非局部遍历Jacobi半群:谱理论、收敛至平衡和收缩性

Patrick Cheridito, P. Patie, A. Srapionyan, A. Vaidyanathan
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引用次数: 7

摘要

本文引入并研究了非局部Jacobi算子,它是经典(局部)Jacobi算子的推广。我们证明了这些算子可以推广到具有唯一不变概率测度的遍历马尔可夫半群的生成上,并研究了它的谱性和收敛性。特别地,我们给出了半群的显定义多项式的级数展开式,它是经典雅可比正交多项式的对应物。此外,我们给出了非自伴随产生子和半群的谱的完整表征。我们证明了半群的方差衰减是具有显式常数的准强制,这提供了谱隙估计的自然推广。经过一段随机预热时间后,半群的熵也呈指数衰减,同时具有超收缩性和超收缩性。我们的证明依赖于局部和非局部Jacobi算子/半群之间的交换恒等式的发展,即纠缠关系,局部Jacobi算子/半群作为将性质转移到非局部的参考对象。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity
In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operator. We show that these operators extend to the generator of an ergodic Markov semigroup with a unique invariant probability measure and study its spectral and convergence properties. In particular, we give a series expansion of the semigroup in terms of explicitly defined polynomials, which are counterparts of the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators/semigroups, with the local Jacobi operator/semigroup serving as a reference object for transferring properties to the non-local ones.
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