The α-dependence of the invariant measure of stochastic real Ginzburg-Landau equation driven by α-stable Lévy processes
The α-dependence of the invariant measure of stochastic real Ginzburg-Landau equation driven by α-stable Lévy processes
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DOI：
10.1016/j.jde.2022.01.024
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发表时间：
2022-03
影响因子：
2.4
通讯作者：
Xianming Liu
中科院分区：
文献类型：
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作者：
Xianming Liu
通过研究柱α-稳定Lévy过程驱动的随机真实的Ginzburg-Landau方程的不变概率测度在极限α→ 2下的收敛性，建立了非高斯噪声随机动力系统与高斯噪声随机动力系统之间的联系.证明了在Wasserstein距离下，当α趋于2时，由柱α稳定Lévy过程驱动的环面上的随机真实的Ginzburg-Landau方程的不变测度收敛于由柱布朗运动驱动的随机真实的Ginzburg-Landau方程的不变测度.我们的战略如下。首先，利用Hairer和Mattingly给出的一个抽象结果，证明了一类Wasserstein距离对于由布朗运动强迫的极限方程所对应的马氏半群的对偶算子是收缩的.然后，通过对圆柱从属布朗运动的随机卷积的一致先验矩估计和一个收敛结果，在α→ 2的极限下，建立了由圆柱从属布朗运动驱动的随机真实的Ginzburg-Landau方程解的强收敛结果.最后，利用Monge-Kantorovich对偶，证明了在Wasserstein距离下，柱α稳定Lévy过程驱动的随机真实的Ginzburg-Landau方程的不变概率测度收敛于布朗运动驱动的随机真实的Ginzburg-Landau方程的不变概率测度.
This paper is intended to establish the connection between stochastic dynamical systems with non-Gaussian noises and stochastic dynamical systems with Gaussian noises, by considering the convergence behavior of invariant probability measures for stochastic real Ginzburg-Landau equation driven by cylindrical α-stable Lévy process, in the limit α→ 2. Indeed, we prove that the invariant measure of stochastic real Ginzburg-Landau equation on torus and driven by cylindrical α-stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by cylindrical Brownian motions under the Wasserstein distance, as α tends to 2. We state our strategy as below. First, by using an abstract result given by Hairer and Mattingly, we prove that a type of Wasserstein distance is contracting for the dual operator of Markov semigroup associated with the limit equation which forced by Brownian motions. Then, through server priori uniform moment estimates and a convergence result on stochastic convolutions of cylindrical subordinated Brownian motions, we establish a strong convergence result on the solution of stochastic real Ginzburg-Landau equation driven by cylindrical subordinated Brownian motions conditioned on the initial data is distributed as an invariant measure, in the limit α→ 2. Finally, by the Monge-Kantorovich duality, we prove that the invariant probability measure of stochastic real Ginzburg-Landau equation driven by cylindrical α-stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by Brownian motions under the Wasserstein distance.