Stochastic Solution of a KPP-Type Nonlinear Fractional Differential Equation

Mathematics Subject Classification: 26A33, 76M35, 82B31A stochastic solution is constructed for a fractional generalization of the KPP (Kolmogorov, Petrovskii, Piskunov) equation. The solution uses a fractional generalization of the branching exponential process and propagation processes which are spectral integrals of Levy processes

A quantum approach to Laplace operators

In this paper, a theory of stochastic processes generated by quantum extensions of Laplacians is developed. Representations of the associated heat semigroups are discussed by means of suitable time shifts. In particular the quantum Brownian motion associated to the Levy-Laplacian is obtained as the usual Volterra-Gross Laplacian using the Cesaro Hilbert space as initial space of our process as well as multiplicity space of the associated white noise

White noise Heisenberg evolution and Evans-Hudson flows

We study white noise Heisenberg equations giving rise to flows which are *-automorphisms of the observable algebra, but not necessarily inner automorphisms. We prove that the causally normally ordered form of these white noise Heisenberg equations are equivalent to Evans–Hudson flows. This gives in particular, the microscopic structure of the maps defining these flows, in terms of the original white noise derivations

Generic quantum Markov semigroups: the Fock case

We introduce the class of generic quantum Markov semigroups. Within this class we study the class corresponding to the Fock case which is further split into four sub-classes each of which contains both bounded and unbounded generators, depending on some global characteristics of the intensities of jumps. For the first two of these classes we find an explicit solution which reduces the problem of finding the quantum semigroup to the calculation of two classical semigroups, one of which is diagonal (in suitable basis) and the other one is triangular (in the same basis). In the bounded case our formula gives the unique solution. In the unbounded case it gives one solution, which we conjecture to be the minimal one

White noise quantum time shifts

In the present paper we extend the notion of quantum time shift, and the related results obtained in \cite{[abo06]}, from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra.\\ This produces quantum extensions of a class of classical L\'evy processes much wider than the usual Brownian motion. In particular this class processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical L\'evy processes with a L\'evy measure with finite variance. Finally we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of L\'evy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way, comes somehow as a surprise