On Second Order Elliptic Equations with a Small Parameter

Freidlin, Mark; Hu, Wenqing

doi:10.1080/03605302.2013.812658

Cited by 5 publications

(4 citation statements)

References 19 publications

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“…The first term in the brackets on the right hand side is equal to zero by (4). Therefore, the whole expression can be made smaller than η/3 by selecting a sufficiently small δ.…”

Section: The Theorem On the Convergence Of The Processesmentioning

confidence: 99%

“…The process Y x t defined by this generator spends a positive proportion of time in d k , akin to a sticky one-dimensional Brownian motion.The problem studied in this paper can be considered as one concerning the long time influence of a small non-degenerate perturbation (∆/2) of a degenerate diffusion (operator L). A related problem was studied in [4]. There, the diffusion matrix of operator L is assumed to be smooth and have rank d − 1 outside of the domains D 1 , ..., D n , and full rank inside the domains.…”

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confidence: 99%

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On diffusions in media with pockets of large diffusivity

Freidlin

Koralov

Wentzell

2019

Probab. Theory Relat. Fields

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We consider diffusion processes in media with pockets of large diffusivity. The asymptotic behavior of such processes is described when the diffusion coefficients in the pockets tend to infinity. The limiting process is identified as a diffusion on the space where each of the pockets is treated as a single point, and certain conditions on the behavior of the process on the boundary of the pockets are imposed. Calculation of various probabilities and expectations related to the limiting process leads to new initial-boundary (and boundary) problems for the corresponding parabolic (and elliptic) PDEs.Our goal is to show that X x,ε t converge, on an appropriate state space, to a limiting family of processes Y x t when ε ↓ 0. First, let us give an intuitive description of the behavior of X x,ε t for small ε. For a small δ > 0, letAssume that x ∈ U, in which case X x,ε t moves as a Brownian motion until it reaches ∂D k for some k. Once the process reaches ∂D k , it will reach D −δ k very soon, and will then move very fast in the interior of D k due to the large parameter ε −1 at L. Next, we need to understand how the process exits D +δ k (if ε is small, the process will reach ∂D k and then go back to D −δ k many times prior to exiting D +δ k ). We will argue that the distribution of the exit point from D +δ k is nearly uniform with respect to the (d − 1)-dimensional volume measure (i.e., the measure corresponding to the volume form on ∂D +δ k ) if δ and ε are small (a small δ is fixed first, and then ε is taken to zero). Thus, if we disregard the time spent inside D k , the process gets almost immediately re-distributed along ∂D k (or rather ∂D +δ k ) upon reaching ∂D k . Processes with somewhat similar behavior were considered in [6]. Instead of a selfadjoint operator multiplied by a large parameter, which amounts to fast motion inside D k , the re-distribution along the boundary in [6] was due to a trapping mechanism: the motion inside D k was assumed to be nearly deterministic with a large vector field pointing inside the domain. The exit times were exponentially long as ε ↓ 0, and the exit was due to large deviations, while now the exit times will tend to zero.Due to fast mixing inside D k , it is impossible to distinguish between different points of D k without time re-scaling when studying the limiting behavior of X x,ε t . Let U ′ be the metric space obtained from U by identifying all points of ∂D k , turning every ∂D k , k = 1, ..., n, into one point d k . The family of limiting processes Y x t , x ∈ U ′ , will be defined in terms of its generator. Since we expect Y x t to coincide with a Wiener process inside U, the generator coincides with 1 2 ∆ on a certain class of functions. The domain of the generator of the limiting process, however, should be restricted by certain boundary conditions to account for non-trivial behavior of Y x t on the boundary of U and for the delay at the points d k .The generator of the limiting process will be carefully defined in the Section 2, where we also formulate the main resul...

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“…The first term in the brackets on the right hand side is equal to zero by (4). Therefore, the whole expression can be made smaller than η/3 by selecting a sufficiently small δ.…”

Section: The Theorem On the Convergence Of The Processesmentioning

confidence: 99%

mentioning

confidence: 99%

On diffusions in media with pockets of large diffusivity

Freidlin

Koralov

Wentzell

2019

Probab. Theory Relat. Fields

Self Cite

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“…To prove part (b) of this Theorem, we shall make use of a modification of Lemma 3.1 in [28,Chapter 8]. This has been used in the works [23], [22], [26], [9], [10], [20], [19]. First, in Lemma 4.9 we show that the family of processes Y ε t is tight in C [0,T ] (R).…”

Section: Our Proof Intuitively Goes As Followsmentioning

confidence: 99%

On the Long-Time Behavior of a Perturbed Conservative System with Degeneracy

2019

J Theor Probab

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We consider in this work a model conservative system subject to dissipation and Gaussian-type stochastic perturbations. The original conservative system possesses a continuous set of steady states, and is thus degenerate. We characterize the longtime limit of our model system as the perturbation parameter tends to zero. The degeneracy in our model system carries features found in some partial differential equations related, for example, to turbulence problems.

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“…This notion of solution (2.4) is motivated by the Feynman-Kac formula. The process B t considered here is a typical example of Markov processes on manifolds with singularity (such as graphs, see [19], [14], [15], [17], [21], [23], [22]).…”

Section: Fkpp Equation and Its Wavefront Propagationmentioning

confidence: 99%

Wave propagation for reaction-diffusion equations on infinite random trees

Fan,

Hu,

Terlov

2019

Preprint

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The asymptotic wave speed for FKPP type reaction-diffusion equations on a class of infinite random metric trees are considered. We show that a travelling wavefront emerges, provided that the reaction rate is large enough. The wavefront travels at a speed that can be quantified via a variational formula involving the random branching degrees d and the random branch lengths ℓ of the tree T d, ℓ . This speed is slower than that of the same equation on the real line R, and we estimate this slow down in terms of d and ℓ. Our key idea is to project the Brownian motion on the tree onto a one-dimensional axis along the direction of the wave propagation. The projected process is a multi-skewed Brownian motion, introduced by Ramirez [31], with skewness and interface sets that encode the metric structure ( d, ℓ) of the tree. Combined with analytic arguments based on the Feynman-Kac formula, this idea connects our analysis of the wavefront propagation to the large deviations principle (LDP) of the multi-skewed Brownian motion with random skewness and random interface set. Our LDP analysis involves delicate estimates for an infinite product of 2 × 2 random matrices parametrized by d and ℓ and for hitting times of a random walk in random environment. By exhausting all possible shapes of the LDP rate function (action functional), the analytic arguments that bridge the LDP and the wave propagation overcome the random drift effect due to multi-skewness.

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On Second Order Elliptic Equations with a Small Parameter

Cited by 5 publications

References 19 publications

On diffusions in media with pockets of large diffusivity

On diffusions in media with pockets of large diffusivity

On the Long-Time Behavior of a Perturbed Conservative System with Degeneracy

Wave propagation for reaction-diffusion equations on infinite random trees

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