This work has two objectives. The first is to begin a mathematical formalism appropriate to treating particles which only interact with each otherindirectly due to hypothesized memory effects in a stochastic medium. More specifically we treat a situation in which a sequence of particles consecutively passes through a region (e.g., a measuring apparatus) in such a way that one particle leaves the region before the next one enters. We want to study a situation in which a particle may interact (...) with other particles that previously passed through the system via disturbances made in the region by these previous particles.Second, we apply the type of stochastic process appearing in this context to the stochastic interpretation of quantum mechanics to obtain a modified version of this interpretation. This version is free of many of the criticisms made against the stochastic interpretation of quantum mechanics. (shrink)

We first define a class of processes which we call regular quantum Markov processes. We next prove some basic results concerning such processes. A method is given for constructing quantum Markov processes using transition amplitude kernels. Finally we show that the Feynman path integral formalism can be clarified by approximating it with a quantum stochastic process.

The paper is concerned with the existence and asymptotic character of the nonlinear boundary value problemdG/dt=F ¦ –¦dF/dt=gG =k 1,G=k 2 as ¦– ¦ o+ The discussion is related to the problem of particle-number fluctuations in the theory of cosmic radiation andG andF denote respectively the probability generating functions for the electron distribution in an electron-initiated and a photon-initiated shower.A solution of the system satisfying the boundary conditions is constructed so that specified limiting conditions are fulfilled.

An alternative model for tunneling processes, based on the capability of the telegrapher's equation to describe stochastic processes, is able to account for delay time results of an optical experiment at the microwave scale, where superluminal behaviors have been evidenced.

There is some irony, and perhaps a bit of gallows humor, in opening a paper in this volume with the claim that “applied ethics” is a misnomer. Yet that claim is true in the following sense. What we need for most of the issues that have sparked the contemporary resurgence of moral and political theory is not the application of ethics as we know it, but the revamping of ethics to make it relevant to the issues we face. It is (...) in our concern with major policy programs that ethics and political philosophy are most commonly rejoined to become a unified enquiry after a nearly complete separation through most of this century. Yet, ethical theories may be shaken to their foundations by our effort to apply them to policy problems. I do not propose to revamp ethics here, but only to show that much ethical theory cannot readily be applied to major policy problems. There are at least three important characteristics of major policy issues in general that may give traditional moral theories difficulties. First, such issues can generally be handled only by institutional intervention; they commonly cannot be resolved through uncoordinated individual action. Theories formulated at the individual level must therefore be recast to handle institutional actions and possibilities. Second, major policy issues typically have complicating strategic interactions between individuals at their bases. Third, they are inherently stochastic in the important sense that they affect large numbers with more or less determinable probabilities. C. H. Waddington calls such issues instances of “the problem of the ethics of stochastic processes.”. (shrink)

In an earlier paper by one of us [K.-E. Hellwig (1981)], elements of discrete quantum stochastic processes which arise when the classical probability space is replaced by quantum theory have been considered. In the present paper a general formulation is given and its properties are compared with those of classical stochastic processes. Especially, it is asked whether such processes can be Markovian. An example is given and similarities to methods in quantum statistical thermodynamics are pointed (...) out. (shrink)

Two different kinds of stochastic processes in Hilbert space used to introduce spontaneous localization into the quantum evolution are investigated. In the processes of the first type, finite changes of the wave function take place instantaneously with a given mean frequency. The processes of the second type are continuous. It is shown that under a suitable infinite frequency limit the discontinuous process transforms itself into the continuous one.

It is shown that: (i) the Onsager-Machlup postulate applies to nonlinear stochastic processes over a time scale that, while being much longer than the correlation times of the random forces, is still much shorter than the time it takes for the nonlinear distortion to become visible; (ii) these are also the conditions for the validity of the generalized Fokker-Planck equation; and (iii) when the fine details of the space-time structure of the stochastic processes are unimportant, the (...) generalized Fokker-Planck equation can be replaced by the ordinary Fokker-Planck equation. (shrink)

The problem of finding optimal sampling schemes has been resolved in two models. The novelty of this study lies in its cost efficiency, specifically, for the applied problems with expensive sampling process. In discussed models, we show that some observations counteract other ones in prediction mechanism. The autocovariance function of underlying process causes mentioned result. Our interesting result is that, although removing neutralizing observations convert sampling scheme to nonredundant case, it causes to worse prediction. A simulation study confirms this matter, (...) too. (shrink)

The measurement of one or more observables can be considered to yield sample points which are in general fuzzy sets. Operationally these fuzzy sample points are the outcomes of calibration procedures undertaken to ensure the internal consistency of a scheme of measurement. By introducing generalized probability measures on σ-semifields of fuzzy events, one can view a quantum mechanical state as an ensemble of probability measures which specify the likelihood of occurrence of any specific fuzzy sample point at some instant. These (...) sample points are the possible outcomes of any infinitely rapid succession of measurements at that instant of any sequence of observables of the system. (shrink)

A formalism developed in previous papers for the description of continual observations of some quantities in the framework of quantum mechanics is reobtained and generalized, starting from a more axiomatic point of view. The statistics of the observations of continuous state trajectories is treated from the beginning as a generalized stochastic process in the sense of Gel'fand. An effect-valued measure and an operation-valued measure on the σ-algebra generated by the cylinder sets in the space of trajectories are introduced. The (...) properties of the characteristic functional for the “operation-valued stochastic process” are discussed and, through a suitableansatz, a significant class of such processes is explicitly constructed, which contains the examples of the preceding papers as particular cases. (shrink)

The idea that quantum randomness can be reduced to randomness of classical fields (fluctuating at time and space scales which are essentially finer than scales approachable in modern quantum experiments) is rather old. Various models have been proposed, e.g., stochastic electrodynamics or the semiclassical model. Recently a new model, so called prequantum classical statistical field theory (PCSFT), was developed. By this model a “quantum system” is just a label for (so to say “prequantum”) classical random field. Quantum averages can (...) be represented as classical field averages. Correlations between observables on subsystems of a composite system can be as well represented as classical correlations. In particular, it can be done for entangled systems. Creation of such classical field representation demystifies quantum entanglement. In this paper we show that quantum dynamics (given by Schrödinger’s equation) of entangled systems can be represented as the stochastic dynamics of classical random fields. The “effect of entanglement” is produced by classical correlations which were present at the initial moment of time, cf. views of Albert Einstein. (shrink)

I develop a notion of nonlinear stochastic integrals for hyperfinite Lévy processes and use it to find exact formulas for expressions which are intuitively of the form $\sum_{s=0}^t\phi(\omega,dl_{s},s)$ and $\prod_{s=0}^t\psi(\omega,dl_{s},s)$ , where l is a Lévy process. These formulas are then applied to geometric Lévy processes, infinitesimal transformations of hyperfinite Lévy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic (...) calculus for processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques. (shrink)

With the rapid development of sensor technology for automated driving applications, the fusion, analysis, and application of multimodal data have become the main focus of different scenarios, especially in the development of mobile edge computing technology that provides more efficient algorithms for realizing the various application scenarios. In the present paper, the vehicle status and operation data were acquired by vehicle-borne and roadside units of electronic registration identification of motor vehicles. In addition, a motion model and an identification system for (...) the single-vehicle lane-change process were established by mobile edge computing and self-organizing feature mapping. Two scenarios were modeled and tested: lane change with no vehicles in the target lane and lane change with vehicles in the target lane. It was found that the proposed method successfully identified the stochastic parameters in the process of driving trajectory simulation, and the standard deviation between simulation and the measured results obeyed a normal distribution. The proposed methods can provide significant practical information for improving the data processing efficiency in automated driving applications, for solving single-vehicle lane-change applications, and for promoting the formation of a closed loop from sensing to service. (shrink)

We introduce a framework that can be used to model both mathematics and human reasoning about mathematics. This framework involves stochastic mathematical systems (SMSs), which are stochastic processes that generate pairs of questions and associated answers (with no explicit referents). We use the SMS framework to define normative conditions for mathematical reasoning, by defining a “calibration” relation between a pair of SMSs. The first SMS is the human reasoner, and the second is an “oracle” SMS that can (...) be interpreted as deciding whether the question–answer pairs of the reasoner SMS are valid. To ground thinking, we understand the answers to questions given by this oracle to be the answers that would be given by an SMS representing the entire mathematical community in the infinite long run of the process of asking and answering questions. We then introduce a slight extension of SMSs to allow us to model both the physical universe and human reasoning about the physical universe. We then define a slightly different calibration relation appropriate for the case of scientific reasoning. In this case the first SMS represents a human scientist predicting the outcome of future experiments, while the second SMS represents the physical universe in which the scientist is embedded, with the question–answer pairs of that SMS being specifications of the experiments that will occur and the outcome of those experiments, respectively. Next we derive conditions justifying two important patterns of inference in both mathematical and scientific reasoning: (i) the practice of increasing one’s degree of belief in a claim as one observes increasingly many lines of evidence for that claim, and (ii) abduction, the practice of inferring a claim’s probability of being correct from its explanatory power with respect to some other claim that is already taken to hold for independent reasons. (shrink)

Scientists often think of the world as a dynamical system, a stochastic process, or a generalization of such a system. Prominent examples of systems are the system of planets orbiting the sun or any other classical mechanical system, a hydrogen atom or any other quantum–mechanical system, and the earth’s atmosphere or any other statistical mechanical system. We introduce a general and unified framework for describing such systems and show how it can be used to examine some familiar philosophical questions, (...) including the following: how can we define nomological possibility, necessity, determinism, and indeterminism; what are symmetries and laws; what regularities must a system display to make scientific inference possible; how might principles of parsimony such as Occam’s Razor help when we make such inferences; what is the role of space and time in a system; and might they be emergent features? Our framework is intended to serve as a toolbox for the formal analysis of systems that is applicable in several areas of philosophy. (shrink)

The text is concerned with a class of two-sided stochastic processes of the form. Here is a two-sided Brownian motion with random initial data at time zero and is a function of. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when is a jump process. Absolute continuity of under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue (...) measure,, and on with we verify i.e. where the product is taken over all coordinates. Here is the divergence of with respect to the initial position. Crucial for this is the temporal homogeneity of in the sense that,, where is the trajectory taking the constant value. By means of such a density, partial integration relative to a generator type operator of the process is established. Relative compactness of sequences of such processes is established. (shrink)

The aim of this paper is to study the monotonicity properties with respect to the probability distribution of the state processes, of optimal decisions in bandit decision problems. Orderings of dynamic discrete projects are provided by extending the notion of stochastic dominance to stochastic processes.

In a small handful of papers in theoretical population genetics, John Gillespie (2000a, 2000b, 2001) argues that a new stochastic process he calls "genetic draft" is evolutionarily more significant than genetic drift. This case study of chance in evolution explores Gillespie's proposed stochastic evolutionary force and sketches the implications of Gillespie's argument for philosophers' explorations of genetic drift.

Over the last 40 years or so, there has been an explosion of cultural evolution research in anthropology and archaeology. In each discipline, cultural evolutionists investigate how interactions between individuals translate into group level patterns, with the aim of explaining the diachronic dynamics and diversity of cultural traits. However, while much attention has been given to deterministic processes, we contend that current evolutionary accounts of cultural change are limited because they do not adopt a systematic stochastic approach. First, (...) we show that, in contrast with the intense debates in ecology and population genetics, the importance of stochasticity in evolutionary processes has generated little discussion in the sciences of cultural evolution to date. Second, we speculate on the reasons, both ideological and methodological, why that should be so. Third, we highlight the inadequacy of genetically-inspired stochastic models in the context of cultural evolution modelling, and ask which fundamental stochastic processes might be more relevant to take up. We conclude that the field of cultural evolution would benefit from a stochastic revolution. For that to occur, stochastic models ought to be developed specifically for cultural data and not through a copy-pasting of neutral models from population genetics or ecology. (shrink)

We formulate from first principles a theory of stochastic processes in configuration space. The fundamental equations of the theory are an equation of motion which generalizes Newton's second law and an equation which expresses the condition of conservation of matter. Two types of stochastic motion are possible, both described by the same general equations, but leading in one case to classical Brownian motion behavior and in the other to quantum mechanical behavior. The Schrödinger equation, which is derived (...) here with no further assumption, is thus shown to describe a specific stochastic process. It is explicitly shown that only in the quantum mechanical process does the superposition of probability amplitudes give rise to interference phenomena; moreover, the presence of dissipative forces in the Brownian motion equations invalidates the superposition principle. At no point are any special assumptions made concerning the physical nature of the underlying stochastic medium, although some suggestions are discussed in the last section. (shrink)

Following up on previous results by Falmagne, this paper investigates possible mechanisms explaining how preference relations are created and how they evolve over time. We postulate a preference relation which is initially empty and becomes increasingly intricate under the influence of a random environment delivering discrete tokens of information concerning the alternatives. The framework is that of a class of real-time stochastic processes having interlinked Markov and Poisson components. Specifically, the occurence of the tokens is governed by a (...) Poisson process, while the succession of preference relations is a Markov process. In an example case, the preference relations are the various possible semiorders on the set of alternatives. Asymptotic results are obtained in the form of the limit probabilities of any semiorder. The arguments extend to a much more general situation including interval orders, biorders and partial orders. The results provide (up to a small number of parameters) complete quantitative predictions for panel data of a standard type, in which the same sample of subjects has been asked to compare the alternatives a number of times. (shrink)

Cerebellar Purkinje cells generate two distinct types of spikes, complex and simple spikes, both of which have conventionally been considered to be highly irregular, suggestive of certain types of stochastic processes as underlying mechanisms. Interestingly, however, the interspike interval structures of complex spikes have not been carefully studied so far. We showed in a previous study that simple spike trains are actually composed of regular patterns and single interspike intervals, a mixture that could not be explained by a (...) simple rate-modulated Poisson process. In the present study, we systematically investigated the interspike interval structures of separated complex and simple spike trains recorded in anaesthetized rats, and derived an appropriate stochastic model. We found that: (i) complex spike trains do not exhibit any serial correlations, so they can effectively be generated by a renewal process, (ii) the distribution of intervals between complex spikes exhibits two narrow bands, possibly caused by two oscillatory bands (0.5–1 and 4–8 Hz) in the input to Purkinje cells and (iii) the regularity of regular patterns and single interspike intervals in simple spike trains can be represented by gamma processes of orders, which themselves are drawn from gamma distributions, suggesting that multiple sources modulate the regularity of simple spike trains. (shrink)

The replicator dynamics and Moran process are the main deterministic and stochastic models of evolutionary game theory. The models are connected by a mean-field relationship—the former describes the expected behavior of the latter. However, there are conditions under which their predictions diverge. I demonstrate that the divergence between their predictions is a function of standard techniques used in their analysis and of differences in the idealizations involved in each. My analysis reveals problems for stochastic stability analysis in a (...) broad class of games, demonstrates a novel domain of agreement between the dynamics, and indicates a broader moral for evolutionary modeling. (shrink)

The aim of this paper is to study the monotonicity properties with respect to the probability distribution of the state processes, of optimal decisions in bandit decision problems. Orderings of dynamic discrete projects are provided by extending the notion of stochastic dominance to stochastic processes.

Creativity is generally thought to be the production of things that are novel and valuable. Humans are unique in the extent of their creativity, which plays a central role in innovation and problem solving, as well as in the arts. But what are the cognitive sources of novelty? More particularly, what are the cognitive sources of stochasticity in creative production? I will argue that they belong to two broad categories. One is associative, enabling the selection of goal-relevant ideas that have (...) become activated by happenstance in an unrelated context. The other relies on selection processes that leverage stochastic fluctuations in neural activity. At the same time, I will address a central puzzle, which is to understand how the outputs of stochastic processes can nevertheless generally fall within task constraints. While the components appealed to in the accounts that I offer are well established, the ways in which I combine them together are new. (shrink)

A theoretical framework for the reinforcement learning of aesthetic biases was recently proposed based on brain circuitries revealed by neuroimaging. A model grounded on that framework accounted for interesting features of human aesthetic biases. These features included individuality, cultural predispositions, stochastic dynamics of learning and aesthetic biases, and the peak-shift effect. However, despite the success in explaining these features, a potential weakness was the linearity of the value function used to predict reward. This linearity meant that the learning process (...) employed a value function that assumed a linear relationship between reward and sensory stimuli. Linearity is common in reinforcement learning in neuroscience. However, linearity can be problematic because neural mechanisms and the dependence of reward on sensory stimuli were typically nonlinear. Here, we analyze the learning performance with models including optimal nonlinear value functions. We also compare updating the free parameters of the value functions with the delta rule, which neuroscience models use frequently, vs. updating with a new Phi rule that considers the structure of the nonlinearities. Our computer simulations showed that optimal nonlinear value functions resulted in improvements of learning errors when the reward models were nonlinear. Similarly, the new Phi rule led to improvements in these errors. These improvements were accompanied by the straightening of the trajectories of the vector of free parameters in its phase space. This straightening meant that the process became more efficient in learning the prediction of reward. Surprisingly, however, this improved efficiency had a complex relationship with the rate of learning. Finally, the stochasticity arising from the probabilistic sampling of sensory stimuli, rewards, and motivations helped the learning process narrow the range of free parameters to nearly optimal outcomes. Therefore, we suggest that value functions and update rules optimized for social and ecological constraints are ideal for learning aesthetic biases. (shrink)

We consider a stochastic process which is described by a continuous-time Markov chain on only short time-scales and constrained to conserve a number of hidden quantities on long time-scales. We assume that the transition matrix of the Markov chain is given and the conserved quantities are known to exist, but not explicitly given. To study the stochastic dynamics we propose to use the principle of stationary entropy production. Then the problem can be transformed into a variational problem for (...) a suitably defined “action” and with time-dependent Lagrange multipliers. We show that the stochastic dynamics can be described by a Schrödinger equation, with Lagrange multipliers playing the role of phases, whenever the transition matrix is symmetric or the detailed balance condition is satisfied, the system is not too far from the equilibrium and the number of the conserved quantities is large. (shrink)