langevin described Brownian motion through a force balance containing drag and rapidly fluctuating molecular forces. In modern notation, the Langevin equation $m\dot{v}=-\gamma v+\xi (t)$ combines damping with a random force. At thermal equilibrium, the relation between fluctuations and drag gives the long-time diffusion coefficient $D=k_{B}T/\gamma$, where $k_B$ is the Boltzmann constant, $T$ is temperature, and $\gamma$ is the linear drag coefficient. His 1908 note recovers the displacement law developed by Albert Einstein through this mechanical route, connecting irregular microscopic forces with a predictable statistical diffusion law.^[ref]

By Paul Langevin. Comptes rendus 146, 530–533.

Original French note read: session 9 March 1908, Einstein displacement formula and force-based route; distinguish later stochastic-calculus notation.