  We study the statistically invariant structures of the nonlinear generalized
Langevin equation (GLE) with a power-law memory kernel. For a broad class of
memory kernels, including those in the subdiffusive regime, we construct
solutions of the GLE using a Gibbsian framework, which does not rely on
existing Markovian approximations. Moreover, we provide conditions on the decay
of the memory to ensure uniqueness of statistically steady states, generalizing
previous known results for the GLE under particular kernels as a sum of
exponentials.

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