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Shenglan Yuan

Publications and source records attributed to Shenglan Yuan.

26 records · Page 2Linked to original sources
arXiv2021

Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical α-stable Lévy processes

In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical α-stable Lévy processes via modulation or amplitude equations. We study SPDEs with a cubic nonlinearity, where the deterministic equation is close to a change of stability of the trivial solution. The natural separation of time-scales close to this bifurcation allows us to obtain an amplitude equation describing the essential dynamics of the bifurcating pattern, thus reducing the original infinite dimensional dynamics to a simpler finite-dimensional effective dynamics. In the presence of a multiplicative stable Lévy noise that preserves the constant trivial solution we study the impact of noise on the approximation. In contrast to Gaussian noise, where non-dominant pattern are uniformly small in time due to averaging effects, large jumps in the Lévy noise might lead to large error terms, and thus new estimates are needed to take this into account.

math.DS↗
arXiv2021

Stochastic turbulence for Burgers equation driven by cylindrical Lévy process

This work is devoted to investigating stochastic turbulence for the fluid flow in one-dimensional viscous Burgers equation perturbed by Lévy space-time white noise with the periodic boundary condition. We rigorously discuss the regularity of solutions and their statistical quantities in this stochastic dynamical system. The quantities are such as the moment estimate, the structure function and the energy spectrum of the turbulent velocity field. Furthermore, we provide qualitative and quantitative properties of the stochastic Burgers equation when the kinematic viscosity ν tends towards zero. The inviscid limit describes the strong stochastic turbulence.

math.PR↗
arXiv2021

Stochastic Bifurcation in Single-Species Model Induced by α-Stable Levy Noise

Bifurcation analysis has many applications in different scientific fields, such as electronics, biology, ecology, and economics. In population biology, deterministic methods of bifurcation are commonly used. In contrast, stochastic bifurcation techniques are infrequently employed. Here we establish stochastic P-bifurcation behavior of (i) a growth model with state-dependent birth rate and constant death rate, and (ii) a logistic growth model with state-dependent carrying capacity, both of which are driven by multiplicative symmetric stable Levy noise. Transcritical bifurcation occurs in the deterministic counterpart of the first model, while saddle-node bifurcation takes place in the logistic growth model. We focus on the impact of the variations of the growth rate, the per capita daily adult mortality rate, the stability index, and the noise intensity on the stationary probability density functions of the associated non-local Fokker-Planck equation. In the first model, the bifurcation parameter is the ratio of the population birth rate to the population death rate. In the second model, the bifurcation parameter corresponds to the sensitivity of carrying capacity to change in the size of the population near equilibrium. In each case, we show that as the value of the bifurcation parameter increases, the shape of the steady-state probability density function changes and that both stochastic models exhibit stochastic P-bifurcation. The unimodal density functions become more peaked around deterministic equilibrium points as the stability index increases. While an increase in any one of the other parameters has an effect on the stationary probability density function. That means the geometry of the density function changes from unimodal to flat, and its peak appears in the middle of the domain, which means a transition occurs.

math.DS↗
arXiv2019

Action functionals for stochastic differential equations with Lévy noise

By using large deviation theory that deals with the decay of probabilities of rare events on an exponential scale, we study the longtime behaviors and establish action functionals for scaled Brownian motion and Lévy processes with existing finite exponential moments. Based on extended contraction principle, Legendre transform and Lévy symbols, we derive the action functionals for stochastic differential equations driven by Lévy processes.

math.DS↗
arXiv2019

Invariant foliations for stochastic dynamical systems with multiplicative stable Levy noise

This work deals with the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Levy noise. We first establish the existence of stable and unstable foliations for this system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Finally, we illustrate our results in an example.

math.DS↗
arXiv2019

Characterization of the Most Probable Transition Paths of Stochastic Dynamical Systems with Stable Lévy Noise

This work is devoted to the investigation of the most probable transition path for stochastic dynamical systems driven by either symmetric $α$-stable Lévy motion ($0<α<1$) or Brownian motion. For stochastic dynamical systems with Brownian motion, minimizing an action functional is a general method to determine the most probable transition path. We have developed a method based on path integrals to obtain the most probable transition path of stochastic dynamical systems with symmetric $α$-stable Lévy motion or Brownian motion, and the most probable path can be characterized by a deterministic dynamical system.

math.DS↗
arXiv2017

Slow manifolds for stochastic systems with non-Gaussian stable Lévy noise

This work is concerned with the dynamics of a class of slow-fast stochastic dynamical systems with non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, eliminating the fast variables to reduce the dimension of these coupled dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps understand long time dynamics. The approximation of slow manifolds with error estimate in distribution are also considered.

math.DS↗