From scaling to large deviations: a doubly stochastic continuous time random walk model for non-Gaussian transport

Brownian yet non-Gaussian diffusion is a paradigmatic example of transport in complex systems: the mean squared displacement grows linearly in time, while the displacement distribution remains non-Gaussian over experimentally relevant time scales. This coexistence challenges the identification of normal diffusion with Gaussian statistics, since higher-order fluctuations, rare events and microscopic transport mechanisms remain visible in the full propagator. A central aim of this thesis is to determine under which conditions self-similar scaling emerges, when higher cumulants are instead suppressed by concentration mechanisms of large-deviation type, and how rare large displacements are encoded in the far tails. We investigate these questions through continuous time random walks and diffusing diffusivity models, using subordination methods and excess kurtosis as the main diagnostic tool. We first analyse diffusing diffusivity and derive the time-dependent kurtosis through subordination, identifying the finite short-time limiting value associated with local self-similarity. We then study renewal continuous time random walks with several waiting-time laws, including exponential, Mittag-Leffler, Pareto Type II and one-sided stable laws represented through Wright functions. For power-law-tailed waiting times, the Mittag-Leffler law plays a special role as the universal fixed point associated with thinning limits and fractional diffusion. The kurtosis is then used to quantify how different waiting-time laws approach the asymptotic fractional-diffusion scaling regime and how non-universal transient corrections arise from subleading terms in their Laplace transforms. In the second part of the thesis we develop a doubly stochastic continuous time random walk with an exponential conditional waiting-time law and a stochastic jump rate governed by a Cox-Ingersoll-Ross process. This construction encodes environmental activity directly in the random clock of the walk. In the continuum limit, the model recovers the main signatures of diffusing diffusivity, including a linear mean squared displacement, a non-Gaussian-to-Gaussian crossover, and the finite plateau-like short-time limit of the excess kurtosis. At finite numbers of jumps, however, the discrete event structure produces additional terms in the kurtosis and breaks the ideal self-similar behaviour predicted by continuous descriptions at ultra-short times. Finally, we show that the cusp-like central peak predicted by some continuous superstatistical descriptions is not universal, but depends on the microscopic jump kernel. For Gaussian jumps, the regular part of the propagator remains smooth at the origin, whereas non-smooth kernels can transmit a cusp. The large-displacement tails are analysed through a saddle-point approach, revealing a dominant exponential decay with an algebraic correction controlled by the shape parameter of the stochastic rate. Overall, the results clarify how microscopic granularity, temporal disorder and environmental fluctuations combine to shape Brownian yet non-Gaussian transport.