Localization results for densities associated with stable small-noise diffusions
This paper concerns asymptotic properties of the stationary density associated with small-noise diffusion processes, such as considered in the well-known work of \textit{A. D. Ventcel} and \textit{M. I. Frejdlin} [Usp. Mat. Nauk 25, No.1(151), 3-55 (1970; Zbl 0291.34042); English translation in Russ. Math. Surv. 25, 1-55 (1970)]. We assume that the origin is a globally attracting asymptotically stable equilibrium point of the underlying deterministic flow. For a bounded domain D, containing the origin, we derive estimates which establish the asymptotic independence, as the size of the noise vanishes, of the equilibrium density in D from the coefficients of the process outside D. These results are applied to generalize a result of \textit{S.-Y. Sheu} [SIAM J. Math. Anal. 17, 451-460 (1986; Zbl 0599.60073)] on an asymptotic representation of the equilibrium density.
scientific article; zbMATH DE number 4007373
scientific article