×

On quantum extensions of the Azéma martingale semigroup. (English) Zbl 0956.46044

Azéma, J. (ed.) et al., Séminaire de probabilités XXIX. Berlin: Springer-Verlag. Lect. Notes Math. 1613, 1-16 (1995).
From the introduction: We study some quantum extensions of classical Markovian semigroups related to the Azéma martingales with parameter \(\beta\) (\(\beta\neq 0\), \(\beta\neq -1\)). The formal infinitesimal generator given by \[ ({\mathcal L}_0f)(x)= (\beta x)^{-2}(f(cx)- f(x)- \beta xf'(x)) \] on bounded smooth functions \(f\) can be written formally as follows \[ {\mathcal L}(m_f)= Gm_f+ L^*m_f L+ m_fG^*, \] where \(m_f\) denotes the multiplication operator by \(f\), the operator \(G\) is the infinitesimal generator of a strongly continuous contraction semigroup on \(L^2(\mathbb{R};\mathbb{C})\) (see Section 2) and \(L\) is related to \(G\) by the formal condition \(G+ G^*+ L^*L= 0\). The associated minimal quantum dynamical semigroup can be easily constructed. We show that this semigroup is conservative if \(\beta< \beta_*\) and it is not if \(\beta> \beta_*\) where \(\beta_*\) is the unique solution of the equation \[ \exp(\beta)+ \beta+ 1=0. \] Therefore it is a natural conjecture that the minimal quantum dynamical semigroup is a ultraweakly continuous extension to \({\mathcal B}(h)\) of the Azéma martingale semigroup when \(\beta< \beta_*\). However, we can not prove this fact because the characterization of the classical infinitesimal generator is not known. The above quantum dynamical semigroup is not such an extension when \(\beta>\beta_*\) because the corresponding classical Markovian semigroup is identity preserving.
We were not able to study the critical case \(\beta= \beta_*\) although it seems reasonable that conservability holds also in this case. In fact, as shown by M. Emery in [Lect. Notes Math. 1372, 66-87 (1989)], the Azéma martingale with parameter \(\beta\) starting from \(x\neq 0\) can hit \(0\) in finite time only if \(\beta>\beta_*\). The operators \(G\) and \(L\) we consider are singular at the point \(0\), hence, in this case, boundary conditions on \(G\) at \(0\) should appear to describe the behaviour of the process.
The cases when \(\beta< \beta_*\) and \(\beta>\beta_*\) are studied in Section 3 by checking a necessary and sufficient condition obtained in [the the first author, Proc. 1989 COSMEX meeting, Szklarska Pouba, 79-95 (1990)]. In Section 5 we apply a sufficient condition for conservativity obtained in [the authors, J. Funct. Anal. 118, No. 1, 131-153 (1993; Zbl 0801.46083)]. This condition yields the previous result when \(\beta\leq -1.5\); since \(\beta_*= -1.278\dots\), it is quite “close” to the necessary and sufficient one.
For the entire collection see [Zbl 0826.00027].

MSC:

46L53 Noncommutative probability and statistics
47D07 Markov semigroups and applications to diffusion processes
60G44 Martingales with continuous parameter
46L07 Operator spaces and completely bounded maps
46L60 Applications of selfadjoint operator algebras to physics
PDFBibTeX XMLCite
Full Text: Numdam EuDML