Pathwise mild solutions for quasilinear stochastic partial differential equations

Approximate solutions for stochastic time-fractional reaction–diffusion equations with multiplicative noise

In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use this equation with multiplicative noise and fractional space. Additionally, we show the influence of the stochastic term on the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation.

The Exact Solutions of Stochastic Fractional-Space Kuramoto-Sivashinsky Equation by Using (G′G)-Expansion Method

In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results.

https://www.mdpi.com/1099-4300/20/1/63/pdf

Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications

Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the classical theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time showing that solutions can only be local-in-time for general quasilinear SPDEs, while they might be global-in-time for special subclasses of problems.

In this article, we will prove the existence, uniqueness and Holder regularity of the solution to the fractional stochastic partial differential equation of the form 

$$\begin{aligned} \frac{\partial }{\partial t}u(t,x)=\mathfrak {D}(x,D)u(t,x)+\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^2 W^H}{\partial t\partial x}(t,x), \end{aligned}$$

where $\mathfrak {D}(x,D)$ denotes the Markovian generator of stable-like Feller process, $f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ is a measurable function, and $\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)$ is a double-parameter fractional noise. In addition, we establish lower and upper Gaussian bounds for the probability density of the mild solution via Malliavin calculus and the new tool developed by Nourdin and Viens (Electron J Probab 14:2287–2309, 2009).

Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L2 if 1/2<H<3/4.

On the Linear Fractional Self-attracting Diffusion

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L^2 if $\frac12<H<\frac3{4}$.

On the linear fractional self-attracting diffusion

In this paper, we discuss a class of neutral retarded stochastic functional differential equations driven by a fractional Brownian motion on Hilbert spaces. We develop a $$C_0$$
 -semigroup theory of the driving deterministic neutral system and formulate the neutral time delay equation under consideration as an infinite-dimensional stochastic system without time lag and neutral item. Consequently, a criterion is presented to identify a strictly stationary solution for the systems considered. In particular, the ergodicity of the strictly stationary solution is studied. Subsequently, the ergodicity behavior of non-stationary solution for the systems considered is also investigated. We present an example which can be explicitly determined to illustrate our theory in the work.

Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion

Let $$B^H$$
 be a fractional Brownian motion with Hurst index $$\frac{1}{2}\le H<1$$
 . In this paper, we consider the linear self-attracting diffusion $$\begin{aligned} X^H_t=B^H_t-\theta \int _0^t\int _0^s \left( X^H_s-X^H_u\right) \mathrm{d}u\mathrm{d}s+\nu t \end{aligned}$$
 with $$X^H_0=0$$
 , where $$\theta >0$$
 and $$\nu \in {{\mathbb {R}}}$$
 are two unknown parameters. The process is an analogue of the linear self-attracting diffusion (Cranston and Le Jan in Math Ann 303:87–93, 1995). We introduce some laws of large numbers associated with the self-attracting diffusion. In particular, we show that the convergences $$\begin{aligned} \frac{1}{\theta T^{3-2H}}\int _0^TY^H_t\mathrm{d}B^H_t,\quad \frac{1}{T^{3-2H}}\int _0^T\left( Y^H_t\right) ^2\mathrm{d}t\longrightarrow \frac{H\theta ^{-2H}}{3-2H} \Gamma (2H) \end{aligned}$$
 hold almost surely and in $$L^2(\Omega )$$
 , as $$T\rightarrow \infty $$
 , where $$Y^H_t=\int ^t_0(X^H_t-X^H_s)\mathrm{d}s$$
 and the integral $$\int _0^TY^H_t\mathrm{d}B^H_t$$
 is the Young integral. As some applications, we study asymptotic behaviors of the least squares estimators of $$\theta $$
 and $$\nu $$
 under the continuous observation.

Litan Yan

Papers

Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

On the Linear Fractional Self-attracting Diffusion

On the linear fractional self-attracting diffusion

Ergodicity and Stationary Solution for Stochastic Neutral Retarded Partial Differential Equations Driven by Fractional Brownian Motion

The Laws of Large Numbers Associated with the Linear Self-attracting Diffusion Driven by Fractional Brownian Motion and Applications