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.
Ann. Probab.
25(4):
1953-1994
(October 1997).
DOI: 10.1214/aop/1023481117