Convergence Rates of Continuous-Time Random Walks to Time-Fractional Diffusions with Unbounded Coefficients
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916066541699072 |
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| author | Sidorenko, Artur Kolokoltsov, Vasilii |
| author_facet | Sidorenko, Artur Kolokoltsov, Vasilii |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31471 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence Rates of Continuous-Time Random Walks to Time-Fractional Diffusions with Unbounded Coefficients Sidorenko, Artur Kolokoltsov, Vasilii Probability Numerical Analysis 60H10, 60H35, 60J60, 35R11 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. |
| title | Convergence Rates of Continuous-Time Random Walks to Time-Fractional Diffusions with Unbounded Coefficients |
| topic | Probability Numerical Analysis 60H10, 60H35, 60J60, 35R11 |
| url | https://arxiv.org/abs/2605.31471 |