Dr Lorenzo ToniazziOffice: Science III, room 217
Marsden Grant Postdoctoral Fellow with Prof Boris Baeumer.
Lecturer for MATH170 Algebra.
Postdoctoral Fellow with Prof Qiang Du, Columbia University.
PhD with Prof Vassili Kolokoltsov, Prof Xue-Mei Li and Dr Roger Tribe, 2018 Warwick University.
Stochastic processes and PDEs, in particular:
- Lévy processes and heavy-tailed random variables
- Non-Markovian dynamics and subdiffusion
- Fractional and nonlocal models
- Numerical methods for nonlocal models
- Herman, J., Johnston, I., & Toniazzi, L. (2020). Space-time coupled evolution equations and their stochastic solutions. Electronic Journal of Probability, 25, 147. doi: 10.1214/20-EJP544
- Du, Q., Toniazzi, L., & Zhou, Z. (2020). Stochastic representation of solution to nonlocal-in-time diffusion. Stochastic Processes & their Applications, 130, 2058-2085. doi: 10.1016/j.spa.2019.06.011
- Toniazzi, L. (2019). Stochastic classical solutions for space–time fractional evolution equations on a bounded domain. Journal of Mathematical Analysis & Applications, 469(2), 594-622. doi: 10.1016/j.jmaa.2018.09.030
- Hernández-Hernández, M. E., Kolokoltsov, V. N., & Toniazzi, L. (2017). Generalised fractional evolution equations of Caputo type. Chaos, Solitons & Fractals, 102, 184-196. doi: 10.1016/j.chaos.2017.05.005
Boundary conditions for nonlocal operators: we connect boundary conditions for backward and forward nonlocal Kolmogorov equations to the underlying onesided Lévy processes restricted to an interval. We study Dirichlet, Neumann and fast-forwarding boundary conditions. The latter is a new nonlocal boundary condition describing particles free to move in and out of the domain. Our integro-differential operators are based on convolution kernels that generalise Riemann-Liouville and Caputo derivatives of order $\alpha\in(1,2)$. Our method is based on continuous embeddings of finite difference numerical schemes. Joint work with Boris Baeumer and Mihály Kovács (arXiv:2012.10864 and 2103.00715).
Censored stable subordinators: we study a stable Lévy subordinator censored on crossing a barrier, and we derive several results for the corresponding inital value problems. In particular we identify a new fractional derivative, a new subdiffusion model and a new relaxation equation with fast algebraic decay. Joint work with Qiang Du and Zirui Xu (arXiv:1906.07296). Recently presented at Wrocław University of Science and Technology (Youtube link).