Research

Research interests

My research interests span different areas of probability theory and its applications. From a theoretical standpoint, my work is particularly focused on the interplay of nonlocal operators, stochastic processes, and random evolutions, especially semi-Markov processes, semi-Markov random evolutions, kinetic processes and nonlocal operators in time. I am also interested in computational aspects: in particular I’m currently working with a PhD Student on simulation methods for semi-Markov processes. Furthermore I study modeling aspects related to this theory, particularly anomalous transport and diffusion, as well as applications of semi-Markov models to mathematical finance, hydrology and neurosciences. During my PhD, I also explored other topics such as non-Euclidean geometry, pseudoprocesses theory, and random processes on fractals.

Some of my talks that have been recorded

• Semi-Markov processes, time-changes and non-local equations (LINK TO SEE THE VIDEO), SISSA MATHLAB  online seminar series on fractional calculus and related topics. August 02, 2024.

• Semi-Markov approach to anomalous diffusion (LINK TO SEE THE VIDEO), on the occasion of the programme "Fractional Differential Equations" at the Isaac Newton Institute for Mathematical Sciences - February 21, 2022.

• Semi-Markov approach to non-local equations (LINK TO SEE THE VIDEO), on the occasion of the workshop Deterministic and Stochastic Fractional Differential Equations and Jump Processes during the programme "Fractional Differential Equations" at the Isaac Newton Institute for Mathematical Sciences - February 21, 2022.

• A semi-Markov integrate and fire model for neuronal activity (LINK TO SEE THE VIDEO ),  on the occasion of the "International Conference on Mathematical Neurosciences", Digital Edition - June 30, 2021.

Editorial duties

Associate Editor for the "Journal of Theoretical Probability"

Coauthors

Giacomo Ascione, Ivan Biočić, Maria Chiara Bovier, Daniel E. Cedeño-Girón, Mirko D'Ovidio, Lorenzo Facciaroni, Sergei Fedotov,  Stefano Ferraris, Alessio Gentile, Mark M. Meerschaert, Enzo Orsingher, Pierre Patie, Enrica Pirozzi, Costantino Ricciuti, Mladen Savov, Enrico Scalas

Publications

Mathematics:

• L. Facciaroni, C. Ricciuti, E. Scalas & B. Toaldo. Random Flights and Anomalous Diffusion: A Non-Markovian Take on Lorentz Processes.
  Submitted. Available on arXiv:2507.02796 

• I. Biočić & B. Toaldo. Harmonic problems arising from continuous time random walks limit processes.
  Submitted. Available on arXiv: 2505.14550 

• I. Biočić, D. E. Cedeño-Girón & B. Toaldo. Sampling inverse subordinators and subdiffusions.
  Submitted. Available on arXiv:2412.15815 

• G. Ascione, E. Scalas, B. Toaldo & L. Torricelli. Time-changed Markov processes and coupled non-local equations.
  Submitted. Available on arXiv:2412.14956 

• L. Facciaroni, C. Ricciuti, E. Scalas & B. Toaldo. Para-Markov chains and related non-local equations
  Fractional Calculus and Applied Analysis, 28: 1071 - 1093, 2025. DOI: 10.1007/s13540-025-00390-9 

• G. Ascione, M. Savov & B. Toaldo. Regularity and asymptotics of densities of inverse subordinators.
  Transactions of the London Mathematical Society, 11(1): e70004, 2024. DOI: 10.1112/tlm3.70004 

• G. Ascione, P. Patie & B. Toaldo. Non-local heat equations with moving boundary.
  Submitted. Available on arXiv: 2203.09850

• E. Scalas & B. Toaldo. Limit theorems for prices of options written on semi-Markov processes.
  Theory of Probability and Mathematical Statistics, 105: 3 - 33, 2021. DOI: 10.1090/tpms/1153  

• C. Ricciuti & B. Toaldo. From Semi-Markov random evolutions to scattering transport and superdiffusion.
  Communications in Mathematical Physics, 401(3): 2999 - 3042, 2023. DOI: 10.1007/s00220-023-04705-w 

• G. Ascione & B. Toaldo. A semi-Markov Leaky integrate-and-fire model.
  Mathematics (Special Issue Stochastic Processes in Neuronal Modeling), 7(11): 1022, 2019. DOI: 10.3390/math7111022

• M. Savov & B. Toaldo. Semi-Markov processes, integro-differential equations and anomalous diffusion-aggregation.
  Annales de l'Institut Henri Poincaré (B) Probabilités and Statistiques, 56(4): 2640 - 2671, 2020. DOI: 10.1214/20-AIHP1053 

• G. Ascione, E. Pirozzi & B. Toaldo. On the exit time from open sets of some semi-Markov processes.
  Annals of Applied Probability, 30(3): 1130 - 1163, 2020. DOI: 10.1214/19-AAP1525

• C. Ricciuti & B. Toaldo. Semi-Markov models and motion in heterogeneous media.
  Journal of Statistical Physics, 169(2): 340 - 361, 2017. Free link; DOI: 10.1007/s10955-017-1871-2

• E. Orsingher, C. Ricciuti & B. Toaldo. On semi-Markov processes and their Kolmogorov's integro-differential equations.
  Journal of Functional Analysis, 275(4): 830 - 868, 2018. DOI: 10.1016/j.jfa.2018.02.011

• E. Orsingher, C. Ricciuti & B. Toaldo. Time-inhomogeneous jump processes and variable order operators.
  Potential Analysis, 45(3): 435 - 461, 2016. DOI: 10.1007/s11118-016-9551-4

• M.M. Meerschaert & B. Toaldo. Relaxation patterns and semi-Markov dynamics.
  Stochastic Processes and their Applications, 129(8): 2850 - 2879, 2019. DOI: 10.1016/j.spa.2018.08.004

• E. Orsingher, C. Ricciuti & B. Toaldo. Population models at stochastic times.
  Advances in Applied Probability, 48(2): 481 - 498, 2016. DOI: 10.1017/apr.2016.11

• B. Toaldo. Lévy mixing related to distributed order calculus, subordinators and slow diffusions.
  Journal of Mathematical Analysis and Applications, 430(2): 1009 - 1036, 2015. DOI: 10.1016/j.jmaa.2015.05.024

• E. Orsingher & B. Toaldo. Counting processes with Bernstein intertimes and random jumps.
  Journal of Applied Probability, 52(4), 1028 - 1044, 2015. DOI: 10.1239/jap/1450802751

• B. Toaldo. Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups.
  Potential Analysis, 42(1): 115 - 140, 2015. DOI: 10.1007/s11118-014-9426-5

• E. Orsingher & B. Toaldo. Pseudoprocesses related to space-fractional higher-order heat-type equations.
  Stochastic Analysis and Applications, 32(4): 619 - 641, 2014. DOI: 10.1080/07362994.2014.911107

• E. Orsingher & B. Toaldo. Pseudoprocesses on a circle and related Poisson kernels.
  Stochastics, 87(4): 664 - 679, 2015. DOI: 10.1080/17442508.2014.991326

• E. Orsingher & B. Toaldo. Space-time fractional equations and the related stable processes at random time.
  Journal of Theoretical Probability, 30(1): 1 - 26, 2017. DOI: 10.1007/s10959-015-0641-9 Here is a Free Link

• M. D'Ovidio, E. Orsingher and B. Toaldo. Time-changed processes governed by space-time fractional telegraph equations.
  Stochastic Analysis and Applications, 32(6): 1009 - 1045, 2014. DOI: 10.1080/07362994.2014.962046

• M. D'Ovidio, E. Orsingher & B. Toaldo. Fractional telegraph-type equations and hyperbolic Brownian motion.
  Statistics and Probability Letters, 89(1): 131 - 137, 2014. DOI: 10.1016/j.spl.2014.02.021

• E. Orsingher & B. Toaldo. Shooting randomly against a line in euclidean and non-euclidean spaces.
  Stochastics, 86(1): 16 - 45, 2014. DOI: 10.1080/17442508.2012.749260

Other:

• M.C. Bovier, S. Fedotov, S. Ferraris, A. Gentile & B. Toaldo. Stochastic model for subsurface water flow in Swiss catchments.
  Advances in Water Resources, 196: 104883, 2025. DOI: 10.1016/j.advwatres.2024.104883 

• P. Palazzo, C. Balucani, K. Barlinn, G. Tsivgoulis, Y. Zhang, L. Zhao, J. DeWolfe, B. Toaldo, E. Stamboulis, F. Vernieri, P.M. Rossini, and A.V. Alexandrov. Association of reversed Robin Hood syndrome with risk of stroke recurrence.
  Neurology, 75(22): 2003-2008, 2010.

Dissertation titled "Libertà e Democrazia nel pensiero di Amartya Sen".