标题： 通过两个马尔可夫生成元在右和左特征向量的双正交基中的谱分解得到的马尔可夫对偶性 显示英文标题

标题： Markov dualities via the spectral decompositions of the two Markov generators in their bi-orthogonal basis of right and left eigenvectors

摘要： 两个马尔可夫过程存在于两个不同的配置空间 $(x,{\tilde x})$ 中，我们通过对两个马尔可夫生成器在其左右特征向量的双正交基中进行谱分解，重新探讨了这两个过程之间的马尔可夫对偶性概念。 在这个公式中，两个生成器应该具有相同的特征值 $(-E)$，该特征值可能是复数，而对偶函数 $\Omega(x,{\tilde x})$ 可以被认为是两个模型的左右特征向量之间的映射。 我们描述了这种谱视角如何有助于更好地理解在同一配置空间中定义的过程之间的两个著名对偶性：时间反转对偶对应于左右特征向量之间的交换，该交换涉及稳态；而在西格蒙德对偶中，左特征向量对应于对偶右特征向量的积分。 然后，我们重点讨论区间 $x \in [0,1] $ 上的 Wright-Fisher 扩散与半无限格 $n \in {\mathbb N}$ 上的 Kingman Markov 跳跃过程之间著名的矩对偶性，以分析它们的特征向量在两个不同配置空间中的关系。 最后，我们讨论了如何利用谱视角构造新的对偶性，并给出了非退化实特征值的情况的例子，其中总是可以在半无限格 $n \in {\mathbb N}$ 上构造一个对偶有向跳跃过程，其跃迁率与特征值相反。 显示英文摘要

摘要： The notion of Markov duality between two Markov processes that can live in two different configurations spaces $(x,{\tilde x})$ is revisited via the spectral decompositions of the two Markov generators in their bi-orthogonal basis of right and left eigenvectors. In this formulation, the two generators should have the same eigenvalues $(-E)$ that may be complex, while the duality function $\Omega(x,{\tilde x})$ can be considered as a mapping between the right and the left eigenvectors of the two models. We describe how this spectral perspective is useful to better understand two well-known dualities between processes defined in the same configuration space: the Time-Reversal duality corresponds to an exchange between the right and the left eigenvectors that involves the steady state, while in the Siegmund duality, the left eigenvectors correspond to integrals of the dual right eigenvectors. We then focus on the famous Moment-Duality between the Wright-Fisher diffusion on the interval $x \in [0,1] $ and the Kingman Markov jump process on the semi-infinite lattice $n \in {\mathbb N}$ in order to analyze the relations between their eigenvectors living in two different configuration spaces. Finally, we discuss how the spectral perspective can be used to construct new dualities and we give an example for the case of non-degenerate real eigenvalues, where one can always construct a dual Directed Jump process on the semi-infinite lattice $n \in {\mathbb N}$, whose transitions rates are the opposite-eigenvalues.