Quasistatic approximation to time-inhomogeneous stochastic differential equations

Abstract This work studies time averages of an observable h (t, Xₜ), where Xₜ is the solution to a time-inhomogeneous stochastic differential equation (SDE) driven by drift, b (t, x), and diffusion, (t,. 5ptx), 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, ) and (t, ) with fixed t ; specifically, we utilize -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.