| Abstract: |
| The Generalized Langevin Equation is commonly used to describe the velocity of microparticles in viscoelastic fluids.
Formally, the Generalized Langevin Equation (GLE) is written
\\begin{align*}
m \\ddot{x}(t)&=-\\gamma \\dot{x}(t)-\\Phi'(x(t))-\\int_{-\\infty}^t \\!\\!\\!\\! K(t-s)\\dot{x}(s)ds +F(t)+\\sqrt{2\\gamma}\\dot{W}(t),
\\end{align*}
where $\\Phi(x)$ is a non-linear potential well, $W(t)$ is a Brownian motion, and $F(t)$ is a stationary, mean zero and Gaussian process satisfying $E(F(t)F(s))=K(t-s)$. Describing the long-term behavior of sub-diffusive GLEs in non-linear potentials is a long-standing open problem. We will look at recent advances in establishing existence and uniqueness of a stationary distribution for an infinite-dimensional Markov representation of the GLE. |
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