☰ Pierre Patie

Pierre Patie



Research overview

I develop a research program at the interface of probability, analysis, operator theory, algebras and computation. A central theme is that symmetry and classification are not only conceptual tools: they can be turned into explicit formulas, structural results, and efficient algorithms for complex stochastic and operator-theoretic systems.

At Cornell, this program connects fundamental mathematics with engineering-relevant computation. It draws together ideas from spectral theory, semigroups, operator algebras, representation theory, and intertwining relations to study long-standing problems from a unified structural perspective, with applications ranging from scaling limits and universality to computational mathematics and questions related to the Riemann hypothesis.

Applications range from finance and insurance, neuroscience to mathematical physics.

My research has been supported by public agencies, philanthropic foundations, and institutional partners in the United States and Europe, including the NSF (current support), the Swiss NSF, FNRS, ARC, CNRS, the AXA Research Fund, and the National Bank of Belgium.




Research Programs

My research is organized around four interconnected mathematical programs at the interface of probability, analysis, operator theory, algebras and computation.

Symmetry, isospectral schemes and universality for semigroups

I study structural methods for operators and semigroups, with emphasis on symmetry and isospectral classification schemes such as intertwining, interweaving, gateway and weak similarity orbits. Structural equivalences are treated as constructive tools for deriving spectral information, transferring results across models, and generating analytically tractable systems.

  • Spectral theory of self-adjoint, non-self-adjoint, and non-local operators
  • Intertwining, interweaving, gateway, and weak similarity orbits
  • Convergence to equilibrium: hypocoercivity, hyper(ultra)contractivity, and cut-off phenomena
  • Representation theory and operator algebras for scaling limits and universality

Potential and fluctuation theory for Markov and related processes

A second line of work develops potential theory and fluctuation theory for Markov and related processes, with particular emphasis on first-passage problems, exit distributions, Martin boundary theory, path functionals, and free-boundary questions.

  • Fluctuation theory, Martin boundary, and exit problems, including jump and non-Markovian models
  • Curve-crossing problems with moving or stochastic boundaries
  • Optimal stopping and related free-boundary problems
  • Subdiffusive models, time-changes, non-local evolution, and fractional Cauchy problems
  • Exponential functionals of Lévy processes and branching processes with immigration

Special functions, asymptotic analysis and zero phenomena

A third program investigates universal analytic structures across apparently different models through asymptotic analysis, Mellin-transform methods, special-function theory, and the study of zeros.

  • Self-similar and related ergodic operators
  • Exponential functionals of Lévy processes
  • Moment problems, Tauberian/Abelian methods, and asymptotic analysis
  • Special functions, determinantal structures, and random-matrix limits
  • Entire-function and zero phenomena, including Laguerre–Pólya, Riemann function, van Dantzig duality, and Lee–Yang-type questions

Exact discretization, numerical algorithms, and applications

A fourth line of work translates structural and probabilistic insight into exact discretization methods and numerical algorithms that preserve the essential operator-theoretic or stochastic structure of the original problem.

  • Spectral algorithms for non-local operators
  • Isospectral gateway/interweaving discretizations linking continuum and lattice models for semigroups
  • Computational methods for stochastic processes and parabolic PDEs
  • High-dimensional PDE algorithms via wavelets


Application Domains

Finance and insurance.
First-passage problems, ruin-type questions, path-dependent functionals, tractable stochastic models, pricing, and risk analysis.
Mathematical physics.
Spectral theory, symmetry, universality, scaling limits, determinantal dynamics, and zero phenomena.
Neuroscience.
Threshold and first-passage models, stochastic dynamics, and semigroup-based computational approaches.
Cross-disciplinary computation.
Structure-preserving algorithms for stochastic processes, non-local operators, and PDEs.


For a thematic organization of papers, see Research Themes.

Publications

Full list of Publications and my ORCID iD iconORCID Profile

Submitted

Some recently published/accepted papers


List of co-authors

    L. Alili, G. Ascione, F. Avram, C. Bartholmé, L. Chaumont, M. Chazal, A. Chee, P. Cheridito, M.C.H. Choi, M. Cissé, C. Constantinescu, G. D'Onofrio, P. Embrechts, R. Frey, X.Y. Han, B. Jarrow, R. Kaufmann, T. Konstantopoulos, A. Kyprianou, Y. Liu, R.L. Loeffen, L. Miclo, J.-C. Pardo, J.L. Pedersen, L. Sacerdote, R. Sarkar, M. Savov, T. Simon, A. Srapionyan, E. Tanré, T. Tian, B. Toaldo, A. Vaidyanathan, S. Vakeroudis, J. Wang, C. Winter, Y. Zhao


PhD Students

  • Andrew Chee (ORIE, Cornell), 2023
    Algebraic, analytical and numerical perspectives on the universe of integrable systems
  • Rohan Sarkar (ORIE, Cornell), 2021
    Continuous and discrete self-similarity via classification schemes of Markov processes, and the van Dantzig problem
  • Jian Wang (ORIE, Cornell), 2020
    Continuous time skip-free Markov process and study of branching process with immigration
  • Aditya Vaidyanathan (CAM, Cornell), 2019
    Contributions to the Stieltjes moment problems and to the intertwining of Markov semigroups
  • Anna Srapyonian (CAM, Cornell), 2019
    Some spectral ideas applied to finance and to self-similar and long-range dependent processes
  • Michael C.H. Choi (ORIE, Cornell), 2017
    Analysis of non-reversible Markov chains
  • Yixuan Zhao (ORIE, Cornell), 2017
    Spectral expansions and excursion theory for non-self-adjoint Markov semigroups with applications in mathematical finance
  • Christopher Van Weverberg (ULB), co-supervised with G. Deelstra, 2015
    Contributions to the study of affine processes with applications in insurance
  • Carine Bartholmé (ULB), 2014
    Self-similarity and exponential functionals of Lévy processes