Monotone and boolean unitary Brownian motions

Abstract

Abstract. The additive monotone (resp. boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (resp. boolean) independent and stationary increments which are distributed according to the arcsine law (resp. Bernoulli law) . We introduce the monotone and boolean unitary Brownian motions and we derive a closed formula for their associated moments. This provides a description of their spectral measures. We prove that, in the monotone case, the multiplicative analog of the arcsine distribution is absolutely continuous with respect to the Haar measure on the unit circle, whereas in the boolean case the multiplicative analog of the Bernoulli distribution is discrete. Finally, we use quantum stochastic calculus to provide a realization of these processes as the stochastic exponential of the correspending additive Brownian motions. IntroductionIn non-commutative probability theory, there exist several natural notions of independence. The most famous ones are the usual independence and the free independence. Other interesting examples are monotone and boolean independence. These allow to define new convolutions for probability measures. The monotone convolutions on the unit circle and the positive half-line were introduced by Bercovici in [2], see also [6]. But the additive monotone Brownian motion and the monotone Fock space where first studied by Muraki, see [9]. The monotone stochastic calculus can also be realized on the symmetric Fock space, see [5]. Bercovici studied also the boolean convolution for probability measures on the positive half-line, but it is not always defined. For probability measures on the unit circle, the boolean convolution is well defined. It was introduced by Franz in [7], see also [3]. The boolean stochastic calculus has been studied by Ben Ghorbal and Schürmann, see [1].The aim of this paper is to point out several connections between classical, free Brownian motions and their counterparts in the monotone and boolean cases. We shall consider two kinds of unitary Brownian motions, the monotone and boolean one. Both are non-commutative unitary processes, that is families of noncommutative random variables which are unitary and are characterized as having 2010 Mathematics Subject Classification. Primary 60J65, 46L51, 46L53; Secondary 65C30.

The Exponential Map in Non-commutative Probability

“…In particular, for the case of discrete measure σ as in Proposition 6, we have a complete description of ν β,σ x Y . The following corollary generalizes Example 3.8 of [Fra08] and the results in Section 2.4 of [Ham15].…”

“…x Y . The following corollary generalizes Example 3.8 of [Fra08] and the results in Section 2.4 of [Ham15]. x Y ˚is purely atomic, and its atoms can be decomposed into N families tpe x jk q jPZ , 1 ď k ď Nu, with x jk the unique solution of the equation…”

Spectral distribution of the free Jacobi process, revisited

“…Let µ 1 , µ 2 ∈ M T and set F µ (z) = 1 z χ µ (z). Then the multiplicative boolean convolution µ = µ 1 × ∪ µ 2 is uniquely determined by (see [14] or [13] for more details)…”

“…Monotone independent initial operators. For µ 1 , µ 2 ∈ M T , the multiplicative monotone convolution µ = µ 1 ⊲ µ 2 is uniquely determined by (see [14] or [12] for more details) χ µ (z) = χ µ 1 χ µ 2 (z) .…”

Boolean convolutions and regular variation

“…The Boolean convolutions of probability measures are also used in studying quantum stochastic calculus, see Ben Ghorbal and Schürmann [2004]. The Boolean Brownian motion and Poisson processes are investigated using Boolean convolutions to study the Fock space in Privault [2001], Hamdi [2015]. We can also observe the connection between Appell polynomials and Boolean theory in Anshelevich [2009].The Boolean stable laws and their relationship with free and classical stable laws were extensively studied in recent works (see Hasebe [2013, 2014], Anshelevich et al [2014], Arizmendi and Hasebe [2016]).…”