The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and Positive Friction

Herzog, David P.; Hottovy, Scott; Volpe, Giovanni

doi:10.1007/s10955-016-1498-8

Cited by 27 publications

(20 citation statements)

References 21 publications

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“…The term adiabatic elimination is also used in literature to describe this phenomenon. The above result was proven rigorously by Hottovy et al [48] and by Herzog et al [82]. Related results were obtained earlier by Hänggi [19] and by Sancho et al [39].…”

Section: Acknowledgmentssupporting

confidence: 70%

Effective drifts in dynamical systems with multiplicative noise: a review of recent progress

2016

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Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling of these phenomena requires access to their microscopic dynamics, whose time scales are typically much shorter than the observable time scales, there is often need to resort to effective mathematical models such as stochastic differential equations (SDEs). In particular, here we consider effective SDEs describing the behavior of systems in the limits when natural time scales become very small. In the presence of multiplicative noise (i.e. noise whose intensity depends upon the system's state), an additional drift term, called noise-induced drift or effective drift, appears. The nature of this noise-induced drift has been recently the subject of a growing number of theoretical and experimental studies. Here, we provide an extensive review of the state of the art in this field. After an introduction, we discuss a minimal model of how multiplicative noise affects the evolution of a system. Next, we consider several case studies with a focus on recent experiments: the Brownian motion of a microscopic particle in thermal equilibrium with a heat bath in the presence of a diffusion gradient; the limiting behavior of a system driven by a colored noise modulated by a multiplicative feedback; and the behavior of an autonomous agent subject to sensorial delay in a noisy environment. This allows us to present the experimental results, as well as mathematical methods and numerical techniques, that can be employed to study a wide range of systems. At the end we give an application-oriented overview of future projects involving noise-induced drifts, including both theory and experiment.

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Section: Acknowledgmentssupporting

confidence: 70%

Effective drifts in dynamical systems with multiplicative noise: a review of recent progress

2016

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“…An SDE can be derived that governs the dynamics of the state, q m t (here and in the sequel we use a superscript to denote the m dependence), in the limit m → 0 and, when γ (σ if the Stratonovich integral is used) is state-dependent, the limiting equation can be shown to involve an additional drift term that was not present in the original system. This was first derived in [15] and has been studied in numerous subsequent works [16][17][18][19][20][21]. See [18] for further references and discussion.…”

Section: Introductionmentioning

confidence: 93%

Phase Space Homogenization of Noisy Hamiltonian Systems

Birrell

Wehr

2018

Ann. Henri Poincaré

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We study the dynamics of an inertial particle coupled to forcing, dissipation, and noise in the small mass limit. We derive an expression for the limiting (homogenized) joint distribution of the position and (scaled) velocity degrees of freedom. In particular, weak convergence of the joint distributions is established, along with a bound on the convergence rate for a wide class of expected values.Mathematics Subject Classification (2010). 60H10, 82C31.

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“…See references in the recent paper [13], where a formula for the noise-induced drift has been established for a large class of systems in Euclidean space of an arbitrary dimension. See also [26], where some of the assumptions made in [13] are relaxed.…”

Section: Introductionmentioning

confidence: 99%

Small Mass Limit of a Langevin Equation on a Manifold

Birrell¹,

Hottovy²,

Volpe³

et al. 2016

Ann. Henri Poincaré

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Abstract. We study damped geodesic motion of a particle of mass m on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as m → 0, its solutions converge to solutions of a limiting equation which includes a noise-induced drift term. A very special case of the main result presents the Brownian motion on the manifold as a limit of inertial systems.Mathematics Subject Classification (2010). 58J65, 60H10, 82C31.

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The Small-Mass Limit for Langevin Dynamics with Unbounded Coefficients and Positive Friction

Cited by 27 publications

References 21 publications

Effective drifts in dynamical systems with multiplicative noise: a review of recent progress

Effective drifts in dynamical systems with multiplicative noise: a review of recent progress

Phase Space Homogenization of Noisy Hamiltonian Systems

Small Mass Limit of a Langevin Equation on a Manifold

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