Some properties of a class of stochastic heat and wave equations with multiplicative Gaussian noises

We study stochastic partial differential equations of the form Lu = ∆u+σ(u)N˙ with particular initial conditions. Here L denotes a first or second order partial differential operator, N˙ a Gaussian noise, ∆ the Laplacian operator, and σ : R → R a Lipschitz continuous function. In the first case we choose Lu := ∂tu, we study the stochastic fractional heat equation of the form ∂tu(t, x) = ∆ α/2u(t, x) +λ σ(u(t, x))N˙ for t > 0, x ∈ B(0, R) ⊂ R d . Here ∆ α/2 is the infinitesimal generator of a symmetric α-stable process killed at the boundary of the ball B(0, R), λ is a positive parameter called the level noise and N˙ denotes space-time white noise when d = 1 or white-colored noise in the case of Riesz Kernel of order β when d ≥ 1. We start to show the existence and uniqueness of solutions, the main task is to study how the second moment of the solution u(t, x) and excitation index of the solution grows as λ tends to infinity for a fixed t > 0. This study was initiated by [KK13] and [KK15]. Our results are significant extensions of those results and that of [FJ14]. In the second case we choose Lu := ∂ttu+2η∂tu, and study the stochastic damped wave equation of the form ∂ttu(t, x)+2η∂tu(t, x) = ∆u(t, x)+σ(u(t, x))W˙ for t > 0, x ∈ R, where η represents the positive damping parameter and W˙ space-time white noise. We study second moment and p-th moment of solution to show the intermittency properties, and existence and uniqueness of solutions will be proved. The study of the intermittency properties in stochastic partial differential equations was initiated by [FK09], our results are significant extension of results in [CJKS13]. In the end, we also show the result of excitation index for the stochastic damped wave equation

On some properties of a class of fractional stochastic heat equations

We consider nonlinear parabolic stochastic equations of the form ∂tu=Lu+λσ(u)ξ˙∂tu=Lu+λσ(u)ξ˙ on the ball B(0,R)B(0,R) , where ξ˙ξ˙ denotes some Gaussian noise and σσ is Lipschitz continuous. Here LL corresponds to a symmetric αα -stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σσ , we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014)

On some properties of a class of fractional stochastic heat equations

We consider nonlinear parabolic stochastic equations of the form ∂tu = Lu + λσ(u) ˙ξ on the ball B(0, R), where ˙ξ denotes some Gaussian noise and σ is Lipschitz continuous. Here L corresponds to a symmetric α-stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on σ, we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014)

On hybrid stochastic population models with impulsive perturbations

This paper considers the dynamic behaviours of a hybrid stochastic population model with impulsive perturbations. The existence of the global positive solution is studied in this paper. Moreover, under some conditions on the noises and impulsive perturbations, the properties of the persistence and extinction, stochastic permanence, global attractivity and stability in distribution are presented. Our results illustrate that impulsive perturbations play a crucial role in these properties. The bounded impulse term will not affect these properties, however, when the impulse term is unbounded, some of the properties, such as the persistence and extinction may be changed significantly. As a part of this paper, a couple of examples and numerical simulations are provided to illustrate our results