Equivalence Between Canonical Gibbs Measures and Stationary Measures for Stochastic Lattice-Gas Model

It is proved that, under appropriate conditions on the jump rate and potential, one- and two-dimensional stochastic lattice-gas models (exclusion process with speed change) have only canonical Gibbs measures as their stationary measures. This extends the previously known result, which treats only a special jump rate and potential.     

Stochastic Nature in Cellular Processes

The importance of stochasticity in cellular processes is increasingly recognized in both theoretical and experimental studies. General features of stochasticity in gene regulation and expression are briefly reviewed in this article, which include the main experimental phenomena, classification, quantization and regulation of noises. The correlation and transmission of noise in cascade networks are analyzed further and the stochastic simulation methods that can capture effects of intrinsic and extrinsic noise are described.     

Correlated Noises of the Stochastic Processes

The combined effect of several noises has been proposed recently. It has displayed that the presence of the correlation between noises can change the properties of the stochastic processes. In this paper, a general effective Fokker-Planck equation (FPE) with correlated noises is derived. The stationary distributions and the moments of two kinetic models under two correlated noise sources are obtained. The effects of the interference of additive and multiplicative noises are analyzed.     

Stochastic Dissipative PDE's and Gibbs Measures

We study a class of dissipative nonlinear PDE's forced by a random force η^{omega( t} , ^x ), with the space variable x varying in a bounded domain. The class contains the 2D Navier–Stokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in t and stationary, short-correlated in time t. In this paper, we confine ourselves to “kick forces” of the form

Quantum States as Probability Measures

We review Misra's representation of quantum states by probability measures on the projective Hilbert space and investigate some properties of that correspondence in detail. Furthermore, the fundamental probabilistic aspect of these results is discussed.     

Filtration of Stochastic Processes Using First-Order Splines

In the present paper, the feasibility of filtration of a stationary stochastic process using first-order splines is examined when the moments of measurements form a Poisson stream of events with a constant intensity. The mean-square filtration error is determined and an approximate algorithm for estimating values of the process in spline nodes is described.     

Stochastic Dynamics of Discrete Curves and Multi-Type Exclusion Processes

This study deals with continuous limits of interacting one-dimensional diffusive systems, arising from stochastic distortions of discrete curves with various kinds of coding representations. These systems are essentially of a reaction-diffusion nature. In the non-reversible case, the invariant measure has in general a non Gibbs form. The corresponding steady-state regime is analyzed in detail, by using a tagged particle together with a state-graph cycle expansion of the probability currents. As a consequence, the constants appearing in Lotka–Volterra equations—which describe the fluid limits of stationary states—can be traced back directly at the discrete level to tagged particle cycles coefficients. Current fluctuations are also studied and the Lagrangian is obtained via an iterative scheme. The related Hamilton–Jacobi equation, which leads to the large deviation functional, is investigated and solved in the reversible case, just for the sake of checking.     

Information Length Analysis of Linear Autonomous Stochastic Processes

When studying the behaviour of complex dynamical systems, a statistical formulation can provide useful insights. In particular, information geometry is a promising tool for this purpose. In this paper, we investigate the information length for n-dimensional linear autonomous stochastic processes, providing a basic theoretical framework that can be applied to a large set of problems in engineering and physics. A specific application is made to a harmonically bound particle system with the natural oscillation frequency ω, subject to a damping γ and a Gaussian white-noise. We explore how the information length depends on ω and γ, elucidating the role of critical damping γ=2ω in information geometry. Furthermore, in the long time limit, we show that the information length reflects the linear geometry associated with the Gaussian statistics in a linear stochastic process.     

Quantum Versus Stochastic Processes and the Role of Complex Numbers

We argue that the complex numbers are an irreducible object of quantum probability: this can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having the complex phases as primitive ingredient implies that we need to accept nonadditive probabilities. This has the desirable consequence of removing constraints of standard theorems about the possibility of describing quantum theory with commutative variables. Motivated by the formalism of consistent histories and keeping an analogy with the theory of stochastic processes, we develop a (statistical) theory of quantum processes: they are characterized by the introduction of a density matrix on phase space paths (it thus includes phase information) and fully reproduces quantum mechanical predictions. We can write quantum differential equations (in analogy to Langevin equation) that could be interpreted as referring to individual quantum systems. We describe the reconstruction theorem by which a quantum process can yield the standard Hilbert space structure if the Markov property is imposed. We discuss the relevance of our results for the interpretation of quantum theory (a sample space is possible if probabilities are nonadditive) and quantum gravity (the Hilbert space arises here after the consideration of a background causal structure).     

Probability Weights and Measures on Finite Effect Algebras

We study probability weights and measures onfinite effect algebras, thus generalizing the existingtheory for orthomodular posets and orthoalgebras. Ourdevelopment proceeds somewhat more generally in that we study weights and measures associatedwith an antichain in the positive cone of a euclideanvector space with the standard partialordering.     

Probability Evolution and Mean First-Passage Time for Multidimensional Non-Markovian Processes

We investigate a multidimensional system described by a set of stochastic differential equations in which the multiplicative noise is assumed to be an O-U noise. With the help of the projection operator technique, we derive an integrodifferential equation for the probability density and an approximate equation for the mean first-passage time (MFPT).Under some approximation, we obtain an effective Fokker-Planck equation and apply the equation to the single mode laser problem. The concrete calculations of MFPT are made with an important example.