Week 09.10.2023 – 15.10.2023

The Random Walk Metropolis (RWM) is a simple and enduring Markov chain-based algorithm for approximate simulation from an intractable ‘target’ probability distribution. In this work, we study quantitatively the convergence of this algorithm, providing non-asymptotic estimates on mixing times, with explicit dependence on dimension and other problem parameters. The results hold at a reasonable level of generality, and are often sharp in a suitable sense.

The focus of the talk will be conceptual rather than technical, with an eye towards enabling intuition for i) which high-level aspects of the target distribution influence the convergence behaviour of RWM, and ii) which concrete properties must be verified in order to obtain a rigorous proof. No prior knowledge of the RWM is required from the audience.

We study solutions of the irregular Stratonovich SDE $dX = X|^\alpha \circ dB$, $\alpha\in (0, 1)$. In particular we construct solutions spending positive time in 0, describe solutions spending zero time in 0, and show how a particular physically natural solution can be singled out by means of an additional external "ambient" noise.

This talk is based on the joint works with G. Shevchenko (Kiev).