On the Neumann problem for PDE’s with a small parameter and the corresponding diffusion processes

Abstract

The diffusion process in a region G ⊂ R 2 governed by the operator L ε = 1 2 u x x + 1 2ε u zz inside the region and undergoing instantaneous co-normal reflection at the boundary is considered. We show that the slow component of this process converges to a diffusion process on a certain graph corresponding to the problem. This allows to find the main term of the asymptotics for the solution of the corresponding Neumann problem in G. The operatorL ε is, up to the factor ε −1 , the result of small perturbation of the operator 1 2 u zz . Our approach works for other operators (diffusion processes) in any dimension if the process corresponding to the non-perturbed operator has a first integral, and the ε-process is non-degenerate on non-singular level sets of this first integral.

SPDEs on narrow domains and on graphs: An asymptotic approach

“…The processZ has been described in terms of its generatorL, which is given by suitable differential operatorsL k within each edge I k = {(x, k) : a k ≤ x ≤ b k } of the graph and by certain gluing conditions at the vertices O i of the graph. More precisely, for each k, the differential operatorL k has the form 5) and the operatorL, acting on functions f defined on the graph Γ, is defined as…”

“…More precisely, it has been proven that for any z ∈ G and any bounded and continuous functional F on C([0, T ]; Γ), with T > 0, lim →0 E z F (Π (·)) =Ē Π(z) F (Z(·)). (1.4) Notice that in [5] the generatorL of the Markov processZ(t) is explicitly described in terms of certain second order differential operators L k , acting in the interior of each edge I k of Γ, and of suitable gluing conditions, given at the vertices of Γ.…”

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

“…such that P |Θ ε t − Θ t | ≤ Cε 3/10 = 1 (32) in finite t. From (29), (30), (31) and (32) we know that T ε X is finite, and thus T ε X ∼ O(ε 9/10 ) and Y ε T ε X ≥ δ. From here we know that whenever X ε 0 ≥ 2δ, the flow will quickly bring the particle to the region Y ≥ δ, and during this process X ε t ≤ 3δ.…”

“…ε k j t (ω)) → EF (Y 0 t (ω))(39)for some j → ∞ and some random element Y 0 t in C [0,T ] (R). We then understand (39) is just saying that Y ε t is weakly-compact under P. We will then make use of Lemma 5.1 in[29]. In fact, Lemma 5.1 in[29] indicates that in order to show weak-compactness of the family of sample paths in Y ε t in C [0,t] (R) under the measure P, it suffices to show, for each δ > 0, weak-compactness of the family of sample pathsY ε,δ t , where Y ε,δ t = Y ε t for σ k−1 ≤ t ≤ τ k , k = 1, 2, ..., N and Y ε,δ t = δ τ k − t τ k − σ k + 2δ t − σ k τ k − σ kfor τ k ≤ t ≤ σ k .…”

On Second Order Elliptic Equations with a Small Parameter

“…If the first integral H(x) has in G ∪ ∂G no critical points, one can describe the lim ε↓0 u ε (x) in the way similar to [10]: One shall introduce a graph G corresponding to the set of connected components of the intersections of the level sets of H(x) within G. A boundary problem on G with appropriate gluing conditions at the vertices can be formulated, and the solution of this problem defines lim ε↓0 u ε (x). If the function H(x) has saddle points inside G, additional branchings in the graph appear.…”