Cultural evolution as a nonstationary stochastic process

We present an individual based model of cultural evolution, where interacting agents are coded by binary strings standing for strategies for action, blueprints for products or attitudes and beliefs. The model is patterned on an established model of biological evolution, the Tangled Nature Model (TNM), where a “tangle” of interactions between agents determines their reproductive success. In addition, our agents also have the ability to copy part of each other's strategy, a feature inspired by the Axelrod model of cultural diversity. Unlike the latter, but similarly to the TNM, the model dynamics goes through a series of metastable stages of increasing length, each characterized by mutually enforcing cultural patterns. These patterns are abruptly replaced by other patterns characteristic of the next metastable period. We analyze the time dependence of the population and diversity in the system, show how different cultures are formed and merge, and how their survival probability lacks, in the model, a finite average life‐time. Finally, we use historical data on the number of car manufacturers after the introduction of the automobile to the market, to argue that our model can qualitatively reproduce the flurry of cultural activity which follows a disruptive innovation. © 2015 Wiley Periodicals, Inc. Complexity 21: 214–223, 2016     

The separability of the Hilbert space generated by a stochastic process

The separability of the Hilbert space generated by a stochastic process is one of the basic assumptions in the time-spectral analysis of stochastic processes. This assumption is either presupposed explicitly or, more often, obtained as a consequence of the assumption of existence of left and right limits of the process for any value of the time parameter. In this paper it is shown that the existence of a left limit only, for each value of the time parameter, is a sufficient condition for the separability of the Hilbert space generated by the process.     

Quantization as geometry in phase space

The geometrical interpretation of quantum mechanics in relativistic phase space proposed by this writer leads, under the sole assumption of a connection from which commutation rules between covariant derivatives ensue that reproduce Heisenberg’s, to an 8-dimensional metric space which contains and generalizesall of first quantization. Some of the new results (to be tested) are: Existence of a maximal proper acceleration; The reality condition on geodesic lines is identical with Bohr-Sommerfeld quantization; The Klein-Gordon and Dirac equations thus generalized give positive mass spectra lying on Regge trajectories; There exists (only in 8 dimensions) a natural supersymmetry (Cartan’s triality); Pure spinors and octonions appear as natural tools for describing particles, and for deeper analyses.     

A phase transition for a stochastic PDE related to the contact process

Summary We consider the one-dimensional heat equation, with a semilinear term and with a nonlinear white noise term. R. Durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions. We show that our equation possesses a phase transition. To be more precise, we assume that the initial function is nonnegative with bounded total mass. If a certain parameter in the equation is small enough, then the solution dies out to 0 in finite time, with probability 1. If this parameter is large enough, then the solution has a positive probability of never dying out to 0. This result answers a question of Durett.Supported by an NSA grant, and by the Army's Mathematical Sciences Institute at Cornell     

Quantum stochastic calculus

An introduction to quantum stochastic calculus in symmetric Fock spaces from the point of view of the theory of stochastic processes. Among the topics discussed are the quantum Itô formula, applications to probability representation of solutions of differential equations, extensions of dynamical semigroups. New algebraic expressions are given for the chronologically ordered exponential functions generated by stochastic semigroups in classical probability theory.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 36, pp. 3–28, 1990.     

The dynamic multilevel assignment problem as a stochastic extremal process

This paper considers the multilevel assignment problem (i.e. the assignment problem where the supply alternatives are ranked in hierarchical levels) under the assumption that the utility components for each pairwise matching are stochastic. A dynamic version of the multilevel stochastic assignment model is developed, where both demand and supply evaluate alternatives according to a stochastic extremal process, i.e. a process where the maximum of a sequence of random variables is taken into account. The probability distributions of the random variables which describe the joint dynamic behaviour of demand and supply are found. It is also shown that the assignment probabilities assume the structure of a nested-logit model.     

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.     

On the path regularity of a stochastic process in a hilbert space,defined by the ito integral

Poisson Processes, by J.F.C. Kingman. Clarendon Press (Oxford University Press), Oxford (1993), 104 pp. $ 39,95 ISBN 0-19-853693-3     

Kitting process in a stochastic assembly system

In small-lot, multi-product, multi-level assembly systems, kitting (or accumulating) components required for assembly plays a crucial role in determining system performance, especially when the system operates in a stochastic environment. This paper analyzes the kitting process of a stochastic assembly system, treating it as an assembly-like queue. If components arrive according to Poisson processes, we show that the output stream departing the kitting operation is a Markov renewal process. The distribution of time between kit completions is also derived. Under the special condition of identical component arrival streams having the same Poisson parameter, we show that the output stream of kits approximates a Poisson process with parameter equal to that of the input stream. This approximately decouples assembly from kitting, allowing the assembly operation to be analyzed separately.     

Complex dynamics in a stochastic internal HIV model

In this paper, we analyze a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modeling. By analyzing the Lyapunov exponent, singular boundary and probability density, some new criteria ensuring stochastic stability, D-bifurcation and P-bifurcation for stochastic internal HIV model are obtained, respectively. Numerical simulation results are given to support the theoretical predictions.     

Chaotic dynamics applied to signal complexity in phase space and in time domain

In the past few years fractal analysis techniques have gained increasing attention in signal and image processing in Medicine. We concentrate on using fractal techniques for analysis of encephalographic data (EEG). Better understanding of general principles that govern discrete dynamics of these signals can help to reveal ‘the signatures' of different physiological and pathological states. Fractal complexity of the signal in time domain, calculated using Higuchi's algorithm, seems to be the simplest method and may also be used in other biomedical applications.     

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.     

Quantum stochastic convolution cocycles III

Every Markov-regular quantum Lévy process on a multiplier C ^*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C ^*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Lévy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C ^*-bialgebra, to locally compact quantum groups and multiplier C ^*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.     

Quantum and non-causal stochastic calculus

In this paper, using the Guichardet space technique, the relationship between Fermion quantum stochastic calculus and non-causal calculus in Segal spaceL ² (H) is discussed, and an anticipating quantum stochastic calculus is naturally given. Subject supported by NSF     

Linearization of controlled stochastic evolution systems in a Hilbert space

We consider the control problem by a nonlinear stochastic evolution equation in a Hilbert space. It is proved that in a small neighborhood of zero a nonlinear problem can be approximated by a linear one with quadratic payoff function. We obtain a relation which allows us to extend the -optimal price to the boundary of the neighborhood.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 528–532, April, 1990.     

A weak stochastic integral in Banach space with application to a linear stochastic differential equation

Cylindrical Wiener processes in real separable Banach spaces are defined, and an approximation theorem involving scalar Wiener processes is given for such processes. A weak stochastic integral for Banach spaces involving a cylindrical Wiener process as integrator and an operator-valued stochastic process as integrand is defined. Basic properties of this integral are stated and proved.A class of linear, time-invariant, stochastic differential equations in real, separable, reflexive Banach spaces is formulated in such fashion that a solution of the equation is a cylindrical process. An existence and uniqueness theorem is proved. A stochastic version of the problem of heat conduction in a ring provides an example.Research supported by National Science Foundation under Grant No. ECS-8005960.     

Probability of a stationary stochastic process staying in a strip

We consider the distribution of the duration of a stochastic process staying in a strip. An example is given of the calculation of the distribution for a Gaussian cosine process.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 128–130, 1988.