Convergence Rates of Continuous-Time Random Walks to Time-Fractional Diffusions with Unbounded Coefficients

We investigate uniform weak convergence rates for probabilistic numerical methods applied to backward time-fractional diffusion equations whose dynamics are driven by diffusions with possibly unbounded coefficients, such as the Geometric Brownian Motion. The fractional structure is represented through a random time-change by the inverse of a stable subordinator. To approximate the underlying fractional dynamics, we combine discrete Markov chain schemes for the diffusion component with heavy-tailed random walk approximations of the time change. Our analysis builds on Feller semigroup techniques and a high-order sensitivity framework for diffusion semigroups based on the Kunita stochastic flows and tensor fields. We derive uniform bounds for all orders of sensitivities, establish a quasi-contraction property for the associated semigroup, and transfer these estimates to the fractional setting via the convolution representation with the inverse subordinator. As a result, under killing conditions which dominate at least the base-space semigroup growth, we obtain weak convergence rates for the combined continuous-time-random-walk scheme to the time-fractional diffusion, with a logarithmic regime before the discount dominates the stronger smooth-space growth.