Processes with free increments.

*(English)*Zbl 0902.60060The article introduces a notion of processes with free independent increments, inspired from the free probability theory of Voiculescu. More precisely, given a von Neumann algebra \({\mathcal A}\) provided with a faithful normal trace \(\tau\), a random variable \(X\) is a self-adjoint element of \({\mathcal A}\), and its distribution \(\mu\) is defined via the spectral representation as the unique probability measure such that for any real Borel function \(f\) it holds \(\tau (f(X))=\int f(\lambda)\mu (d\lambda)\). A process with free additive increments is a map \(t\mapsto X_t\) from \(\mathbb{R}_+\) to the set of self-adjoint elements of \({\mathcal A}\) such that for any finite collection of times \(t_1<\ldots<t_n\), the elements \(X_{t_1},X_{t_2}-X_{t_1},\ldots, X_{t_n}-X_{t_{n-1}}\) form a free family. This means that if we call \(\mu_t\) the distribution of \(X_t\) and \(\mu_{s,t}\) that of \(X_t-X_s\) \((s<t)\), then \(\mu_s\boxplus\mu_{s,t}=\mu_t\), where \(\boxplus\) denotes the free convolution of measures in the sense of Voiculescu.

The multiplicative convolution of measures on the unit circle is also considered. A map \(t\mapsto U_t\) from \(\mathbb{R}_+\) into the set of unitary elements is a unitary process with free increments if for any finite collection \(t_1<\ldots<t_n\) of times, the elements \(U_{t_1},U_{t_2}U_{t_1}^{-1},\ldots, U_{t_n}U_{t_{n-1}}^{-1}\) form a free family. This means that for all \(r<s<t\), \(\mu_s\boxtimes\mu_{s,t}=\mu_t\), \(\mu_{r,s}\boxtimes\mu_{s,t}=\mu_{r,t}\), where \(\boxtimes\) is the free multiplicative convolution of measures on \(\mathbb{T}\) and \(\mu_t\) is the distribution of \(U_t\) and \(\mu_{s,t}\) that of \(U_tU_s^{-1}\) for \(s<t\). It is proved that free processes satisfy a Markov property too. This is indeed one of the main results of the paper which deals with the quantum analogue of Lévy’s theory on processes with independent increments. Moreover, it is proved that for any process \(X\) with free additive increments, there exists a classical Markov process \(Z\), defined on some probability space, with values in \(\mathbb{R}\) which has the same time ordered moments as \(X\).

The multiplicative convolution of measures on the unit circle is also considered. A map \(t\mapsto U_t\) from \(\mathbb{R}_+\) into the set of unitary elements is a unitary process with free increments if for any finite collection \(t_1<\ldots<t_n\) of times, the elements \(U_{t_1},U_{t_2}U_{t_1}^{-1},\ldots, U_{t_n}U_{t_{n-1}}^{-1}\) form a free family. This means that for all \(r<s<t\), \(\mu_s\boxtimes\mu_{s,t}=\mu_t\), \(\mu_{r,s}\boxtimes\mu_{s,t}=\mu_{r,t}\), where \(\boxtimes\) is the free multiplicative convolution of measures on \(\mathbb{T}\) and \(\mu_t\) is the distribution of \(U_t\) and \(\mu_{s,t}\) that of \(U_tU_s^{-1}\) for \(s<t\). It is proved that free processes satisfy a Markov property too. This is indeed one of the main results of the paper which deals with the quantum analogue of Lévy’s theory on processes with independent increments. Moreover, it is proved that for any process \(X\) with free additive increments, there exists a classical Markov process \(Z\), defined on some probability space, with values in \(\mathbb{R}\) which has the same time ordered moments as \(X\).

Reviewer: G.Högnäs (Åbo)

##### MSC:

60J99 | Markov processes |