Class Of Stochastic Processes Research Articles

Entwined paths, difference equations, and the Dirac equation

Entwined space-time paths are bound pairs of trajectories which are traversed in opposite directions with respect to macroscopic time. In this paper we show that ensembles of entwined paths on a discrete space-time lattice are simply described by coupled difference equations which are discrete versions of the Dirac equation. There is no analytic continuation, explicit or forced, involved in this description. The entwined paths are `self-quantizing'. We also show that simple classical stochastic processes that generate the difference equations as ensemble averages are stable numerically and converge at a rate governed by the details of the stochastic process. This result establishes the Dirac equation in one dimension as a phenomenological equation describing an underlying classical stochastic process in the same sense that the Diffusion and Telegraph equations are phenomenological descriptions of stochastic processes.

Classical dynamics of quantum fluctuations

It is shown that the vacuum state of weakly interacting quantum field theories can be described, in the Heisenberg picture, as a linear combination of randomly distributed incoherent paths that obey classical equations of motion with constrained initial conditions. We call such paths "pseudoclassical" paths and use them to define the dynamics of quantum fluctuations. Every physical observable is assigned a time-dependent value on each path in a way that respects the uncertainty principle, but in consequence, some of the standard algebraic relations between quantum observables are not necessarily fulfilled by their time-dependent values on paths. We define "collective observables" which depend on a large number of independent degrees of freedom, and show that the dynamics of their quantum fluctuations can be described in terms of unconstrained classical stochastic processes without reference to any additional external system or to an environment. Our analysis can be generalized to states other than the vacuum. Finally, we compare our formalism to the formalism of coherent states, and highlight their differences.

On Markovian cocycle perturbations in classical and quantum probability

We introduce Markovian cocycle perturbations of the groups of transformations associated with classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. The Markovian cocycle perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed into the sum of two parts. One part in the decomposition is associated with a deterministic stochastic process lying in the past of the initial process, while another part is associated with the noise isomorphic to the initial one. This construction can be considered as some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increments, the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations has been constructed. Using this model, we construct Markovian cocycles transforming the Gaussian state ρ to the Gaussian states equivalent to ρ.

Time operators, innovations and approximations

We present a new approach to the spectral analysis and prediction of such complex systems for which the time evolution is described by a semigroup of operators. This approach is based on an extended concept of time operator and can be interpreted as a shift representation of dynamical systems. The time operator method includes the multiresolution analysis of wavelets as a particular case but can also be applied for a substantially larger class of dynamical systems. Among the examples where shift representation have been explicitly derived are exact endomorphisms, the diffusion equation, generalized shifts associated with the Haar or Faber–Schauder basis and some classes of stochastic processes.

Additive noise, Weibull functions and the approximation of psychometric functions

The Weibull function is frequently chosen to define psychometric functions. Tyler and Chen (Vis. Res. 40 (2000) 3121) criticised the high-threshold postulate implied by the Weibull function and argued that this function implies the assumption of multiplicative noise. It will be shown in this paper that in fact the Weibull function is compatible with the assumption of additive noise, and that the Weibull function may be generalised to the case of detection not being high threshold. The derivations rest, however, on a representation of sensory activity lacking a satisfying degree of generality. Therefore, a more general representation of sensory activity in terms of stochastic processes will be suggested, with detection being defined as a level-crossing process, containing the original representation as a special case. Two classes of stochastic processes will be considered: one where the noise is assumed to be additive, stationary Gaussian, and another resulting from cascaded Poisson processes, representing a form of multiplicative noise. While Weibull functions turn out to approximate well psychometric functions generated by both types of stochastic processes, it also becomes obvious that there is no simple interpretation of the parameters of the fitted Weibull functions. Moreover, corresponding to Tyler and Chen’s discussion of the role of multiplicative noise particular sources of this type of noise will be considered and shown to be compatible with the Weibull. It is indicated how multiplicative noise may be defined in general; however, it will be argued that in the light of certain empirical data the role of this type of noise may be negligible in most detection tasks.

An Ito formula for domain-valued processes driven by stochastic flows

We consider a natural class of stochastic processes taking values in the space of smoothly bounded domains in ℝ n with compact closure. These processes are generated by stochastic flows on ℝ n which are obtained as the solutions of stochastic differential equations on ℝ n . We establish an Ito formula for smooth domain functionals, applied to processes in this class.

A class of interacting marked point processes: rate of convergence to equilibrium

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

A class of interacting marked point processes: rate of convergence to equilibrium

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

Margrabe's option to exchange in a paretian-stable subordinated market

This paper derives the formula for a European option to exchange one asset for another provided that the underlying asset price is logstable. Using the subordination principle of Feller [1] first applied by Hurst, Platen and Rachev [2] to option pricing, we can extend former results of Margrabe [3] to a richer class of stochastic processes which are able to better capture empirical asset return distributions. The obtained option price can be used as a building block in the context of real option valuation.

Processus complexe stochastique non standard en mécanique

We introduce a complex and nonstandard stochastic process, nondiffusive and nonlocal, therefore very different from classical stochastic processes. This process is corresponding at trajectories in a new type of fiber space. Then we define on this fiber space a complex analytical mechanics analogous to classical mechanics, but based on a new “principe de moindre action” and on the complex minplus analysis defined in [1]. This approach can given new models in mechanics and in particular in quantum mechanics.

The case for modeling correlation in manufacturing systems

Manufacturing-related models have traditionally made independence assumptions on associated stochastic processes in order to achieve tractability of analytical models and simplify Monte Carlo models. This paper aims to alert users to potential deleterious implications stemming from unfounded independence assumptions in traditional stochastic models of manufacturing systems. Specifically, it demonstrates the dramatic impact that appreciable autocorrelations can have on manufacturing performance measures through a preliminary study of prediction errors incurred in ignoring dependence. To this end, the study compared performance measures of common manufacturing models with renewal components to their autocorrelated counterparts, drawn from the TES (Transform-Expand-Sample) class. TES models constitute a versatile class of stochastic processes, designed to capture empirical distributions and autocorrelations, simultaneously, and as such, are suitable for both Monte Carlo simulation and analytical modeling of autocorrelated time series. A brief overview of simple TES processes and their generation algorithms is also included.

Modeling quantum measurement probability as a classical stochastic process.

The time-dependent measurement probabilities for the simple two-state quantum oscillator seem to invite description as a classical two-state stochastic process. It has been shown that such a description cannot be achieved using a Markov process. Constructing a more general non-Markov process is a challenging task, requiring as it does the proper generalizations of the Markovian Chapman-Kolmogorov and master equations. Here we describe those non-Markovian generalizations in some detail, and we then apply them to the two-state quantum oscillator. We devise two non-Markovian processes that correctly model the measurement statistics of the oscillator, we clarify a third modeling process that was proposed earlier by others, and we exhibit numerical simulations of all three processes. Our results illuminate some interesting though widely unappreciated points in the theory of non-Markovian stochastic processes. But since quantum theory does not tell us which one of these quite different modeling processes "really" describes the behavior of the oscillator, and also since none of these processes says anything about the dynamics of other (noncommuting) oscillator observables, we can see no justification for regarding any of these processes as being fundamentally descriptive of quantum dynamics. (c) 2001 American Institute of Physics.

Estimation de la période d'un processus à temps continu à représentation autorégressive

We consider the class of stochastic processes which admit an autoregressive representation. Their mean function is periodic and we construct an estimator of the period based on the “minimum distance”. Under regularity conditions we show its consistence and give its asymptotic distribution.

Continuous quantum measurement of two coupled quantum dots using a point contact: A quantum trajectory approach

We obtain the finite-temperature unconditional master equation of the density matrix for two coupled quantum dots (CQD) when one dot is subjected to a measurement of its electron occupation number using a point contact (PC). To determine how the CQD system state depends on the actual current through the PC device, we use the so-called quantum trajectory method to derive the zero-temperature conditional master equation. We first treat the electron tunneling through the PC barrier as a classical stochastic point process (a quantum-jump model). Then we show explicitly that our results can be extended to the quantum-diffusive limit when the average electron tunneling rate is very large compared to the extra change of the tunneling rate due to the presence of the electron in the dot closer to the PC. We find that in both quantum-jump and quantum-diffusive cases, the conditional dynamics of the CQD system can be described by the stochastic Schr\"{o}dinger equations for its conditioned state vector if and only if the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements.

Classical analogs of quasifree quantum stochastic processes given by stochastic states of the quantized electromagnetic field

The mathematical model describing a light beam prepared in an arbitrary quantum optical state is a quasifree quantum stochastic process on the C* algebra of the canonical commutatation relations. For such quantum stochastic processes the concept of stochastic states is introduced. Stochastic quantum states have a classical analog in the following sense: If the light beam is prepared in a stochastic state, one can construct a generalized classical stochastic process, such that the distributions of the quantum observables and the classical random variables coincide. A sufficient algebraic condition for the stochasticity of a quantum state is formulated. The introduced formalism generalizes the Wigner representation from a single field mode to a continuum of modes. For the special case of a single field mode the stochasticity condition provides a new criterion for the positivity of the Wigner function related to the given state. As an example the quantized eletromagnetic field in empty space at temperature T=0 is discussed. It turns out that the corresponding classical stochastic process is not a white noise but a colored noise with a linearly increasing spectrum.

The case for modeling correlation in manufacturing systems

Manufacturing-related models have traditionally made independence assumptions on associated stochastic processes in order to achieve tractability of analytical models and simplify Monte Carlo models. This paper aims to alert users to potential deleterious implications stemming from unfounded independence assumptions in traditional stochastic models of manufacturing systems. Specifically, it demonstrates the dramatic impact that appreciable autocorrelations can have on manufacturing performance measures through a preliminary study of prediction errors incurred in ignoring dependence. To this end, the study compared performance measures of common manufacturing models with renewal components to their autocorrelated counterparts, drawn from the TES (Transform-Expand-Sample) class. TES models constitute a versatile class of stochastic processes, designed to capture empirical distributions and autocorrelations, simultaneously, and as such, are suitable for both Monte Carlo simulation and analytical modeling of autocorrelated time series. A brief overview of simple TES processes and their generation algorithms is also included.

Ergodicity of Band Graph Markov Chains and Their Application to Problems of the Analysis of Existence of a Stationary Regime in a Dynamic Random Multiple Access Communication Network

A concept of the class of band graph Markov chains is introduced. Simple ergodicity criteria for this class of stochastic processes are found. We show how these criteria can be applied to prove the existence of a stationary operation regime in a dynamic access communication network with conflict warning.

Characterization of stochastic processes which stabilize linear companion form systems

The class of stochastic processes is characterized which, as multiplicative noise with large intensity, stabilizes a linear system with companion form d× d-matrix. This includes the characterization of parametric noise which stabilizes the damped inverse pendulum. The proof yields also an expansion of the top Lyapunov exponent in terms of the noise intensity as well as a criterion for a stationary diffusion process permitting a stationary integral and it shows that stabilizing noise averages the Lyapunov spectrum.

Quadratic Bosonic and Free White Noises

We discuss the meaning of renormalization used for deriving quadratic bosonic commutation relations introduced by Accardi and find a representation of these relations on an interacting Fock space. Also, we investigate classical stochastic processes which can be constructed from noncommutative quadratic white noise. We postulate quadratic free white noise commutation relations and find their representation on an interacting Fock space.

Regularity criterion for a mathematical model of the laser

There exists a deep relationship between the nonexplosion conditions for Markov evolution in classical and quantum probability theories. Both of these conditions are equivalent to the preservation of the unit operator (total probability) by a minimal Markov semigroup. In this work, we study the Heisenberg evolution describing an interaction between the chain ofN two-level atoms andn-mode external Bose field, which was considered recently by J. Alli and J. Sewell. The unbounded generator of the Markov evolution of observables acts in the von Neumann algebra. For the generator of a Markov semigroup, we prove a nonexplosion condition, which is the operator analog of a similar condition suggested by R. Z. Khas’minski and later by T. Taniguchi for classical stochastic processes. For the operator algebra situation, this condition ensures the uniqueness and conservativity of the quantum dynamical semigroup describing the Markov evolution of a quantum system. In the regular case, the nonexplosion condition establishes a one-to-one relation between the formal generator and the infinitesimal operator of the Markov semigroup.