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Belinschi et al. [Adv. Math., 226 (2011), 3677--3698] proved that the normaldistribution is freely infinitely divisible. This paper establishes a certainmonotonicity, real analyticity and asymptotic behavior of the density of thefree L\'{e}vy measure. The monotonicity property strengthens the result inHasebe et al. [Int. Math. Res. Not. (2019), 1758--1787] that the normaldistribution is freely selfdecomposable.

On the free L\'{e}vy measure of the normal distribution