2018/12/13 by Bertrand Lods, Lods, Bertrand, Mustapha Mokhtar‐Kharroubi +4
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Boundary (topology) #Bounded function #Combinatorics #Compact space #Ergodic theory #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Invariant (physics) #Invariant measure #Irreducibility #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Stochastic processes and financial applications #math-ph #math.AP #math.MP
paper · pdf · doi:10.48550/arxiv.1812.05397
This preprint supersedes the previous version. In version1, a gap was contained in Lemma A.11. We corrected Lemma A.11 which results now in a new and different kind of result for Theorem 5.1 covering the diffuse case. The main existence result (Theorem 5.6) has been corrected under some additional condition on the accomodation coefficient
openalex publication_date 2018/12/13 · arxiv created 2019/04/06 · arxiv updated 2019/04/09 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
This paper deals with collisionless transport equations in bounded open domains Ω⊂ \Rd (d≥ 2) with C1 boundary ∂ Ω, orthogonally invariant velocity measure \bmm(\d v) with support V⊂ \Rd and stochastic partly diffuse boundary operators H relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C0-semigroups ( UH(t)) t≥ 0 on % L1(Ω× V,\d x ⊗ \bmm(\d v)). We give a general criterion of irreducibility of % ( UH(t)) t≥ 0 and we show that, under very natural assumptions, if an invariant density exists then ( UH(t)) t≥ 0 converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then ( UH(t)) t≥ 0 is sweeping in the sense that, for any density φ, the total mass of UH(t)φ concentrates near suitable sets of zero measure as t→ +∞ . We show also a general weak compactness theorem of interest for the existence of invariant densities. This theorem is based on several results on smoothness and transversality of the dynamical flow associated to ( UH(t)) t≥ 0.