Couplings and Kantorovich contractions with explicit rates for diffusions

Abstract

We consider certain classes of diffusion and McKean-Vlasov processes and provide non-asymptotic quantifications of the longtime behavior using coupling methods. The thesis is divided into three main parts.<br /> In the first part, we consider ℝ^d valued diffusions of type <center>dX_t = b( X_t ) dt + dB_t.</center> Assuming a geometric drift assumption, we establish Kantorovich contractions with explicit contraction rates for the transition kernels. The results are in the spirit of Mattingly and Hairer’s extensions of Harris’ theorem, but do not rely on a small set condition. Instead we use reflection coupling and adjust the underlying cost function of the Kantorovich distance in a very specific way to the diffusion model. The resulting rate is given explicitly in terms of a one-sided Lipschitz bound on the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov processes in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for subgeometric ergodicity assuming a subgeometric drift condition.<br /> In the second part, we show that a related strategy can also be applied for a class of infinite-dimensional and degenerate diffusion processes. Given a separable and real Hilbert space ℍ and a trace-class, symmetric and non-negative operator Ǥ : ℍ→ℍ we examine the equation <center> dX_t = -X_t dt + b( X_t ) dt + v2 dW_t, X₀ = x ∈ ℍ, </center> where (W_t) is a Ǥ--Wiener process on H and b : ℍ→ℍ is Lipschitz. We assume that there is a splitting of ℍ into a finite-dimensional space H^l and its orthogonal complement ℍ^h such that G is strictly positive definite on ℍ^l and the nonlinearity b admits a contraction property on ℍ^h. Assuming a geometric drift condition, we derive a Kantorovich contraction with an explicit contraction rate for the corresponding Markov kernels. Our bounds on the rate are based on the eigenvalues of Ǥ on the space ℍ^l, a Lipschitz bound on b and a geometric drift condition.<br /> In the third part, we present a novel approach of coupling two multidimensional and nondegenerate Itô processes ( X_t ) and ( Y_t ) which follow dynamics with different drifts.<br /> The coupling is sticky in the sense that there is a stochastic process ( r_t ), which solves a one-dimensional stochastic differential equation with a sticky boundary behavior at zero, such that almost surely | X_t - Y_t | ≤ r_t for all t ≥ 0. The coupling is constructed as a weak limit of Markovian couplings. We provide explicit, non-asymptotic and longtime stable bounds for the probability of the event { X_t = Y_t }.