Quantum and non-causal stochastic calculus

Summary The quantum stochastic calculus initiated by Hudson and Parthasarathy, and the non-causal stochastic calculus originating with the papers of Hitsuda and Skorohod, are two potent extensions of the Itô calculus, currently enjoying intensive development. The former provides a quantum probabilistic extension of Schrödinger's equation, enabling the construction of a Markov process for a quantum dynamical semigroup. The latter allows the treatment of stochastic differential equations which involve terms which anticipate the future. In this paper the close relationship between these theories is displayed, and a noncausal quantum stochastic calculus, already in demand from physics, is described.     

Semicircular limits on the free Poisson chaos: Counterexamples to a transfer principle

We establish a class of sufficient conditions ensuring that a sequence of multiple integrals with respect to a free Poisson measure converges to a semicircular limit. We use this result to construct a set of explicit counterexamples showing that the transfer principle between classical and free Brownian motions (recently proved by Kemp, Nourdin, Peccati and Speicher (2012)) does not extend to the framework of Poisson measures. Our counterexamples implicitly use kernels appearing in the classical theory of random geometric graphs. Several new results of independent interest are obtained as necessary steps in our analysis, in particular: (i) a multiplication formula for free Poisson multiple integrals, (ii) diagram formulae and spectral bounds for these objects, and (iii) a counterexample to the general universality of the Gaussian Wiener chaos in a classical setting.     

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

We define stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of It?'s predictable representation theorem, as well as product form and functional form of It?'s formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space. Received: 6 February 1998     

On dissipative stochastic equations in a Hilbert space

Summary A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.     

Exponential Formula for the Reachable Sets of Quantum Stochastic Differential Inclusions

We establish an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) which are locally Lipschitzian with convex values. Our main results partially rely on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable. By applying the exponential formula, we obtain results concerning convergence of the discrete approximations of the reachable set of the QSDI. This extends similar results of Wolenski[20] Wolenski, P.R. 1990. The exponential formula for the reachable set of a Lipschitz differential inclusion. SIAM J. Control Optim., 28(5): 1148–1161.  [Google Scholar] for classical differential inclusions to the present noncommutative quantum setting.     

Stochastic generalized porous media and fast diffusion equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N-functions in the theory of Orlicz spaces.     

Stochastic dilations of uniformly continuous completely positive semigroups

For an arbitrary uniformly continuous completely positive semigroup ( _t :t0) on the space of bounded operators on a Hilbert space, we construct a family (U(t)t0) of unitary operators on a Hilbert space and a conditional expectation from to, such that, for arbitraryt0,. The unitary operatorsU(t) satisfy a stochastic differential equation involving a noncommutative generalisation of infinite dimensional Brownian motion. They do not form a semigroup.Part of this work was completed when the first author was visiting research associate at the Center for Relativity, Physics Department, The University of Texas at Austin, Austin, TX 78712, U.S.A., supported in part by NSF PHY 81-01381.     

A different quantum stochastic calculus for the Poisson process

Summary We show that a gradient operator defined by perturbations of the Poisson process jump times can be used with its adjoint operator instead of the annihilation and creation operators on the Poisson-Charlier chaotic decomposition to represent the Poisson process. The quantum stochastic integration and the Itô formula are developed accordingly, leading to commutation relations which are different from the CCR. An analog of the Weyl representation is defined for a subgroup ofSL(2, ), showing that the exponential and geometric distributions are closely related in this approach.     

Weyl quantization for semidirect products

Let G be the semidirect product V?K where K is a connected semisimple non-compact Lie group acting linearily on a finite-dimensional real vector space V. Let O be a coadjoint orbit of G associated by the Kirillov-Kostant method of orbits with a unitary irreducible representation π of G. We consider the case when the corresponding little group K₀ is a maximal compact subgroup of K. We realize the representation π on a Hilbert space of functions on Rn where n=dim(K)−dim(K₀). By dequantizing π we then construct a symplectomorphism between the orbit O and the product R2n×O^′ where O^′ is a little group coadjoint orbit. This allows us to obtain a Weyl correspondence on O which is adapted to the representation π in the sense of [B. Cahen, Quantification d'une orbite massive d'un groupe de Poincaré généralisé, C. R. Acad. Sci. Paris Série I 325 (1997) 803-806]. In particular we recover well-known results for the Poincaré group.     

Continuity of the occupation density for anticipating stochastic integral processes

In this paper we prove the existence of a continuous local time for an anticipating process which is composed of an indefinite Skorohod integral and an absolutely continuous term.The work of P. Imkeller was done during his visit to the CRM of Barcelona.Partially supported by the DGICYT grant number PB90-0452.     

A mathematical flat integral realization and a large deviation result for the free Euclidean field

In this paper, we give a nonstandard construction of the free Euclidean field via S-white noise. This provides a flat integral realization of the free Euclidean field measure, which extends N. J. Cutland's flat integral representation of Wiener measure. Moreover, we show how a Cameron-Martin type formula for translations of the free field measure and a Schilder type large deviation principle for the scalar free field measure can be deduced from our nonstandard construction.SFB 237 Essen-Bochum-Düseldorf; BiBoS-Research Centre; CERFIM, Locarno, Switzerland.     

A class of elliptic pseudo differential operators generating symmetric Dirichlet forms

It is shown that a special class of symmetric elliptic pseudo differential operators do generate a Feller semigroup and therefore a non-local Dirichlet form.     

Discrete wick calculus and stochastic functional equations

We develop Wick calculus over finite probability spaces and prove that there is a one-to-one correspondence between the solutions of Wick stochastic functional equations and the solutions of the deterministic functional equations obtained by turning off the noise. We also point out some possible applications to ordinary and partial stochastic differential equations.This research is supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap a.s. (STATOIL).     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

Bernstein processes and Pauli-type equations

We use ideas from a previous paper by the author to construct a Markov Bernstein process, whose probability density is the product of the solutions of the (imaginary time) Schrödinger-equation and its adjoint equation, associated to a class of Pauli-type Hamiltonians. A path integral representation of these solutions is obtained as well as the associated regularised Newton equations.     

Stability inL 1 of some integral operators

A class of Markov operators appearing in biomathematics is investigated. It is proved that these operators are asymptotic stable inL ¹, i.e. lim _n P ⁿ f=0 forfL ¹ and f(x) dx=0.     

Stochastic Volterra equations driven by cylindrical Wiener process

In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of α-times resolvent families. Both authors are supported partially by project “Proyecto Anillo: Laboratorio de Analisis Estocastico; ANESTOC”.     

Stochastic partial differential equations in Hölder spaces

Summary We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.The work of the first author was supported in part by the NSF grant DMS-91-01360     

The analytic operator-valued Feynman integral via additive functionals of Brownian motion

The definition of the analytic operator-valued Feynman integral discussed by G. W. Johnson is extended to potentials given by a class of generalized signed measures described in terms of additive functions associated with Dirichlet forms.