Intermittency and Physical Properties of Stochastic Partial Differential Equations

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This project studies physical properties of the solutions to stochastic partial differential equations (SPDEs), which have been a growing topic of research in probability over the last decades. These equations are partial differential equations involving a random component, known as the noise. They mathematically model dynamical systems in a wide spectrum of fields. Physics is the most important of these and was the origin of the use of SPDEs. These equations appear for instance in models for growth at interfaces (the Kardar-Parisi-Zhang equation), models for the movement of galaxies, and polymer models, and also in models of randomly forced propagation of waves, such as the movement of DNA in a fluid. SPDEs also appear in biology, for instance in predator-prey models and models for growth of bacterial populations. They are also a tool of importance in the modeling of interest rates in mathematical finance. This project focuses on study of the property of intermittency for solutions to SPDEs: the fact that the solution develops high-valued peaks (atypical of the average behavior) concentrated on small spatial regions. This phenomenon and its conjectured connection to turbulence and chaos has been very carefully described by physicists, who conjectured many results, only a very few of which are mathematically proved. This research project aims to obtain a more profound understanding of this phenomenon. This field being relatively young, some effort will also be invested into popularizing these ideas among the scientific community, in particular among students and future researchers. More specifically, this project will focus on equations related to the Kardar-Parisi-Zhang (KPZ) equation of physics, which is the central component of the KPZ universality class, a class of probabilistic models appearing in many instances of interface growth phenomena, such as percolation of a liquid in a porous medium, development of a population of bacteria, or the movement of cars in heavy traffic. In order to establish and understand the intermittency of solutions, several new techniques, such as stochastic Young inequalities, moment estimates, and scaling properties have been developed in recent years. These techniques have been successful in specific cases, but more general and more precise results are still sought. Knowledge of the quantitative behavior of the moments of the solution to the SPDE, as well as almost-sure behavior of this solution, are extremely important when it comes to careful study the peaking phenomenon, for instance the position, size, movement, and fractal dimension of the peaks. The project will develop new methods, such as the use of Galton-Watson type processes. The main objective is to carefully understand the impact of the noise on the physical properties of the solution. Thus, the project aims to compare the impact of different types of noise (e.g., white, colored, fractional) on properties of the associated equations.