This thesis represents a small contribution to our understanding of metastable patterns in various stochastic models from physics and biology.
By a \emph{metastable pattern}, we mean a pattern that appears to persist in a regular fashion on some timescale, but disappears or undergoes an irregular change on a longer timescale.
Metastable patterns frequently result from stochastic perturbations of patterns that are stable without perturbation.
In this thesis, we study stochastic perturbations of stable spatiotemporal patterns in several classes of PDE and integral equations.
In particular, we address two major questions:
\begin{enumerate}[Q1.]
\item When perturbed by noise, for how long does a pattern that is stable without noise persist?
\item How does the stochastic perturbation affect the average behaviour of a pattern on the timescale where it appears to persist?
\end{enumerate}
To address these questions, we pursue two lines of inquiry: the first based on the theory of \emph{quasi-ergodic measures}, and the second based on \emph{phase decomposition techniques}.
In our first line of inquiry we present novel, rigorous connections between metastability of general infinite dimensional stochastic evolution systems and the spectral properties of their sub-Markov generators using the theory of quasi-ergodic measures.
To do so, we develop a novel $L^p$-approach to the study of quasi-ergodic measures.
We are then able to draw conclusions about the metastability of travelling waves and other patterns in a class of stochastic reaction-diffusion equations.
For instance, we obtain a rigorous definition of the \emph{quasi-asymptotic speed}~of a travelling wave in a stochastic PDE.
We moreover find that stochastic perturbations of amplitude $\sigma>0$ cause the quasi-asymptotic speed of certain travelling waves to deviate from the deterministic wave speed by a constant that is approximately proportional to $\sigma^2$.
In our second line of inquiry, the dynamics of our (infinite dimensional) stochastic evolution system are projected onto a finite dimensional manifold that captures some property of a metastable pattern.
While most previous studies using phase reduction techniques have used the \emph{variational phase}, we take an approach based on the \emph{isochronal phase}, inspired by classical work on finite dimensional oscillatory systems.
When the pattern in question is a travelling wave, the isochronal phase captures the position of the wave at a given point in time.
By exploiting the regularity properties of the isochronal phase, we are able to prove several novel results about the metastable behaviour of the reduced dynamics in the small noise regime in a very large class of stochastic evolution systems.
These results allow us to moreover compute the noise-induced changes in the speed of stochastically perturbed travelling waves and other patterns.
The results we obtain using this approach are numerically precise, and may be applied to a very general class of stochastic evolution systems.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:88675 |
| Date | 15 December 2023 |
| Creators | Adams, Zachary P. |
| Contributors | Max Planck Institute für Mathematik in den Naturwissenschaft |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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