On the limit distribution for stochastic differential equations driven by cylindrical non-symmetric $α$-stable Lévy processes

由标准对称$\alpha$-稳定柱形Lévy过程驱动的SPDE：存在性、Lyapunov泛函与Itô公式

SPDEs driven by standard symmetric $α$-stable cylindrical Lévy processes: existence, Lyapunov functionals and Itô formula

本文研究希尔伯特空间中标准对称$\alpha$-稳定柱形噪声驱动的随机偏微分方程，证明温和解存在性、给出解的长期正则性与Itô公式，针对柱形噪声特性发展了新的论证方法。

Transition density estimates for diagonal systems of SDEs driven by cylindrical $α$-stable processes

We consider the system of stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $α$-stable process $Z_t$ in $\mathbb{R}^d$. We assume that $A(x) = (a_{ij}(x))$ is diagonal and $a_{ii}(x)$ are bounded away from zero, from infinity and Hölder continuous. We construct transition density $p^A(t,x,y)$ of the process $X_t$ and show sharp two-sided estimates of this density. We also prove Hölder and gradient estimates of $x \to p^A(t,x,y)$. Our approach is based on the method developed by Chen and Zhang.

Strong Feller property for SDEs driven by multiplicative cylindrical stable noise

We consider the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $α$-stable process $Z_t$ in $R^d$, where $α\in (0,1)$ and $d \ge 2$. We assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. We show that for any fixed $γ\in (0,α)$ the semigroup $P_t$ of the process $X_t$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-γ/α} |x - y|^γ ||f||_\infty$ for arbitrary bounded Borel function $f$. Our approach is based on Levi's method.

关于由柱面稳定过程驱动的随机波动方程的小质量极限

On the small mass limit of stochastic wave equation driven by cylindrical stable process

本文研究了由柱面$α$-稳定噪声驱动的随机波动方程的小质量极限，通过分裂技术和紧性证明，证明了其解收敛于随机热方程。

Linear Evolution Equations with Cylindrical Lévy Noise: Gradient Estimates and Exponential Ergodicity

Explicit coupling property and gradient estimates are investigated for the linear evolution equations on Hilbert spaces driven by an additive cylindrical Lévy process. The results are efficiently applied to establish the exponential ergodicity for the associated transition semigroups. In particular, our results extend recent developments on related topic for cylindrical symmetric $α$-stable processes.

无限维中与柱面稳定Lévy噪声相关的HJB方程的温和解

Mild solutions of HJB equations associated with cylindrical stable Lévy noise in infinite dimensions

本文研究了可分Hilbert空间中由柱面稳定噪声驱动的无限维随机系统的最优控制问题，证明了相关HJB方程温和解的存在性与唯一性，为后续验证定理及最优性充分条件的建立奠定了基础。

Structural properties of semilinear SPDEs driven by cylindrical stable processes

We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities.

Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $α$-stable processes

We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$ {\rm d} X_t=σ(t, X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $( Z_t)_{t\geq 0}$ is a $d$-dimensional cylindrical $α$-stable process with $α\in (0, 2)$, $σ(t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is bounded $β$-Hölder continuous in $x$ uniformly in…

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 d…