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Exponential convergence in L p ‐Wasserstein distance for diffusion processes without uniformly dissipative drift
Author(s) -
Luo Dejun,
Wang Jian
Publication year - 2016
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201500351
Subject(s) - mathematics , dissipative system , exponential function , constant (computer programming) , convex function , diffusion , convergence (economics) , regular polygon , exponential decay , mathematical analysis , geometry , physics , quantum mechanics , computer science , nuclear physics , economics , thermodynamics , programming language , economic growth
By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups( P t ) t ≥ 0with respect to the standard L p ‐Wasserstein distance for all p ∈ [ 1 , ∞ ) . In particular, we show that for the Itô stochastic differential equation d X t = d B t + b ( X t )d t , if the drift term b is such that for any x , y ∈ R d ,⟨ b ( x ) − b ( y ) , x − y ⟩ ≤K 1| x − y | 2 ,| x − y | ≤ L ;− K 2| x − y | 2 ,| x − y | > Lholds with some positive constants K 1 , K 2 and L > 0 , then there is a constant λ : = λ ( K 1 , K 2 , L ) > 0 such that for all p ∈ [ 1 , ∞ ) , t > 0 and x , y ∈ R d ,W p ( δ x P t , δ y P t ) ≤ C e − λ t / p| x − y | 1 / p ,if| x − y | ≤ 1 ;| x − y | ,if| x − y | > 1where C : = C ( K 1 , K 2 , L , p ) is a positive constant. This improves the main result in [14][A. Eberle, ] where the exponential convergence is only proved for the L 1 ‐Wasserstein distance.