Recent progress on the small parameter exit problem<sup>†</sup>

Day, Martin V.

doi:10.1080/17442508708833440

Cited by 85 publications

(106 citation statements)

References 16 publications

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“…using probabilistic arguments and they show that p may be written (2) p= [ <p(x)dP(x), 'ail where the probability measure P is supported by the set of minimizing points on dQ of the so-called quasi-potential function The proof relies on the large deviations principle. Recently Day [8] has given an exact formula for P, namely he has proved the following formula, given earlier by Matkowsky and Schuss [23], (j^.nRe-'I'ydj.nRe-'f, where R solves a linear first order equation (the main difficulty being to give a sense to this equation which coefficients are not smooth). We refer to [8] for the numerous historical developments about this problem, and we restrict ourself to the particular situation we are interested in, namely the PDE approach.…”

mentioning

confidence: 99%

“…Recently Day [8] has given an exact formula for P, namely he has proved the following formula, given earlier by Matkowsky and Schuss [23], (j^.nRe-'I'ydj.nRe-'f, where R solves a linear first order equation (the main difficulty being to give a sense to this equation which coefficients are not smooth). We refer to [8] for the numerous historical developments about this problem, and we restrict ourself to the particular situation we are interested in, namely the PDE approach. From a purely PDE point of view, Kamin [ 19] has proved that ue converges to a constant.…”

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confidence: 99%

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Perturbed Dynamical Systems with an Attracting Singularity and Weak Viscosity Limits in Hamilton-Jacobi Equations

Perthame¹

1990

Transactions of the American Mathematical Society

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Abstract.We give a new PDE proof of the Wentzell-Freidlin theorem concerning small perturbations of a dynamical system ue = tp on dSl.We prove that, if b has a single attractive singular point, ue converges uniformly on compact subsets of Í2 , and with an exponential decay, to a constant p., and we determine p . We also treat the case of Neumann boundary condition. In order to do so, we perform the asymptotic analysis for some ergodic measure which leads to a study of the viscosity limit of a Hamilton-Jacobi equation. This is achieved under very general assumptions by using a weak formulation of the viscosity limits of these equations. Résumé.Nous donnons une nouvelle preuve, par des méthodes EDP, du théorème de Wentzell-Freidlin concernant les petites perturbations d'un système dynamique: Leue = - §Awe -b-Vue = 0 dans Q, ue = tp sur dSï.

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confidence: 99%

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confidence: 99%

Perturbed Dynamical Systems with an Attracting Singularity and Weak Viscosity Limits in Hamilton-Jacobi Equations

Perthame¹

1990

Transactions of the American Mathematical Society

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“…Optical rays in an inhomogeneous medium may focus, bounce off caustics, and cross one another. The same turns out to be true for optimal trajectories, if extended sufficiently far from an attractor [4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 81%

The birth of a cusp: The unfolding of a ‘boundary catastrophe’

Maier

Stein

2000

AIP Conference Proceedings

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Abstract. We study a phenomenon, analogous to a phase transition, that takes place in the theory of noise-activated transitions between attractors. Large fluctuations away from any attractor tend to be concentrated along certain optimal trajectories, and the flow field of optimal trajectories may contain focal points, or cusps. Optimal trajectories may be viewed as Hamiltonian trajectories, and a cusp may be viewed as a cusp catastrophe in the Lagrangian manifold that the optimal trajectories trace out. As the parameters of a noise-driven system are changed, a cusp may emerge from a saddle point of the underlying deterministic dynamics. This corresponds to a cusp catastrophe being formed at the boundary of the Lagrangian manifold, and moving inward. It is possible to find a nonpolynomial normal form that unfolds the corresponding 'boundary catastrophe', in a space of higher dimensionality. Just as the quartic normal form for a cusp catastophe resembles the scaling form for a classical (Ginzburg-Landau) phase transition, so the normal form for a boundary catastrophe resembles the scaling form for a nonclassical phase transition. Both have continuously varying exponents.

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“…The success of [25] prompted several efforts to put those results on a rigorous basis: [19], [6], [29]. These developments all assumed the nontangential case, n · b < 0.…”

Section: Boundary Local Time Formulationmentioning

confidence: 99%

“…These developments all assumed the nontangential case, n · b < 0. In [5] the probabilistic approach of [6] was refined for use in the context of characteristic boundaries. This is the method we describe in this section.…”

Section: Boundary Local Time Formulationmentioning

confidence: 99%