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It is well known that in quantum gravity, the very geometry of space and timeis subject to continual fluctuation. The mathematical formulation for this oldtheory is still lacking. This article formulates this more than forty-year-oldtheory of quantum gravity. On the other hand, recent attention has been paid tofractional gravity. Although this type of gravity leads to brilliant results,we have no deep reason for applying it other than that it works. In this paper,it is demonstrated that quantum gravity equations become fractional whenspace-time fluctuations are taken into account. Therefore, here a reasonableroot and argument for applying fractional gravity are found: ``Fractionalquantum gravity is generated by stochastic fluctuations of space-time''. Forclarification, Einstein-Hilbert theory along with a scalar field isinvestigated in deformed and non-deformed minisuperspaces. The resultsillustrate a transition from decelerated to accelerated expansion(late-time-accelerated expansion).

Formulating the fluctuations of space-time and a justification for applying fractional gravity