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## Dr Lorenzo Toniazzi

Office: Science III, room 217
Phone: 479-4109
Email: ltoniazzi@maths.otago.ac.nz

Current:

Marsden Grant Postdoctoral Fellow with Prof Boris Baeumer.

Lecturer for MATH170 Algebra (S1).

Previous:

Postdoctoral Fellow with Prof Qiang Du, Columbia University.

PhD with Prof Vassili Kolokoltsov, Prof Xue-Mei Li and Dr Roger Tribe, 2018 Warwick University.

#### MATH170 extra material

The linear algebra applications tutorials for MATH170 can be run online from the repository ltoniazzi/Algebra_applications.

#### Research interests

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

#### Publications

• 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

#### Recent preprints

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).