2018/05/24 by В. И. Богачев, Bogachev, V. I., A. V. Shaposhnikov +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Markov Chains and Monte Carlo Methods #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1805.09467
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‖L1(μ) for all α<1/4, which is a weakened L1-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of -x are absolutely continuous with respect to Gaussian measures.