Diffusion Processes on Graphs and the Averaging Principle

Freidlin, Mark; Wentzell, Alexander D.

doi:10.1214/aop/1176989018

Cited by 165 publications

(194 citation statements)

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“…It is well known [6,7] that if all critical points of ψ are nondegenerate and no two of them lie on the same level set of ψ, then the functions τ Au converge uniformly to a limitτ u which is constant on the level sets of ψ and satisfies an asymptotic Freidlin problem on the Reeb graph of the function ψ. If Ω is simply connected and ψ has a single (nondegenerate) critical point (in which case either ψ > 0 or ψ < 0 on Ω, and we will assume without loss of generality the former), then we have the explicit formula…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Exit Times of Diffusions with Incompressible Drift

Iyer¹,

Novikov²,

Ryzhik³

et al. 2010

SIAM J. Math. Anal.

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Abstract. Let Ω ⊂ R n be a bounded domain, and for x ∈ Ω let τ (x) be the expected exit time from Ω of a diffusing particle starting at x and advected by an incompressible flow u. We are interested in the question which flows maximize τ L ∞ (Ω) , that is, they are most efficient in the creation of hotspots inside Ω. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow u ≡ 0 maximizes τ L ∞ (Ω) . We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, τ L ∞ (Ω) is maximized by the zero flow on the ball.

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Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Exit Times of Diffusions with Incompressible Drift

Iyer¹,

Novikov²,

Ryzhik³

et al. 2010

SIAM J. Math. Anal.

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“…The case when the fat graph supports a Laplacian of Neumann type is better understood and the limit of all types of spectra as well as of resonances has been worked out [FW93,RS01,KuZ01,EP05,EP07,G08,EP08]. Moreover, convergence of resolvents etc.…”

Section: Introductionmentioning

confidence: 99%

Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds

Exner

Post

2009

J. Phys. A: Math. Theor.

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Abstract. We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schrödinger operators can approximate non-trivial vertex couplings. The latter include not only the δ-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric δ ′ -couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal.

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“…Then one could expect that the law of of the process H(Y ) will converge. However, we cannot obtain a convergence for H(Y ) itself if H is not injective but for a projection of Y on the "orbit space" Γ defined as the space of connected level sets of H. Previous convergence results have been obtained, notably by Freildin and Wentzell in several papers (see the new edition of their book [6] for a review, and the previous articles [9,8]). …”

Section: Introductionmentioning

confidence: 95%

Averaging principle for diffusion processes via Dirichlet forms

Barret

Renesse

2014

Potential Anal

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Abstract. We study diffusion processes driven by a Brownian motion with regular drift in a finite dimension setting. The drift has two components on different time scales, a fast conservative component and a slow dissipative component. Using the theory of Dirichlet form and Mosco-convergence we obtain simpler proofs, interpretations and new results of the averaging principle for such processes when we speed up the conservative component. As a result, one obtains an effective process with values in the space of connected level sets of the conserved quantities. The use of Dirichlet forms provides a simple and nice way to characterize this process and its properties.

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Diffusion Processes on Graphs and the Averaging Principle

Cited by 165 publications

References 0 publications

Exit Times of Diffusions with Incompressible Drift

Exit Times of Diffusions with Incompressible Drift

Approximation of quantum graph vertex couplings by scaled Schrödinger operators on thin branched manifolds

Averaging principle for diffusion processes via Dirichlet forms

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