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We study the statistically invariant structures of the nonlinear generalizedLangevin equation (GLE) with a power-law memory kernel. For a broad class ofmemory kernels, including those in the subdiffusive regime, we constructsolutions of the GLE using a Gibbsian framework, which does not rely onexisting Markovian approximations. Moreover, we provide conditions on the decayof the memory to ensure uniqueness of statistically steady states, generalizingprevious known results for the GLE under particular kernels as a sum ofexponentials.

Gibbsian dynamics and the generalized Langevin equation