Gibbsian dynamics and the generalized Langevin equation

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