Quasistatic approximation to time-inhomogeneous stochastic differential equations

Abstract This work studies time averages of an observable $h(t,X_t)$ , where $X_t$ is the solution to a time-inhomogeneous stochastic differential equation (SDE) driven by drift, b ( t , x ), and diffusion, $\sigma(t{,}{\kern.5pt}x)$ , that change sufficiently slowly in time. In this quasistatic regime we derive an approximation to the time average that is computable from properties of the time-homogeneous SDEs driven by $b(t,\cdot)$ and $\sigma(t,\cdot)$ with fixed t ; specifically, we utilize $\log$ -Sobolev inequalities for the instantaneous invariant distribution and generator for each t . We obtain explicit non-asymptotic error bounds on this quasistatic approximation, both in the form of concentration inequalities and bounds on the expected value. The error bounds demonstrate a competition between the speed of convergence to the instantaneous invariant distributions and their rate of change, matching the intuition that underlies the quasistatic approximation.