James H. Nolen | Scholars@Duke

James H. Nolen
Associate Professor of Mathematics

I study partial differential equations and probability, which have been used to model many phenomena in the natural sciences and engineering. In some cases, the parameters for a partial differential equation are known only approximately, or they may have fluctuations that are best described statistically. So, I am especially interested in differential equations modeling random phenomena and whether one can describe the statistical properties of solutions to these equations. Asymptotic analysis has been a common theme in much of my research. Current research interests include: reaction diffusion equations, homogenization of PDEs, stochastic dynamics, interacting particle systems.

Office Hours

Mondays 2:30-4:00
Wednesdays, 10:30-12:00

Current Appointments & Affiliations

Associate Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2016

Contact Information

243 Physics Bldg, Durham, NC 27708
Box 90320, Durham, NC 27708-0320
nolen@math.duke.edu (919) 660-2862
http://math.duke.edu/~nolen/

Background
Education, Training, & Certifications
- Ph.D., University of Texas at Austin 2006
- B.S., Davidson College 2000
Duke Appointment History
- Assistant Professor of Mathematics, Mathematics, Trinity College of Arts & Sciences 2009 - 2015
Academic Positions Outside Duke
- Postdoctoral Scholar, Mathematics Department, Stanford University. 2006 - 2008
Research
Selected Grants
- CAREER: Research and training in stochastic dynamics awarded by National Science Foundation 2014 - 2020
- Analysis of Fluctuations awarded by National Science Foundation 2010 - 2015
Fellowships, Supported Research, & Other Grants
- NSF Postdoctoral Research Fellowship awarded by National Science Foundation 2006 - 2008
Publications & Artistic Works
Selected Publications
- Academic Articles
  - Lu, J., Lu, Y., & Nolen, J. (2019). Scaling limit of the Stein variational gradient descent: The mean field regime. Siam Journal on Mathematical Analysis, 51(2), 648–671. https://doi.org/10.1137/18M1187611
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  - Cristali, I., Ranjan, V., Steinberg, J., Beckman, E., Durrett, R., Junge, M., & Nolen, J. (2018). Block size in geometric(P)-biased permutations. Electronic Communications in Probability, 23. https://doi.org/10.1214/18-ECP182
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  - Mourrat, J. C., & Nolen, J. (2017). Scaling limit of the corrector in stochastic homogenization. Annals of Applied Probability, 27(2), 944–959. https://doi.org/10.1214/16-AAP1221
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  - Nolen, J., Roquejoffre, J. M., & Ryzhik, L. (2017). Convergence to a single wave in the Fisher-KPP equation. Chinese Annals of Mathematics. Series B, 38(2), 629–646. https://doi.org/10.1007/s11401-017-1087-4
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  - Gloria, A., & Nolen, J. (2016). A Quantitative Central Limit Theorem for the Effective Conductance on the Discrete Torus. Communications on Pure and Applied Mathematics, 69(12), 2304–2348. https://doi.org/10.1002/cpa.21614
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  - Hamel, F., Nolen, J., Roquejoffre, J. M., & Ryzhik, L. (2016). The logarithmic delay of KPP fronts in a periodic medium. Journal of the European Mathematical Society, 18(3), 465–505. https://doi.org/10.4171/JEMS/595
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  - Nolen, J. (2016). Normal approximation for the net flux through a random conductor. Stochastics and Partial Differential Equations: Analysis and Computations, 4(3), 439–476. https://doi.org/10.1007/s40072-015-0068-4
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  - Bhamidi, S., Hannig, J., Lee, C. Y., & Nolen, J. (2015). The importance sampling technique for understanding rare events in Erdős-Rényi random graphs. Electronic Journal of Probability, 20. https://doi.org/10.1214/EJP.v20-2696
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  - Lu, J., & Nolen, J. (2015). Reactive trajectories and the transition path process. Probability Theory and Related Fields, 161(1–2), 195–244. https://doi.org/10.1007/s00440-014-0547-y
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  - Huckemann, S., Mattingly, J. C., Miller, E., & Nolen, J. (2015). Sticky central limit theorems at isolated hyperbolic planar singularities. Electronic Journal of Probability, 20. https://doi.org/10.1214/EJP.v20-3887
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  - Nolen, J., Roquejoffre, J. M., & Ryzhik, L. (2015). Power-Like Delay in Time Inhomogeneous Fisher-KPP Equations. Communications in Partial Differential Equations, 40(3), 475–505. https://doi.org/10.1080/03605302.2014.972744
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  - Hotz, T., Huckemann, S., Le, H., Marron, J. S., Mattingly, J. C., Miller, E., … Skwerer, S. (2013). Sticky central limit theorems on open books. The Annals of Applied Probability, 23, 2238–2258. https://doi.org/10.1214/12-AAP899
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  - Nolen, J. (2013). Normal approximation for a random elliptic equation. Probability Theory and Related Fields, 1–40. https://doi.org/10.1007/s00440-013-0517-9
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  - Hamel, F, Nolen, J, Roquejoffre, JM, & Ryzhik, L. (2012). A short proof of the logarithmic Bramson correction in Fisher-KPP equations(Accepted). Networks and Heterogeneous Media ,. (Academic Article) Link to Item
  - Matic, I, & Nolen, J. (2012). A Sublinear Variance Bound for Solutions of a Random Hamilton-Jacobi Equation. Journal of Statistical Physics , 149 (2), 342-361. Full Text
  - Mellet, A, & Nolen, J. (2012). Capillary drops on a rough surface. Interfaces and Free Boundaries , 14 (2), 167-184. Full Text
  - Nolen, J, Roquejoffre, JM, Ryzhik, L, & Zlatoš, A. (2012). Existence and Non-Existence of Fisher-KPP Transition Fronts. Archive for Rational Mechanics and Analysis , 203 (1), 217-246. Full Text
  - Cardaliaguet, P., Nolen, J., & Souganidis, P. E. (2011). Homogenization and Enhancement for the G-Equation. Archive for Rational Mechanics and Analysis, 199(2), 527–561. https://doi.org/10.1007/s00205-010-0332-8
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  - Nolen, J., & Novikov, A. (2011). Homogenization of the G-equation with incompressible random drift in two dimensions. Communications in Mathematical Sciences, 9(2), 561–582. https://doi.org/10.4310/CMS.2011.v9.n2.a11
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  - Nolen, J. (2011). An invariance principle for random traveling waves in one dimension. Siam Journal on Mathematical Analysis , 43 (1), 153-188. Full Text
  - Nolen, J. (2011). A central limit theorem for pulled fronts in a random medium. Networks and Heterogeneous Media , 6 (2), 167-194. Full Text
  - Nolen, J., Xin, J., & Yu, Y. (2009). Bounds on front speeds for inviscid and viscous G-equations. Methods and Applications of Analysis, 16(4).
    Link to Item
  - Nolen, J., & Papanicolaou, G. (2009). Fine scale uncertainty in parameter estimation for elliptic equations. Inverse Problems, 25(11). https://doi.org/10.1088/0266-5611/25/11/115021
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  - Mellet, A., Nolen, J., Roquejoffre, J. M., & Ryzhik, L. (2009). Stability of generalized transition fronts. Communications in Partial Differential Equations, 34(6), 521–552. https://doi.org/10.1080/03605300902768677
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  - Nolen, J., & Ryzhik, L. (2009). Traveling waves in a one-dimensional heterogeneous medium. Annales De L’Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis, 26(3), 1021–1047. https://doi.org/10.1016/j.anihpc.2009.02.003
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  - Nolen, J., & Xin, J. (2009). KPP Fronts in 1D Random Drift. Discrete and Continuous Dynamical Systems B, 11(2).
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  - Nolen, J., & Xin, J. (2009). Asymptotic spreading of KPP reactive fronts in incompressible space-time random flows. Annales De L’Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis, 26(3), 815–839. https://doi.org/10.1016/j.anihpc.2008.02.005
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  - Nolen, J., & Xin, J. (2009). KPP fronts in a one-dimensional random drift. Discrete and Continuous Dynamical Systems Series B, 11(2), 421–442. https://doi.org/10.3934/dcdsb.2009.11.421
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  - Nolen, J., & Xin, J. (2008). Variational principle and reaction-diffusion front speeds in random flows. Iciam07 Proceedings, 1040701–1040702.
  - Nolen, J., & Xin, J. (2008). Computing reactive front speeds in random flows by variational principle. Physica D: Nonlinear Phenomena, 237(23), 3172–3177. https://doi.org/10.1016/j.physd.2008.04.024
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  - Nolen, J., Papanicolaou, G., & Pironneau, O. (2008). A framework for adaptive multiscale methods for elliptic problems. Multiscale Modeling and Simulation, 7(1), 171–196. https://doi.org/10.1137/070693230
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  - Nolen, J., & Xin, J. (2007). Variational Principle of KPP Front Speeds in Temporally Random Shear Flows with Applications. Communications in Mathematical Physics, 269, 493–532.
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  - Nolen, J., & Xin, J. (2005). Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete and Continuous Dynamical Systems, 13(5), 1217–1234.
  - Nolen, J., & Xin, J. (2005). A variational principle based study of KPP minimal front speeds in random shears. Nonlinearity, 18(4), 1655–1675. https://doi.org/10.1088/0951-7715/18/4/013
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  - Nolen, J., Rudd, M., & Xin, J. (2005). Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds. Dynamics of Pde, 2, 1–24.
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  - Boye, D. M., Valdes, T. S., Nolen, J. H., Silversmith, A. J., Brewer, K. S., Anderman, R. E., & Meltzer, R. S. (2004). Transient and persistent spectral hole burning in Eu3+-doped sol-gel produced SiO2 glass. Journal of Luminescence, 108(1–4), 43–47. https://doi.org/10.1016/j.jlumin.2004.01.008
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  - Nolen, J., & Xin, J. (2004). Min-Max Variational Principles and Fronts Speeds in Random Shear Flows. Methods and Applications of Analysis, 11(4), 635–644.
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  - Nolen, J., & Xin, J. (2003). Reaction-diffusion front speeds in spatially-temporally periodic shear flows. Multiscale Modeling and Simulation, 1(4), 554–570. https://doi.org/10.1137/S1540345902420234
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  - Boye, D. M., Silversmith, A. J., Nolen, J., Rumney, L., Shaye, D., Smith, B. C., & Brewer, K. S. (2001). Red-to-green up-conversion in Er-doped SiO2 and SiO2-TiO2 sol-gel silicate glasses. Journal of Luminescence, 94–95, 279–282. https://doi.org/10.1016/S0022-2313(01)00301-5
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  - Nolen, J. H., Cohn, S., Iyer, G., & Pego, R. (n.d.). Anomalous diffusion in one and two dimensional combs.
    Link to Item
  - Nolen, J. H., Lu, J., & Lu, Y. (n.d.). Scaling limit of the Stein variational gradient descent: the mean field regime. Siam Journal on Mathematical Analysis.
    Link to Item
  - Nolen, J., Roquejoffre, J.-M., & Ryzhik, L. (n.d.). Refined long-time asymptotics for Fisher–KPP fronts. Communications in Contemporary Mathematics, 1850072–1850072. https://doi.org/10.1142/s0219199718500724
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- Conference Papers
  - Nolen, J., Pavliotis, G. A., & Stuart, A. M. (2012). Multiscale modelling and inverse problems. In Lecture Notes in Computational Science and Engineering (Vol. 83, pp. 1–34). https://doi.org/10.1007/978-3-642-22061-6_1
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Teaching & Mentoring
Recent Courses
- MATH 340: Advanced Introduction to Probability 2019
- MATH 641: Probability 2019
- MATH 740: Advanced Introduction to Probability 2019
- MATH 790-90: Minicourse in Advanced Topics 2019
- MATH 356: Elementary Differential Equations 2018
- BIOLOGY 490: Topics in Biology 2017
- MATH 490: Topics in Mathematics 2017

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Education, Training, & Certifications

Duke Appointment History

Academic Positions Outside Duke

Selected Grants

Fellowships, Supported Research, & Other Grants

Selected Publications

Academic Articles

Conference Papers

Recent Courses