带乘性噪声的Langevin动力系统的本质m-耗散性和亚耗散性
Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise
摘要：
我们提供了一个完整的阐述$L^2$ -希尔伯特空间hypovivity定理的退化Langevin动力学与乘性噪声，研究的长期行为的强连续压缩半群解决抽象柯西问题的相关向后Kolmogorov算子。Dolbeault，Mouhot和Schmeiser在相应的Fokker-Planck框架下证明了具有常数扩散矩阵的Langevin动力学的假设性，Grothaus和Stilgenbauer在Kolmogorov向后设置下证明了该假设性。我们将这些结果推广到弱可微的扩散系数矩阵，在相应的随机微分方程中引入乘性噪声。收敛的收敛速率是根据这些系数的选择和给出外力的势显式计算的。为了得到抽象Cauchy问题的解，我们首先利用Bogachev，Krylov和Röckner的椭圆正则性结果和技巧证明了希尔伯特空间上非退化椭圆Dirichlet算子的本质自伴性.利用算子扰动理论得到了Kolmogorov算子的本质m-耗散性，推广了Conrad和Grothaus的m-耗散性结果.我们强调，选择Kolmogorov方法是自然的，广义Dirichlet形式的理论意味着随机表示的Langevin半群的过渡内核的扩散过程提供了一个鞅解的Langevin方程乘性噪声。此外，我们证明了即使是一个弱解是获得这种方式。
英文摘要：
We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and Röckner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.
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