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We prove variation and oscillation $L^p$-inequalities associated withfractional derivatives of certain semigroups of operators and with the familyof truncations of Riesz transforms in the inverse Gaussian setting. We alsostudy these variational $L^p$-inequalities in a Banach-valued context byconsidering Banach spaces with the UMD-property and whose martingale cotype isfewer than the variational exponent. We establish $L^p$-boundedness propertiesfor weighted difference involving the semigroups under consideration.

Variation and oscillation for harmonic operators in the inverse Gaussian setting