Joe P. Chen's Research Page

Research papers | Professional travel | Undergraduate research | Scientific organization | Funding

From particle systems to (S)PDEs on self-similar and random networks

     
Click on each image for a high-res version. From left to right:

Research tutorials & lectures at Cornell (June 2017) and Bielefeld (July 2017)

My research focuses on the analysis of probability models, as well as the relevant (stochastic) PDEs, on self-similar and random geometries. The overarching goal is to prove limit theorems that describe various laws of nature that take place in these nonsmooth spaces, such as heat flow, wave propagation, charged particles in an electromagnetic field, and fluid dynamics.

From the mathematics point of view, I carry out two interconnected lines of research, one analytic and the other probabilistic.

From the physics point of view, the example models that interest me fall into one of two broad categories: The former category includes the exclusion process, the Gaussian free field, sandpile models, and aggregation models. The latter category involves the spectral analysis of electric or magnetic Schrödinger operators, as well as the associated PDEs. While the discrete models may seem disparate, all of them are ultimately grounded in the study of calculus on graphs, random walks, Markov chains, and electric network theory.

PDF My research in 1 slide. (September '16, presented at the "meet-and-greet" session during Colgate's NASC divisional colloquium.)

PDF 5-page description of my research, pitched at the general scientific audience. (Written in January '17 and updated in February '17. It is NOT up-to-date. An updated statement will appear in Fall '17.)


Research papers & presentations (see my arXiv page for the full list of papers)

indicates work involving undergraduate student(s)
Last updated: October 10, 2017

Most recent papers (Fall 2017)

From non-symmetric particle systems to non-linear PDEs on fractals (a review), with Michael Hinz and Alexander Teplyaev.
To appear in the proceedings for the 2016 conference Stochastic Partial Differential Equations & Related Fields in honor of Michael Röckner's 60th birthday, Bielefeld (2017+)
This is a summary of a series of four papers concerning the hydrodynamic limit of the boundary-driven exclusion process on a resistance space (the Sierpinski gasket being the model space).

Part I: The moving particle lemma for the exclusion process on a weighted graph
Electron. Commun. Probab. 22 (2017), paper no. 47.
I prove in this paper a Sobolev-type energy inequality for the exclusion process, which is analogous to Thomson's (or Dirichlet's) principle for random walks / electric networks. It builds upon the marvelous "octopus inequality" proved by Caputo, Liggett, and Richthammer in 2009.
PDF A 15-minute presentation on this work, targeted towards undergraduates.

Part II: Local ergodicity in the exclusion process on an infinite weighted graph
Submitted
In this paper I prove, on every strongly recurrent weighted graph, the coarse-graining arguments needed to pass from the microscopic observables (in the exclusion process) to the corresponding macroscopic averages. The two-blocks estimate is based on the moving particle lemma established in Part I.
NEW! PDF An hour-long talk on the moving particle lemma and local ergodicity in the exclusion process, delivered at the CUNY Probability Seminar.

Parts III & IV, concerning the hydrodynamic limit theorems and existence/uniqueness/regularity of solutions to semilinear PDEs on SG, are being written in collaboration with Michael Hinz and Alexander Teplyaev.
PDF A 20-minute talk on semilinear evolution equations on resistance spaces, delivered at the Northeast Analysis Network conference in Syracuse.

Internal DLA on Sierpinski gasket graphs, with Wilfried Huss, Ecaterina Sava-Huss, and Alexander Teplyaev.
Submitted
(An informal 1-sentence summary of this paper: "You can grow a crystal on a fractal, and yet the crystal will be round almost surely." Illustrated by this GIF animation.)
See also the shape theorem for divisible sandpiles on SG by Huss and Sava-Huss.
In an upcoming paper with Jonah Kudler-Flam, we show a universal limit shape for four cellular automata models on SG, and identify the precise order of fluctuations in each model.
PDF A 20-minute talk on limit shape universality of Laplacian growth models on SG, delivered at the conference "Analysis & Geometry on Graphs & Manifolds" at Universität Potsdam, Germany. (Open in Adobe Acrobat to run the simulations.)

Power dissipation in fractal AC circuits, with Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew, and Alexander Teplyaev. (2015 UConn math REU fractals group)
J. Phys. A: Math. Theor. 50 325205 (2017)

We give a "fractal spin" on the classic infinite ladder circuit discussed in the Feynman Lectures on Physics. For the Feynman-Sierpinski ladder circuit (pictured) we can rigorously prove the convergence of the effective impedances using the dynamics of Möbius transformations.
A refinement of our results (concerning the energy measure) on the F-S ladder circuit was attained recently by Patricia Alonso-Ruiz.

List of publications (through early 2017)

Regularized Laplacian determinants of self-similar fractals, with Alexander Teplyaev and Konstantinos Tsougkas. Submitted
Wave equations on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials, with Ulysses Andrews, Grigory Bonik, Richard W. Martin, and Alexander Teplyaev. J. Fourier Anal. Appl. 23 (2017) 994-1027. (Click here for the wave animations described in the paper.)
Stabilization by Noise of a \(\mathbb{C}^2\)-Valued Coupled System, with Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O'Connell, and Fan Ny Shum. (2015 UConn math REU stochastics group) Stoch. Dyn. 17 (2017) 1750046.
Singularly continuous spectrum of a self-similar Laplacian on the half-line, with Alexander Teplyaev. J. Math. Phys. 57 052104 (2016).
Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces, with Stanislav Molchanov and Alexander Teplyaev. J. Phys. A: Math. Theor. 48 395203 (2015).
Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs, with Baris Evren Ugurcan. Stoch. Proc. Appl. 125 (2015) 4632-4673.
Periodic billiard orbits of self-similar Sierpinski carpets, with Robert Niemeyer. J. Math. Anal. Appl. 416 (2014) 969-994.
Quantum Theory of Cavity-Assisted Sideband Cooling of Mechanical Motion, with Florian Marquardt, Aashish Clerk, and Steven M. Girvin. Phys. Rev. Lett. 99 093902 (2007).

Papers in preparation (2017)

Strong shape theorems in cellular automata models on the Sierpinski gasket, with Jonah Kudler-Flam.
Part III: Semilinear evolution equations on resistance spaces, with Michael Hinz and Alexander Teplyaev.
Part IV: Hydrodynamic limit of the boundary-driven exclusion process on the Sierpinski gasket, with Michael Hinz and Alexander Teplyaev.
Anderson localization on infinite fractal lattices, with Stanislav Molchanov and Alexander Teplyaev.

Funding

As of September 2017, my research is supported in part by the Simons Foundation (via a Collaboration Grant for Mathematicians, 2017-2022) and the Research Council of Colgate University.


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