The asymptotic behaviors of the principal eigenvalue and the corresponding normalized eigenfunction of the operator $G^\varepsilon f = (\varepsilon/2)\triangle f + g \triangledown f +(l/\varepsilon)f$ for small $\varepsilon$ are studied. Under some conditions, the first order expansions for them are obtained. Two applications to risk-sensitive control problems are also mentioned.
"Asymptotics for the principal eigenvalue and eigenfunction of a nearly first-order operator with large potential." Ann. Probab. 25 (4) 1953 - 1994, October 1997. https://doi.org/10.1214/aop/1023481117