Event № 663
In non-commutative probability there are several well known notions of independence. In 2003, Muraki's classification, which states that there are exactly five independences coming from universal (natural) products, seemingly settled the question of what independences can be considered. But after Voiculescu's invention of bi-free independence in 2014, the question came up again. The key idea that allows to define a new notion of independence with all the features of the universal independences that appear in Muraki's classification is to consider ``two-faced'' (i.e. pairs of) random variables.In the talk, we define bi-monotone independence, a new example of an independence for two-faced random variables. We establish a corresponding central limit theorem and use it to describe the joint distribution of monotone and antimonotone Brownian motion on monotone Fock space, which yields a canonical example of a quantum stochastic process with bi-monotonely independent increments.