Probabilistic analog of Banach fixed-point theorem

Question

Are there probabilistic analogs of Banach fixed-point theorem? For example, is there a notion of probabilistic contractive mappings that gives rise to a probabilistic fixed-point theorem? Sorry for the open-endedness of this question; this is more of a shower thought question than one with a particular context.

Answer 1

Perhaps the following is the style you're looking for.

Consider an irreducible, discrete-time Markov chain on a metric space $(\Omega, d)$, with transition kernel $P$. Then, the curvature $\kappa$ of the Markov chain is defined by $$ \kappa := \min_{x,y \in \Omega} \bigl( 1 - W(P_{x,\cdot}, P_{y,\cdot}) / d(x,y) \bigr), $$ where $P_{z,\cdot}$ is the law of the chain after one step started from $z \in \Omega$ and $W(\mu, \nu)$ is the Wasserstein/transportation distance between meaures $\mu$ and $\nu$ on $\Omega$.

If $\kappa > 0$, then there is contraction at rate $\kappa$: there is a single-step coupling such that $$ \mathbb E\bigl( d(X_t, Y_t) \mid (X_{t-1}, Y_{t-1}) \bigr) \le (1 - \kappa) d(X_{t-1}, Y_{t-1}). $$ Iterating this, $$ \mathbb E_{x,y}\bigl( d(X_t, Y_t) \bigr) \le (1 - \kappa)^t d(x,y) \le (1 - \kappa)^t \operatorname{diam}(\Omega). $$ In particular, there is a unique invariant distribution $\pi$. Taking $Y_0 \sim \pi$, this shows exponential convergence to equilibrium.

In this way, you can think of $P$ as a contactive mapping, converging to the unique solution to $\mu P = \mu$.

I think that $\kappa \ge 0$ should also be sufficient in many scenarios. The idea is that you define $D_t := d(X_t, Y_t)$, then $(D_t)_{t\ge0}$ is a non-negative super-martingale, absorbed at $0$. If there are enough fluctuations, then this should converge to $0$.

Curvature for Markov chains was introduced by Ollivier and Joulin in the last '00s. It is highly related to the path-coupling technique introduced by Bubley and Dyer in the last '90s. Coupling can still occur when there is no contraction (ie, $\kappa = 0$), under certain conditions.