Quadratic transportation inequalities for SDEs with measurable drift

Abstract

Let X be the solution of the multidimensional stochastic differential equationwhere W is a standard Brownian motion. In our main result we show that when b is measurable and σ is in an appropriate Sobolev space, the law of X satisfies a uniform quadratic transportation inequality.

Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions

“…As above, the linear growth of b yields the well-posedness assumption (1). We next check condition (2). The standard argument using linear growth and Gronwall's inequality, as in the proof of Corollary 2.7, yields…”

“…holds for some γ < ∞, as well as square-integrability as in (2.4) but with µ ⊗2 in place of P (2) . This condition (4.33), however, seems difficult to check in practice.…”

“…The proof, given in Section 6.2, uses a known method based on Girsanov's theorem [17,76,60] to deduce that µ satisfies a quadratic transport inequality with respect to the sup-norm; see also [3,2] for recent results on quadratic transport inequalities for SDEs with irregular coefficients, which may be useful in deriving additional corollaries. Interestingly, the (tensorized) quadratic transport inequality is useful not only for checking assumption (3) of Theorem 2.2, but also for transferring the main estimate from entropy to Wasserstein distance, yielding the first inequality (2.13).…”

“…Our main Theorem 2.2 imposes (1.12) as an assumption, and various tractable sufficient conditions are given in Section 2.2, including cases for which not even qualitative propagation of chaos was previously known (see Theorem 2.10). Kantorovich duality gives rise to an enlightening sufficient condition for (1.12), which is to assume that there exists a metric ρ on R d for which the following two properties hold: (i) There exists γ < ∞ such that µ satisfies the transport inequality W 2 1,ρ (•, µ t ) ≤ γH(• | µ t ) for all t, where the 1-Wasserstein distance W 1,ρ is defined using ρ.…”