Invariant measures associated to degenerate elliptic operators

Abstract

This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of R n . For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. A uniqueness result for the invariant measure is obtained in the class of all probability measures that are absolutely continuous with respect to Lebesgue's measure. A sufficient condition for the existence of such a measure is also provided.

Border Avoidance: Necessary Regularity for Coefficients and Viscosity Approach

“…The following proposition gathers some useful properties of compact domains of class C 2,1 (cf. [9], [15]). PROPOSITION 2.3.…”

“…Although stated and proven under the assumption of K being a compact domain of class C 2,1 , all our results of Sections 4 and 5 are entirely based on Proposition 2.3 (and not on K itself ). They can be extended, without any particular effort to piecewise C 2,1 -smooth domains (following [9]).…”

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Motivated by the result of invariance of regular-boundary open sets in [9] and multi-stability issues in gene networks, our paper focuses on three closely related aims. First, we give a necessary local Lipschitz-like condition in order to expect invariance of open sets (for deterministic systems).

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Maximal regularity for gradient systems with boundary degeneracy

“…has a unique regular solution X (t, x), which is global in time because the set O is invariant under the flow, as can be checked using the criteria in [11], [3] (here is where we need the C 3 regularity). Therefore, for any λ > 0 and any f ∈ C 2 (O) the problem…”

Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

“…Assume that(8) holds, Ω is of class C 2 and the Fichera condition is not satisfied in an open subset Γ of ∂Ω (in the induced topology), that is,…”