Talagrand concentration inequalities for stochastic partial differential equations

Abstract

One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations. Here, we develop a similar theory for some stochastic partial differential equations.

A note on transportation cost inequalities for diffusions with reflections

“…In [46], Talagrand proved that a standard Gaussian measure on R d satisfies T 2 (C) with C = 1. Afterwards, TCI inequalities were established for discrete-time Markov chains, [30,37,40]; for discrete-time stationary processes, [31]; for stochastic ordinary differential equations driven by Brownian motion [11,16,34,49] and by more general noise [8,44,39]; for stochastic partial differential equations, [9,25,50], and for neutral stochastic equations (which depend on past history) [3,28,8]. Applications include model selection in statistics [32], risk theory [27], order statistics [6], information theory [5,38], and randomized algorithms [17].…”

Talagrand’S Transportation Inequality for Spdes with Locally Monotone Drifts

“…P x . It is analogous to [9,Theorem 5.6] in the setting of finite-dimensional Brownian motion, [13,Lemma 3.1] in the setting of the space-time white noise on R 2 and [31, Lemma 3.1] in the setting of Gaussian noise white in time and colored in space, hence we omit its proof.…”

“…the L 2 -metric for the stochastic heat equations driven by the space-time white noise and driven by the fractional-white noise. Khoshnevisan and Sarantsev [13] established the T 2 pCq for more general SPDEs driven by the space-time white noise under the uniform and L 2 -norm. S. Shang and T. Zhang [32] established the T 2 pCq w.r.t.…”

Talagrand Inequality on Free Path Space and Application to Stochastic Reaction Diffusion Equations

“…Recently, by using the Girsanov transformation argument developed from [4], the Talagrand inequality was established on the path space for solutions of stochastic reaction diffusion equations with deterministic initial values, see [10], [15]. In this paper, we aim to extend this result to the case with random initial values.…”