Cosmology probes the largest scales of the Universe, and quantum gravity the smallest. In this thesis, within the framework of quantum gravity phenomenology, we explore how stochastic effects in the dynamics of the dark sector can bridge those regimes.

We first study covariant Brownian motion as an effective signature of spacetime discreteness. We derive the process from first principles via Fokker-Planck and Langevin perspectives, and show its uniqueness when minimally coupled to curvature. Applied to dark matter particles, the resulting stochastic dark matter is governed by a single diffusion rate. We solve for the linear perturbations semi-analytically and numerically, showing that the model suppresses the matter power spectrum at small scales and offers a resolution to the $S_8$ discrepancy between CMB and low-redshift probes.

We then extend the framework to massless particles. We solve the modified Boltzmann equation analytically in an FRW spacetime, showing that diffusion impacts long-wavelength modes most strongly. We show that a future detection of a stochastic gravitational-wave background with LISA can improve current CMB bounds on the diffusion and drift parameters by over twelve orders of magnitude.

Finally, we examine Everpresent $\Lambda$, a stochastic model of dark energy. A key feature of this model is that $\Lambda$ fluctuates and tracks the dominant energy density in order to resolve the cosmological-constant puzzle without fine-tuning. Accounting for initial conditions, we find hints that the nonlocality in the model is capable of producing inflationary expansion for many e-folds. A subset of Everpresent-$\Lambda$ realisations fit SN Ia data better than $\Lambda$CDM, but the model struggles at describing CMB. We also study the allowed fluctuations of dark-energy density in a more model-independent manner, and find that some variation, especially prior to recombination, can lead to improvements over $\Lambda$CDM; but the favoured fluctuations are smaller than is typical in Everpresent $\Lambda$.