Brownian survival and Lifshitz tail in perturbed lattice disorder

From the Lifshitz tail to the quenched survival asymptotics in the trapping problem

The survival problem for a diffusing particle moving among random traps is considered. We introduce a simple argument to derive the quenched asymptotics of the survival probability from the Lifshitz tail effect for the associated operator. In particular, the upper bound is proved in fairly general settings and is shown to be sharp in the case of the Brownian motion in the Poissonian obstacles. As an application, we derive the quenched asymptotics for the Brownian motion in traps distributed according to a random perturbation of the lattice.

Existence of a unique solution and invariant measures for the stochastic Landau--Lifshitz--Bloch equation

The Landau--Lifshitz--Bloch equation perturbed by a space-dependent noise was proposed in Garanin 1991 as a model for evolution of spins in ferromagnatic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain $D\subset \mathbb R^d$, $d=1,2,3$, we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases $d=1,2$ we prove uniqueness of pathwise solutions and the existence of invariant measures.

Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation

The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space-dependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere $\mathbb S^2$. The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert equation.

随机扰动Landau-Lifshitz-Baryakhtar方程的适定性与不变测度

Well-posedness and invariant measures for the stochastically perturbed Landau-Lifshitz-Baryakhtar equation

本文研究带Stratonovich型噪声的SLLBar方程初边值问题，得到不同维度与初值条件下的适定性结果及不变测度存在性结论，三维情形仅获鞅解存在性。

随机Landau-Lifshitz-Gilbert方程的松弛最优控制

Relaxed optimal control for the stochastic Landau-Lifshitz-Gilbert equation

针对受实值Wiener过程扰动的随机Landau-Lifshitz-Gilbert方程，研究人员引入依赖控制与解的一般控制算子及非凸代价泛函，用Young测度理论松弛控制问题，结合紧性论证与Jakubowski版Skorohod定理证明松弛问题存在最优控制。

随机 Landau-Lifshitz-Bloch 方程的最优控制

Optimal control of the stochastic Landau-Lifshitz-Bloch equation

本文研究 1、2、3 维随机 Landau-Lifshitz-Bloch 方程的最优控制问题，利用 Young 测度理论将其松弛化，证明了弱鞅解与弱松弛最优控制的存在性，并在 1、2 维中证明了逐路径唯一性及强解存在性。