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The model of binary aggregation with constant kernel is subjected to stochastic resetting: aggregates of any size explode into monomers at independent stochastic times. These resetting times are Poisson distributed, and the rate o...
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The model of binary aggregation with constant kernel is subjected to stochastic resetting: aggregates of any size explode into monomers at independent stochastic times. These resetting times are Poisson distributed, and the rate of the process is called the resetting rate. The master equation yields a Bernoulli-type equation in the generating function of the concentration of aggregates of any size, which can be solved exactly. This resetting prescription leads to a non-equilibrium steady state for the densities of aggregates, which is a function of the size of the aggregate, rescaled by a function of the resetting rate. The steady-state density of aggregates of a given size is maximized if the resetting rate is set to the quotient of the aggregation rate by the size of the aggregate (minus one).
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An elementary proof of the theorem of Chung-Doob-Meyer on the existence of a progressively measurable modification of a measurable adapted process is given. It is shown how this result can be applied to the construction of the It6...
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An elementary proof of the theorem of Chung-Doob-Meyer on the existence of a progressively measurable modification of a measurable adapted process is given. It is shown how this result can be applied to the construction of the It6 integral with respect to a Brownian motion.
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Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. Colombeau stochastic processes with independent values and stationary Colombeau stochastic processes are studied with special ...
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Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. Colombeau stochastic processes with independent values and stationary Colombeau stochastic processes are studied with special attention paid to the property of translational invariance of generalized functions. Processes with stationary increments are characterized via stationarity of their gradient. Gaussian stationary solutions are analyzed for linear stochastic partial differential equations with generalized constant coefficients in the framework of Colombeau stochastic processes. (C) 2019 Published by Elsevier Inc.
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In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I-X is bounded ...
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In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I-X is bounded and strongly additive in L-E(p) (in particular, if I-X is bounded in L-E(p) and c(0) not subset of E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I-X to be bounded and have bounded semivariation are superfluous. For processes with p-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.
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Quasiconvex stochastic processes are introduced. A characterization of pairs of stochastic processes that can be separated by a quasiconvex stochastic process and a stability theorem for quasiconvex processes are given.
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This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrixvalued ...
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This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrixvalued affine processes in finance, including multi-asset option pricing with stochastic volatility and correlation structures, and fixed-income models with stochastically correlated risk factors and default intensities.
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Functionals of a stochastic process Y(t) model many physical time-extensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y(t) is a normal diffusive process, their st...
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Functionals of a stochastic process Y(t) model many physical time-extensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y(t) is a normal diffusive process, their statistics are obtained as the solution of the celebrated Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y(t) is non-Brownian, e.g. an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate powerlaw or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Levy process whose Laplace exponent is directly related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in numerous experiments on biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both space-and timedependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the FeynmanKac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are of particular relevance for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. An additional aim of this paper is to provide a pedagogical introduction to Levy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated proc
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This paper is dedicated to the analysis of determination of volume and anisotropy field distributions using the complex transverse susceptibility (TS). We consider in our study two-dimensional arrays of particles as magnetic patte...
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This paper is dedicated to the analysis of determination of volume and anisotropy field distributions using the complex transverse susceptibility (TS). We consider in our study two-dimensional arrays of particles as magnetic patterned media. It is shown that the amplitude of excitatory ac field and the interactions between particles play an important role.
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Network growth can be framed as a competition for edges among nodes in the network. As with various other social and physical systems, skill (fitness) and luck (random chance) act as fundamental forces driving competition dynamics...
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Network growth can be framed as a competition for edges among nodes in the network. As with various other social and physical systems, skill (fitness) and luck (random chance) act as fundamental forces driving competition dynamics. In the context of networks, cumulative advantage (CA)-the rich-get-richer effect-is seen as a driving principle governing the edge accumulation process. However, competitions coupled with CA exhibit non-trivial behavior and little is formally known about duration and intensity of CA competitions. By isolating two nodes in an ideal CA competition, we provide a mathematical understanding of how CA exacerbates the role of luck in detriment of skill. We show, for instance, that when nodes start with few edges, an early stroke of luck can place the less skilled in the lead for an extremely long period of time, a phenomenon we call 'struggle of the fittest'. We prove that duration of a simple skill and luck competition model exhibit power-law tails when CA is present, regardless of skill difference, which is in sharp contrast to the exponential tails when fitness is distinct but CA is absent. We also prove that competition intensity is always upper bounded by an exponential tail, irrespective of CA and skills. Thus, CA competitions can be extremely long (infinite mean, depending on fitness ratio) but almost never very intense. The theoretical results are corroborated by extensive numerical simulations. Our findings have important implications to competitions not only among nodes in networks but also in contexts that leverage socio-physical models embodying CA competitions.
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We study the final state of a susceptible-infected-susceptible (SIS) process whose running time is an exponentially distributed random variable. The population in which the spreading evolves is assumed to be homogeneously mixed. W...
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We study the final state of a susceptible-infected-susceptible (SIS) process whose running time is an exponentially distributed random variable. The population in which the spreading evolves is assumed to be homogeneously mixed. We show that whenever the state dependent normalized infection rates are on average smaller than the corresponding curing rates, the final prevalence of the process vanishes in the large population size limit, irrespectively of the mean running time of the process. We show how this statement implies similar results concerning the time evolution of the SIS and the modified SIS processes as well as the steady state of the modified SIS process. In the case of the usual SIS model, for suitably low values of the mean running time, the absence of a non-vanishing prevalence is found in the large population size limit, even if the normalized infection rate exceeds the curing rate.
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