arXiv subjects

Nhu N. Nguyen

Publications and source records attributed to Nhu N. Nguyen.

6 recordsLinked to original sources

arXiv2025

Analysis of Reaction-Diffusion Predator-Prey System under Random Switching

This paper investigates the long-term dynamics of a reaction-diffusion predator-prey system subject to random environmental fluctuations modeled by Markovian switching. The model is formulated as a hybrid system of partial differential equations (PDEs), where the switching between different ecological regimes captures the randomness in environmental conditions. We derive a critical threshold parameter that determines whether the predator species will eventually go extinct or persist. We further characterize the system's asymptotic behavior by providing a detailed pathwise description of the omega-limit set of solutions. This analysis reveals how the effects of random switching shape the distribution and long-term coexistence of the species. Numerical simulations are provided to validate and illustrate the theoretical findings, highlighting transitions between different dynamical regimes. To the best of our knowledge, this is the first work that rigorously analyzes a spatially diffusive predator-prey model under Markovian switching, thereby bridging the gap between spatial ecology and stochastic hybrid PDE systems.

math.AP↗

arXiv2024

Second-Order Fast-Slow Stochastic Systems

This paper focuses on systems of nonlinear second-order stochastic differential equations with multi-scales. The motivation for our study stems from mathematical physics and statistical mechanics, for examples, Langevin dynamics and stochastic acceleration in a random environment. Our effort is to carry out asymptotic analysis to establish large deviations principles. Our focus is on obtaining the desired results for systems under weaker conditions. When the fast-varying process is a diffusion, neither Lipschitz continuity nor linear growth needs to be assumed. Our approach is based on combinations of the intuition from Smoluchowski-Kramers approximation, and the methods initiated in [34] relying on the concepts of relatively large deviations compactness and the identification of rate functions. When the fast-varying process is under a general setup with no specified structure, the paper establishes the large deviations principle of the underlying system under the assumption on the local large deviations principles of the corresponding first-order system.

math.PR↗

arXiv2021

Exponential tightness of a family of Skorohod integrals

Exponential tightness of a family of Skorohod integrals is studied in this paper. We first provide a counterexample to illustrate that in general the exponential tightness with speed $ε$ similar to Itô integral does not hold, even for any speed $ε^α$ with $α> 0$. Then, some characterizations of this subject are given. Application is also provided to illustrate the proposed results.

math.PR↗

arXiv2021

Stochastic Functional Kolmogorov Equations II: Extinction

This work, Part II, together with its companion Part I develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the problems, it is natural to divide our contributions into two parts to answer a long-standing question in biology and ecology. What are the minimal conditions for long-term persistence and extinction of a population? Part I of our work provides characterization of persistence, whereas in this part, extinction is the main focus. The main techniques used in this paper are combination of the newly developed functional It^o formula and a dynamical system approach. Compared to the study of stochastic Kolmogorov systems without delays, the main difficulty is that infinite dimensional systems have to be treated. The extinction is characterized after investigating random occupation measures and examining behavior of functional systems around boundaries. Our characterizations of long-term behavior of the systems reduces to that of Kolmogorov systems without delay when there is no past dependence. A number of applications are also examined.

math.PR↗

arXiv2021

Stochastic Functional Kolmogorov Equations (I): Persistence

This work (Part (I)) together with its companion (Part (II) [45]) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the results, it seems to be instructive to divide our contributions to two parts. In contrast to the existing literature, our effort is to advance the knowledge by allowing delay and past dependence, yielding essential utility to a wide range of applications. A long-standing question of fundamental importance pertaining to biology and ecology is: What are the minimal necessary and sufficient conditions for long-term persistence and extinction (or for long-term coexistence of interacting species) of a population? Regardless of the particular applications encountered, persistence and extinction are properties shared by Kolmogorov systems. While there are many excellent treaties of stochastic-differential-equation-based Kolmogorov equations, the work on stochastic Kolmogorov equations with past dependence is still scarce. Our aim here is to answer the aforementioned basic question. This work, Part (I), is devoted to characterization of persistence, whereas its companion, Part (II) [45], is devoted to extinction. The main techniques used in this paper include the newly developed functional It^o formula and asymptotic coupling and Harris-like theory for infinite dimensional systems specialized to functional equations. General theorems for stochastic functional Kolmogorov equations are developed first. Then a number of applications are examined to obtain new results substantially covering, improving, and extending the existing literature. Furthermore, these conditions reduce to that of Kolmogorov systems when there is no past dependence.

math.PR↗

arXiv2021

Large Deviations Principles for Langevin Equations in Random Environment and Applications

In contrast to the study of Langevin equations in a homogeneous environment in the literature, the study on Langevin equations in randomly-varying environments is relatively scarce. Almost all the existing works require random environments to have a specific formulation that is independent of the systems. This paper aims to consider large deviations principles (LDPs) of Langevin equations involving a random environment that is a process taking value in a measurable space and that is allowed to interact with the systems, without specified formulation on the random environment. Examples and applications to statistical physics are provided. Our formulation of the random environment presents the main challenges and requires new approaches. Our approach stems from the intuition of the Smoluchowski-Kramers approximation. The techniques developed in this paper focus on the relation between the solutions of the second-order equations and the associate first-order equations. We obtain the desired LDPs by showing a family of processes enjoy the exponential tightness and local LDPs with an appropriate rate function.

math.PR↗