Log-Sobolev-type inequalities for solutions to stationary Fokker–Planck–Kolmogorov equations

Abstract

We prove that every probability measure µ satisfying the stationary Fokker-Planck-Kolmogorov equation obtained by a µ-integrable perturbation v of the drift term −x of the Ornstein-Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure γ and for the density f = dµ/dγ the integral of f | log(f + 1)| α against γ is estimated via v L 1 (µ) for all α < 1/4, which is a weakened L 1 -analog of the logarithmic Sobolev inequality. This yields that stationary measures of infinitedimensional diffusions whose drifts are integrable perturbations of −x are absolutely continuous with respect to Gaussian measures. A generalization is obtained for equations on Riemannian manifolds. MSC: primary 35J15; Secondary 35B65Keywords: Ornstein-Uhlenbeck operator, stationary Fokker-Planck-Kolmogorov equation, integrable drift, logarithmic Sobolev inequality, Gaussian measureThe latter bound admits an infinite-dimensional version. To this end we write the drift b in the form b(x) = −x + v(x).If v = 0, then the only solution in the class of probability measures is the standard Gaussian measure γ with density (2π) −d/2 exp(−|x| 2 /2). Hence it is natural to express µ 1

Regularity of solutions to Kolmogorov equations with perturbed drifts

“…The sequence {f n } is a martingale with respect to the Gaussian measure γ and the sequence of σ-algebras generated by the projections to R n . According to [15], this sequence is uniformly integrable, hence converges in L 1 (γ) to some function f ∈ L 1 (γ). It is readily seen that µ = f • γ. Convergence also holds in all L p (γ).…”

“…Although the condition |b| ∈ L 1 (µ) is not enough for the membership of ̺ in the Sobolev class W 1,1 (R d ), it implies some weaker version of the logarithmic Sobolev inequality (see [15]). Sufficient conditions for the boundedness of ̺ can be found in [20], [7], [12] along with some other bounds (see [9] for a survey).…”

Владимир Игоревич Богачев (К Шестидесятилетию Со Дня Рождения)

“…Лишь в недавней работе [35] дано отрицательное решение этой проблемы путем построения примера, в котором плотность решения имеет неинтегрируемый градиент. Однако в [37] показано, что интегрируемость коэффициента сноса все же влечет некий аналог логарифмического неравенства Соболева. Надо отметить, что при изучении уравнений Фоккера-Планка-Колмогорова возникает немало задач, относящихся 204 МАТЕМАТИЧЕСКАЯ ЖИЗНЬ к классам Соболева и их обобщениям в конечномерных и бесконечномерных пространствах, и здесь В. И. Богачевым также были получены яркие результаты.…”