Constraint Ornstein-Uhlenbeck bridges

Mazzolo, Alain

doi:10.1063/1.5000077

Cited by 19 publications

(20 citation statements)

References 26 publications

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“…Methods like Doob's h-transform [41] were found intractable due to the conditioning on both X and A [42,43]. An alternative method, proposed for a generalized Brownian bridge [44][45][46], could not be extended to excursions, as it is restricted to Gaussian processes. As we show now, the two large-deviation techniques that describe the two tails of f (ξ), allow one to evaluate p A at small and large A.…”

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confidence: 99%

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Airy distribution: Experiment, large deviations, and additional statistics

Agranov

Zilber

Smith

et al. 2020

Phys. Rev. Research

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The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations -the Donsker-Varadhan formalism and the optimal fluctuation method -manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the Ferrari-Spohn distribution, which has previously appeared in other problems, and we identify the reason for this coincidence.

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confidence: 99%

“…This explains the coincidence of the Gaussian fluctuations in this regime, Eqs. ( 17) and (18), with those of a Brownian bridge conditioned on A [45,46].…”

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confidence: 99%

Airy distribution: Experiment, large deviations, and additional statistics

Agranov

Zilber

Smith

et al. 2020

Phys. Rev. Research

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“…The probability that a Brownian motion reaches a level a for the first time at a given time is an event of probability zero. As mentioned in [8,9], conditioning with respect to a set of sample paths of probability zero requires special care. Despite the technical complexities generated by conditioning with such events of zero measure, the resulting diffusion is indeed well defined.…”

Section: Driftless Casementioning

confidence: 99%

“…This is the case of the Brownian bridge, where the constraint of returning to zero (or any another value) at a fixed time has clearly a zero measure. Conditioning a subtle object like a diffusion on events of zero probability is not an harmless task [8,9], but this approach is a fruitful method since it sheds light on the important process studied by Krapivsky and Redner but also makes the simulations easy. The next step is to understand when the conditioning is so strong that it erases the original drift of the diffusion.…”

Section: Introductionmentioning

confidence: 99%

Strongly constrained stochastic processes: the multi-ends Brownian bridge

Larmier¹,

Mazzolo²,

Zoia³

2019

J. Stat. Mech.

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In a recent article, Krapivsky and Redner (J. Stat. Mech. 093208 (2018)) established that the distribution of the first hitting times for a diffusing particle subject to hitting an absorber is independent of the direction of the external flow field. In the present paper, we build upon this observation and investigate when the conditioning on the diffusion leads to a process that is totally independent of the flow field. For this purpose, we adopt the Langevin approach, or more formally the theory of conditioned stochastic differential equations. This technique allows us to derive a large variety of stochastic processes: in particular, we introduce a new kind of Brownian bridge ending at two different final points and calculate its fundamental probabilities. This method is also very well suited for generating statistically independent paths. Numerical simulations illustrate our findings.

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“…Specifically, we analyze the error in the estimation of the particle density in terms of a finite number of computational particles of finite size, a special case of kernel density estimation, with a focus on the bias-variance tradeoff [25,26]. We also analyze the error in the electric field computed from the charge density, showing that certain negative correlations in the density noise lead to properties of the electric field related to the Ornstein-Uhlenbeck bridge [27][28][29], a generalization of the Brownian bridge [30], a Brownian process with boundary conditions at each end. We concentrate on a 1D (one-dimensional) electrostatic (ES) formulation with periodic boundary conditions, with overall charge neutrality and immobile ions, leaving generalizations such as to higher dimensions and electromagnetic models for future work.…”

Section: Introductionmentioning

confidence: 99%

Noise and error analysis and optimization in particle-based kinetic plasma simulations

Evstatiev,

Finn,

Shadwick

et al. 2021

Preprint

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In this paper we analyze the noise in macro-particle methods used in plasma physics and fluid dynamics, leading to approaches for minimizing the total error, focusing on electrostatic models in one dimension. We begin by describing kernel density estimation for continuous values of the spatial variable x, expressing the kernel in a form in which its shape and width are represented separately. The covariance matrix C(x, y) of the noise in the density is computed, first for uniform true density. The band width of the covariance matrix is related to the width of the kernel. A feature that stands out is the presence of constant negative terms in the elements of the covariance matrix both on and off-diagonal. These negative correlations are related to the fact that the total number of particles is fixed at each time step; they also lead to the property ´C(x, y) dy = 0. We investigate the effect of these negative correlations on the electric field computed by Gauss's law, finding that the noise in the electric field is related to a process called the Ornstein-Uhlenbeck bridge, leading to a covariance matrix of the electric field with variance significantly reduced relative to that of a Brownian process.For non-constant density, ρ(x), still with continuous x, we analyze the total error in the density estimation and discuss it in terms of bias-variance optimization (BVO). For some characteristic length l, determined by the density and its second derivative, and kernel width h, having too few particles within h leads to too much variance; for h that is large relative to l, there is too much smoothing of the density. The optimum between these two limits is found by BVO. For kernels of the same width, it is shown that this optimum (minimum) is weakly sensitive to the kernel shape.We repeat the analysis for x discretized on a grid. In this case the charge deposition rule is determined by a particle shape. An important property to be respected in the discrete system is the exact preservation of total charge on the grid; this property is necessary to ensure that the electric field is equal at both ends, consistent with periodic boundary conditions. We find that if the particle shapes satisfy a sum rule, the particle charge deposited on the grid is conserved exactly. Further, if the particle shape is expressed as the convolution of a kernel with another kernel that satisfies the sum rule, then the particle shape obeys the sum rule. This property holds for kernels of arbitrary width, including widths that are not integer multiples of the grid spacing.We show results relaxing the approximations used to do BVO optimization analytically, by doing numerical computations of the total error as a function of the kernel width, on a grid in x. The comparison between numerical and analytical results shows good agreement over a range of particle shapes.We discuss the practical implications of our results, including the criteria for design and implementation of computationally efficient particles that take advantage of the developed theory.

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Constraint Ornstein-Uhlenbeck bridges

Cited by 19 publications

References 26 publications

Airy distribution: Experiment, large deviations, and additional statistics

Airy distribution: Experiment, large deviations, and additional statistics

Strongly constrained stochastic processes: the multi-ends Brownian bridge

Noise and error analysis and optimization in particle-based kinetic plasma simulations

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