Product structure of heat phase space and branching Brownian motion

TL;DR

This paper introduces a novel formalism for analyzing Brownian processes with variable particle numbers by leveraging the product structure of heat phase space, connecting it to quantum field theory methods.

Contribution

It develops a ring-based formalism for Brownian processes, enabling exact solutions for branching Brownian motion using quantum field theory techniques.

Findings

Exact Dyson-Schwinger equations for binary branching Brownian motion

Ring-valued quantum field theory framework for Brownian processes

Application of quantum field machinery to non-differentiable Brownian paths

Abstract

A generical formalism for the discussion of Brownian processes with non-constant particle number is developed, based on the observation that the phase space of heat possesses a product structure that can be encoded in a commutative unit ring. A single Brownian particle is discussed in a Hilbert module theory, with the underlying ring structure seen to be intimately linked to the non-differentiability of Brownian paths. Multi-particle systems with interactions are explicitly constructed using a Fock space approach. The resulting ring-valued quantum field theory is applied to binary branching Brownian motion, whose Dyson-Schwinger equations can be exactly solved. The presented formalism permits the application of the full machinery of quantum field theory to Brownian processes.