Introduction to Stochastic Calculus with Applications

Klebaner, Fima C.

doi:10.1142/p110

Cited by 185 publications

(166 citation statements)

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“…Equations of the form (35) are typically referred to as Langevin equations and yield solutions that should be interpreted in a statistical sense (Klebaner 1998). It is well known that the integration of white noise,  , yields the Weiner process,  (a scaling limit of a random walk).…”

Section: Damped Driven Evolutionmentioning

confidence: 99%

Dynamical Evolution of Multi-Resonant Systems: The Case of Gj 876

Batygin

Deck

Holman

2015

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The GJ 876 system was among the earliest multi-planetary detections outside of the Solar System, and has long been known to harbor a resonant pair of giant planets. Subsequent characterization of the system revealed the presence of an additional Neptune mass object on an external orbit, locked in a three body Laplace mean motion resonance with the previously known planets. While this system is currently the only known extrasolar example of a Laplace resonance, it differs from the Galilean satellites in that the orbital motion of the planets is known to be chaotic. In this work, we present a simple perturbative model that illuminates the origins of stochasticity inherent to this system and derive analytic estimates of the Lyapunov time as well as the chaotic diffusion coefficient. We then address the formation of the multi-resonant structure within a protoplanetary disk and show that modest turbulent forcing in addition to dissipative effects is required to reproduce the observed chaotic configuration. Accordingly, this work places important constraints on the typical formation environments of planetary systems and informs the attributes of representative orbital architectures that arise from extended disk-driven evolution.

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Section: Damped Driven Evolutionmentioning

confidence: 99%

Dynamical Evolution of Multi-Resonant Systems: The Case of Gj 876

Batygin

Deck

Holman

2015

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“…In this case, notice that ∆(t) is a Gaussian process with mean 0 and variance σ 2 t 0 e −2rs ds. If we can prove that (3.12) then by Lemma 3.5 it follows that From Theorem 7.32 of Klebaner [6] , we see that X i e −rσi ≥ u (3.13) and Theorem 3.1 is proved.…”

Section: Proof Of Theorem 21mentioning

confidence: 94%

The Finite-time Ruin Probability for the Jump-Diffusion Model with Constant Interest Force

Jiang

Yan

2006

Acta Mathematicae Applicatae Sinica, English Series

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“…46 by u m (t) and average based on Ito calculus [48], we can derive the following differential equation…”

Section: Langevin Equation For the Velocitymentioning

confidence: 99%

Stochastic Shell Model for Turbulent Mixing of Multiple Scalars with Mean Gradients and Differential Diffusion

Xia

Vaithianathan

Collins

2010

Flow Turbulence Combust

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In this paper, we develop a shell model for the velocity and scalar concentrations that, by design, is consistent with the eddy damped quasi-normal Markovian (EDQNM) model for multiple mixing scalars. We review the realizable form of the EDQNM model derived by Ulitsky and Collins (J Fluid Mech 412:303-329, 2000), which forms the basis for the shell model. The equations governing the velocity and scalar within each shell are stochastic ordinary differential equations with drift and diffusion terms chosen so that the velocity variance, velocity-scalar cross correlations, and scalar-scalar cross correlations within each shell precisely match the EDQNM model predictions. Consequently, shell averages can be thought of as a representation of the discrete three-dimensional spectrum. An advantage the shell model has over the original EDQNM equations is that the sum of each realization over the shells is a model for the fine-grained, joint velocity/scalar probability density function (PDF). Indeed, this provides some of the motivation for the development of the model. We cannot exploit this feature in the present study of the mixing of two scalars with uniform mean gradients, as the PDF is a joint Gaussian throughout (and hence the correlation matrix completely defines the distribution). The model is capable of predicting Lagrangian correlation functions This paper is dedicated to my colleague, Professor Stephen B. Pope, on the occasion of his 60th birthday. Prof. Pope's pioneering research into the use of probability density function (PDF) methods for chemically reacting turbulent flows has served as inspiration to a generation of scientists and engineers, and indeed motivates the present development of a multi-scale turbulent mixing model.690 Flow Turbulence Combust (2010) 85:689-709for the scalar, scalar dissipation and velocity. We find the predictions of the model are in good qualitative agreement with direct numerical simulations by Yeung (J Fluid Mech 427:241-274, 2001). Eventually we will apply the shell model to scalars that are initially highly non-Gaussian (e.g., double delta function) and observe the relaxation towards a Gaussian. As the shell model contains information on the spectral distribution of the scalar field, the relaxation rate will depend upon the length and time scales of the turbulence and the scalar fields, as well as the molecular diffusivities of the species. The full capabilities of the PDF predictions of the model will be the subject of a future publication.

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Introduction to Stochastic Calculus with Applications

Cited by 185 publications

References 0 publications

Dynamical Evolution of Multi-Resonant Systems: The Case of Gj 876

Dynamical Evolution of Multi-Resonant Systems: The Case of Gj 876

The Finite-time Ruin Probability for the Jump-Diffusion Model with Constant Interest Force

Stochastic Shell Model for Turbulent Mixing of Multiple Scalars with Mean Gradients and Differential Diffusion

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