Analysis of dissipative dynamics on noncommutative spaces and statistical inference of continuous time network stochastic processes

TL;DR

This thesis explores dissipative dynamics on noncommutative spaces, quantum stochastic processes, and classical statistical models for multivariate graph processes, revealing new insights into equilibrium decay, Lie algebra representations, and long-memory graph modeling.

Contribution

It introduces models of dissipative dynamics on quantum spaces, noncommutative Lie algebra representations, and a classical multivariate graph process with consistent estimation methods.

Findings

Models decay to equilibrium algebraically without Poincare inequality

Noncommutative Lie algebra representations via creation and annihilation operators

Consistent and asymptotically normal estimators for multivariate graph processes

Abstract

In this thesis, we analyse the generalisations of the Ornstein-Uhlenbeck (OU) semigroup and study them in both quantum and classical setups. In the first three chapters, we analyse the dissipative dynamics on noncommutative/quantum spaces, in particular, the systems with multiparticle interactions associated to CCR algebras. We provide various models where the dissipative dynamics are constructed using noncommutative Dirichlet forms. Some of our models decay to equilibrium algebraically and the Poincare inequality does not hold. Using the classical representation of generators of nilpotent Lie algebras, we provide the noncommutative representations of Lie algebras in terms of creation and annihilation operators and discuss the construction of corresponding Dirichlet forms. This introduces the opportunity to explore quantum stochastic processes related to Lie algebras and nilpotent Lie algebras. Additionally, these representations enable the investigation of the noncommutative analogue of hypoellipticity. In another direction, we explore the potential for introducing statistical models within a quantum framework. In this thesis, however, we present a classical statistical model of multivariate Graph superposition of OU (Gr supOU) process which allows for long(er) memory in the modelling of sparse graphs. We estimate these processes using generalised method of moments and show that it yields consistent estimators. We demonstrate the asymptotic normality of the moment estimators and validate these estimators through a simulation study.