Research

The main focus of my research has been asymptotic problems arising from Branching Processes, Branching Diffusions and related Dynamical Systems. The central theme in most of my work is the use of spectral theory in addition with techniques from probability, stochastic differential equations, parabolic partial differential equations, homogenization theory and dynamical systems to obtain the asymptotics of these stochastic processes that are valid up-to the domain of large deviations.

Publications

Journal Articles in preparation:

• A. Bonafede, P. Hebbar, P. Pherwani, "Birth and death processes with deterministic mass growth and symmetric mass division among offspring", in collaboration with undergraduate students, pre-print available upon request.

• P. Hebbar, K. Fernando, Higher order asymptotics for large deviations -- Part III.

Published Journal Articles

• D.Dolgopyat, P. Hebbar, L. Koralov, M. Perlman, "Multi-type branching processes with time-dependent branching rates", Journal of Applied Probability, 55(3), 701-727. doi:10.1017/jpr.2018.46, (2018)

• P. Hebbar, K. Fernando, Higher order asymptotics for large deviations -- Part I, Asymptotic Analysis, vol. Pre-press, pp. 1-39, (2020). doi:10.3233/ASY-201602.

• P. Hebbar, L. Koralov, J. Nolen,  Asymptotic behavior of branching diffusion processes in periodic media, Electronic Journal of Probability, vol. 25, paper no. 126, 40 pp doi:10.1214/20-EJP527 (2020)

• P. Hebbar, K. Fernando, Higher order asymptotics for large deviations -- Part II,  Stochastics and Dynamics, Vol. 21, No. 05, 2150025, (2021) 

Ph.D Dissertation

• Branching diffusion processes in periodic media, Thesis, August 2019.

Book Reviews

•  Hebbar P. "Testing the waters of differential equations", Physics Today, Volume 73, Issue 7, Page 54 (2020). doi.org/10.1063/PT.3.4525

Summer Research Project with undergraduate students at Grinnell College - Summer 2024

Exploring Random Walks

In this project, I will mentor undergraduate students in the probability techniques and computational simulations of random walks. Random walks are fundamental stochastic processes used to model various phenomena in diverse fields such as physics, biology, finance, and computer science. This undergraduate research project aims to investigate the dynamics and properties of random walks, focusing on both theoretical analysis and computational simulations. The project will begin with an introduction to the basics of random walks, including simple one-dimensional walks and their extensions to higher dimensions. Students will explore the concept of discrete and continuous random walks and understand how different types of steps (e.g., symmetric, asymmetric) influence the behavior of the walk. Next, the research will delve into more advanced topics such as random walks in random environments and branching random walks. Students will read the recent developments in this field, including research papers and  will study the concepts of recurrence, transience, and convergence to stationary distributions, gaining insight into the long-term behavior of such random processes. Furthermore, the project will involve computational experiments using programming languages like Python or MATLAB to simulate random walks in various settings. This hands-on approach will allow students to validate theoretical results, visualize the behavior of random walks, and explore practical applications such as modeling diffusion processes.

Summer Research Project with undergraduate students at Duke University - Summer 2021 

Galton-Watson Processes with Mass 

We will study Branching process and branching diffusions which are stochastic (random) processes that model birth and death of certain organisms (particles). In this project, we study a cell growth model which consists of a continuous time supercritical branching process with an extra parameter(mass). Namely we will assume that the mass of the particles grows according to a deterministic differential equation between the exponentially distributed splitting moments and that in the moment of the division the mass of the particle is divided in a random proportion between k offspring.   The model described above gives an approximation of biological cell growth process and it is interesting due to the new mathematical questions that can be analyzed. We plan to read the recent literature in this field and pose and make progress towards new and open problems. 

Talks

• • Colloquium, Carleton College, Northfield MN, April 30th 2024.

  • Probability and Analysis Seminar, Iowa State University, Ames IA, March 29th, 2023.

  • Probability and Statistics Seminar, Lehigh University, Bethlehem, March 24th, 2023.

  • Probability Seminar, Louisiana State University, March 20th, 2023.

  • Workshop on Reaction Diffusion Equations, University of Maryland, College Park, March 9th, 2023.

  • 2022 Union College Mathematics Conference, Session on Stochastic Analysis and Applications, Schenectady, June 4th, 2022.

  • Colloquium, Rutgers University - Camden, November 12th, 2021.

  • Colloquium, North Carolina Central University (NCCU), October 26th, 2021.

  • Computational & Applied Mathematics Seminar, University of North Carolina Charlotte (UNCC), October 15th, 2021.

  • SURE: Speakers and Undergraduate Research Engagement, Virginia Tech, USA, October 11th, 2021.

  • CRM-ISM Probability Seminar, Centre de Recherches Mathematiques (CRM) Montreal, Canada, November 4th, 2020.

  • Probability and Statistical Physics Seminar, University of Chicago, October 23nd, 2020.

  • Probability Seminar, University of Virginia, USA, February 5th, 2020.

  • National Webinar on Galton-Watson Processes, MES Garware College, Pune, India, October 5th, 2020. (slides, video excerpt)

  • Probability  Seminar, Duke University, USA, September 26th, 2019.

  •  Conference -- Perturbation Techniques in Stochastic Analysis and Its Applications, CIRM, France, March 14th 2019.

  • Probability Seminar , University of Washington, USA (March 4th 2019).

  • Probability/Math Finance Seminar , Carnegie Mellon University, USA (January 28th 2019).

  • Probability Seminar , Boston University, USA (January 24th 2019).

  • Probability Seminar, Pennsylvania State University, USA (January 11th 2019).

  • Poster talk at the Seminar on Stochastic Processes , Brown University, Providence, USA (May 10, 2018). 

  • GWU-SIAM Conference on Applied Mathematics, George Washington University, Washington D.C, USA (April 29, 2017). 

  • "Multi-type branching processes with time-dependent branching rates" , Workshop on Dynamical Systems and Related Topics, University of Maryland, College Park, USA (March 31 - April 2, 2017) (slides).