A singular 1-D Hamilton–Jacobi equation, with application to large deviation of diffusions

Deng, Xiaoya; Feng, Jian; Liu, Yong

doi:10.4310/cms.2011.v9.n1.a14

Cited by 14 publications

(18 citation statements)

References 7 publications

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“…This approach via the Hamilton-Jacobi equation has been carried out in [23] for Levy processes on R d , systems with multiple time scales and for stochastic equations in infinite dimensions. In [18], the LDP for a diffusion process on (0, ∞) is treated with singular behaviour close to 0.…”

Section: Introductionmentioning

confidence: 99%

Large Deviations for Finite State Markov Jump Processes with Mean-Field Interaction Via the Comparison Principle for an Associated Hamilton–Jacobi Equation

Kraaij

2016

J Stat Phys

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We prove the large deviation principle for the trajectory of a broad class of mean field interacting Markov jump processes via a general analytic approach based on viscosity solutions. Examples include generalized Ehrenfest models as well as Curie-Weiss spin flip dynamics with singular jump rates.The main step in the proof of the large deviation principle, which is of independent interest, is the proof of the comparison principle for an associated collection of Hamilton-Jacobi equations.Additionally, we show that the large deviation principle provides a general method to identify a Lyapunov function for the associated McKean-Vlasov equation. µ n (t) := 1 n i≤n δ σi(t) ,

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Section: Introductionmentioning

confidence: 99%

Large Deviations for Finite State Markov Jump Processes with Mean-Field Interaction Via the Comparison Principle for an Associated Hamilton–Jacobi Equation

Kraaij

2016

J Stat Phys

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“…In the approach to large deviations by [14] and [12], also applied more recently in e.g. [5,10,19], the distributional information of the process at finite n related to the large deviations is encoded in the solutions f n of a class of Hamilton-Jacobi equation f − λH n f = h, λ > 0, h ∈ C b (E n ). Given that the Hamiltonians H n have a natural limiting upper bound H † and lower bound H ‡ , semi-relaxed limits f and f of f n , see (5.1) below, give a sub-solution and super-solution to…”

Section: A General Framework For Hamilton-jacobi Equations With a Boumentioning

confidence: 99%

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

Kraaij

Mahé

2020

Stochastic Processes and their Applications

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We prove Freidlin-Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth-death processes, Galton-Watson trees, epidemic SI models, and prey-predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton-Jacobi equations. The notable feature for these Hamilton-Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton-Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.We start with a collection of preliminary properties of L, L and their Legendre duals.Lemma 2.11. Let {H J } J∈J(E) be a generating set of Hamiltonians.(a) For each x ∈ E the map v → L(x, v) is the convex hull of the Lagrangians v → L J (x, v), i.e. the largest convex function that lies below all L J 's;H ‡ (y, ∇f(y)), establishing (5.3).Step 2: We construct a solution to the differential inclusion (5.2), for which we use Lemma D.4. Note that the Lemma assumes growth bounds on the size of F f , that are not necessarily satisfied.

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“…In the papers [6,8,16] one of the main steps in proving the large deviation principle was proving directly the existence of an operator H such that H ⊆ LIM n H n ; in other words, verifying that, for all (f, g) ∈ H, there are f n ∈ H n such that LIM n f n = f and LIM n H n f n = g (the notion of LIM is introduced in Definition A.4). Here it is hard to follow a similar strategy.…”

Section: A2 Operator Convergence For a Projected Processmentioning

confidence: 99%

Path-space moderate deviations for a Curie–Weiss model of self-organized criticality

Collet¹,

Gorny²,

Kraaij³

2020

Ann. Inst. H. Poincaré Probab. Statist.

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The dynamical Curie-Weiss model of self-organized criticality (SOC) was introduced in [15] and it is derived from the classical generalized Curie-Weiss by imposing a microscopic Markovian evolution having the distribution of the Curie-Weiss model of SOC [5] as unique invariant measure. In the case of Gaussian single-spin distribution, we analyze the dynamics of moderate fluctuations for the magnetization. We obtain a path-space moderate deviation principle via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations. Our result shows that, under a peculiar moderate space-time scaling and without tuning external parameters, the typical behavior of the magnetization is critical.Keywords: moderate deviations · interacting particle systems · mean-field interaction · selforganized criticality · Hamilton-Jacobi equation · perturbation theory for Markov processes 2 Model and main result Notation and definitionsBefore starting with the main contents of the paper, we introduce some notation. We start with the definition of good rate-function and of large deviation principle for a sequence of random variables.Definition 2.1. Let (X n ) n∈N * be a sequence of random variables on a Polish space X . Furthermore, consider a function I : X → [0, ∞] and a sequence (r n ) n∈N * of positive numbers such that

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A singular 1-D Hamilton–Jacobi equation, with application to large deviation of diffusions

Cited by 14 publications

References 7 publications

Large Deviations for Finite State Markov Jump Processes with Mean-Field Interaction Via the Comparison Principle for an Associated Hamilton–Jacobi Equation

Large Deviations for Finite State Markov Jump Processes with Mean-Field Interaction Via the Comparison Principle for an Associated Hamilton–Jacobi Equation

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

Path-space moderate deviations for a Curie–Weiss model of self-organized criticality

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