Browsing by Author "Mattingly, Jonathan C"

Noise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.

The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.

We explore the interplay among probability, stochastic analysis, and dynamical systems through two lenses: (1) absolute continuity of singular stochastic partial differential equations (SPDEs); (2) Bayesian inference on dynamical systems.

In the first part, we prove that up to a certain singular regime, the law of the stochastic Burgers equation at a fixed time is absolutely continuous with respect to the corresponding stochastic heat equation with the nonlinearity removed. The results follow from an extension of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time. To deal with the singularity, we introduce a novel decomposition in the spirit of Da Prato-Debussche and Gaussian chaos decomposition in singular SPDEs, by separating out the noise into different levels of regularity, along with a number of renormalization techniques. The number of levels in this decomposition diverges to infinite as we move to the stochastic Burgers equation associated with the KPZ equation. This result illustrates the fundamental probabilistic structure of a class of singular SPDEs and a notion of ``ellipticity'' in the infinite-dimensional setting.

In the second part, we establish connections between large deviations and a class of generalized Bayesian inference procedures on dynamical systems. We show that posterior consistency can be derived using a combination of classical large deviation techniques, such as Varadhan's lemma, conditional/quenched large deviations, annealed large deviations, and exponential approximations. We identified the divergence term as the Donsker-Varadhan relative entropy rate, also related to the Kolmogorov-Sinai entropy in ergodic theory. As an application, we prove new quenched/annealed large deviation asymptotics and a new Bayesian posterior consistency result for a class of mixing stochastic processes. In the case of Markov processes, one can obtain explicit conditions for posterior consistency, when estimates for log-Sobolev constants are available, which makes our framework essentially a black box. We also recover state-of-the-art posterior consistency on classical dynamical systems with a simple proof. Our approach has the potential of proving posterior consistency for a wide range of Bayesian procedures in a unified way.

Machine learning is a powerful branch of mathematics and statistics that allows the automation of tasks that would otherwise require humans a long time to perform. Two particular fields of machine learning that have been developing in the last two decades are dimensionality reduction and semi-supervised learning.

Dimensionality reduction is a powerful tool in the analysis of high dimensional data by reducing the number of variables under consideration while approximately preserving some quantity of interest (usually pairwise distances). Methods such as Principal Component Analysis (PCA) or Isometric Feature Mapping (ISOMAP) do this do this by embedding the data, equipped with a nonnegative, symmetric, similarity kernel or adjacency matrix into Euclidean space and finding a linear subspace or low dimensional submanifold which best fits the data, respectively.

When the data takes the form of network data, how to perform such dimensionality reduction intrinsically, without resorting to an embedding, that can be extended to the case of nonnegative, non-symmetric adjacency matrices remains an important open problem. In the first part of my dissertation, using current techniques in local spectral clustering to partition the network using a Markov process induced by the adjacency matrix, we deliver an intrinsic dimensionality reduction of the network in terms of a non-Markov process on a reduced state space that approximately preserves transitions of the original Markov process between clusters. By iterating the process, one obtains a family of non-Markov processes on successively finer state spaces representing the original network ands its diffusion at different scales, which can be used to approximate the law of the original process at a particular time scale. We give applications of this theory to a variety of synthetic data sets and evaluate its performance accordingly.

Next, consider the case of detecting astronomical phenomenon solely in terms of the light intensities observed. There already exists a large database of prior recorded phenomena that has been categorized by humans as a function of the observed light intensity. Given these so-called class labels then, how can we automate the procedure of extending these class labels to the massive amount of data that is currently being observed? This is the problem of concern in semi-supervised learning.

In the second part of my thesis, we consider data sets in which relations between data points are more complex than simply pairwise. Examples include gene networks where the the data points are random variables, and similarities of a subset are measured by non-independence of the corresponding random variables. Such data sets can be illustrated as a hypergraph, and the natural question for diagnosis becomes: how does one perform transductive inference (a particular form of semi-supervised learning)? Using the simple case of pairwise and threewise similarities, we construct a reversible random walk on undirected edges induced by threewise relations (faces). By pulling the random walk back to a random walk on the vertex set and mixing it with the random walk induced by pairwise similarities, we perform diffusive transductive inference. We present applications and results of this technique, any analyze its performance on a variety of data sets.

Geospatial statistical approaches must frequently confront the problem of correctlypartitioning a group of geographical sub-units, such as counties, states, or precincts,into larger blocks which share information. Since the space of potential partitions isquite large, sophisticated approaches are required, particularly when this partitioninginteracts with other parts of a larger model, as is frequent with Bayesian inference.Authors such as Balocchi et al. (2021) provide stochastic search algorithms whichprovide certain theoretical guarantees about this partition in the context of Bayesianmodel averaging. We borrow tools from Herschlag et al. (2020) to examine a potentialapproach to sampling these clusters efficiently using a Markov Chain Monte Carlo(MCMC) approach.

The connection between elliptic stochastic diffusion processes and partial differential equations is rich and well understood. This connection is not very well understood when the stochastic differential equation takes values in an infinite dimensional space such as a function space. In this case, the diffusion is a stochastic partial differential equation (SPDE) and the notion of ellipticity is ambiguous. We establish a sufficient condition on the diffusion coefficient of a class of nonlinear SPDEs, which is analogous to the nondegeneracy condition in finite dimensions, that allows for the existence of a Markov transition density that is absolutely continuous with respect to an infinite dimensional Gaussian measure.

In the second part of this work, we consider a two-station queueing network that processes K job types. The first station in this network is a polling station, and we assume that the second station is operating under any nonidling service discipline. We consider diffusion-scaled versions of many of the processes governing this system, and we show that the scaled two-dimensional total workload process converges to Brownian motion in a wedge. We also show that the scaled immediate workload process for station 2 does not converge, but admits an averaging principle.

The solution to the Navier Stokes equations on the 2D torus with stochastic forcing that is white noise in time, coloured in space has a Fourier series representation whose coordinates satisfy a countable system of Stochastic Differential Equations. Inspired by the structure of these equations, we construct a finite system of stochastic differential equations with a similar structure and explore the conditions under which the system has an invariant distribution. \\

Our main tool to prove existence of invariant distributions are Lyapunov functions, or more generally Lyapunov pairs. In particular, we construct the Lyapunov pairs piecewise over different regions and then use mollifiers to unify these disparate characterisations. We also apply some results from Algebraic Geometry and Matrix Perturbation Theory to study and exploit the geometry of the problem in high dimensions. \\

The combination of these methods allowed us to prove that a large class of the equations we constructed have an invariant distributions. Furthermore we have explicit tail estimates for these invariant distributions.