Bootstrapping Nonequilibrium Stochastic Processes

TL;DR

This paper introduces a bootstrap framework using positivity of probability measures to derive rigorous bounds on expectations in nonequilibrium stochastic processes on infinite lattices, applicable to both invariant measures and time evolution.

Contribution

It develops linear and semidefinite programming methods based on positivity and invariance to analyze nonequilibrium processes, providing new bounds on critical rates and temporal properties.

Findings

Lower bounds on critical infection rates.

Two-sided bounds on infection half-life.

Bounds on temporal correlation length.

Abstract

We show that bootstrap methods based on the positivity of probability measures provide a systematic framework for studying both synchronous and asynchronous nonequilibrium stochastic processes on infinite lattices. First, we formulate linear programming problems that use positivity and invariance property of invariant measures to derive rigorous bounds on their expectation values. Second, for time evolution in asynchronous processes, we exploit the master equation along with positivity and initial conditions to construct linear and semidefinite programming problems that yield bounds on expectation values at both short and late times. We illustrate both approaches using two canonical examples: the contact process in 1+1 and 2+1 dimensions, and the Domany-Kinzel model in both synchronous and asynchronous forms in 1+1 dimensions. Our bounds on invariant measures yield rigorous lower bounds on critical rates, while those on time evolutions provide two-sided bounds on the half-life of the infection density and the temporal correlation length in the subcritical phase.