Exponential convergence in L p ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

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.