Publications

• • Obituary: P.W. Millar, 1939-2024 (with Carl Mueller, Maria Eulalia Vares, and Guenther Walther)

IMS Bulletin, Vol.54, Issue 3:10-11.

• • Small-ball constants, and exceptional flat points of SPEs (with Kunwoo Kim and Carl Mueller)

Electr. J. Probab. 29(180):1-31.

• Instantaneous everywhere-blowup of parabolic SPDEs (with Mohammud Foondun and Eulalia Nualart)

Probab. Th. Rel. Fields 190:601-624.

• On the valleys of the stochastic heat equation (with Kunwoo Kim and Carl Mueller)

Ann. Applied Probab. 34(1B):1177-1198.

• • Dissipation in parabolic SPDEs II: Oscillation and decay of the solution (with Kunwoo Kim and Carl Mueller)

Ann. Inst. Henri Poincaré 59(3):1610-1641.

• Optimal regularity of SPDEs with additive noise (with Marta Sanz-Solé)

Electr. J. Probab. 28:1-31.

• Phase analysis of a family of stochastic reaction-diffusion equations (with Kunwoo Kim and Carl Mueller)

Electr. J. Probab. 28:1-66.

• Central limit theorems for spatial averages of the stochastic heat equation via Malliavin calculus (with Le Chen, David Nualart, and Fei Pu)

Stoch. and Partial Diff. Eq. 11(1):122-176.

• Central limit theorems for parabolic stochastic partial differential equations
  (w/ Le Chen, David Nualart, and Fei Pu) Ann. Instit. H. Poincaré 58 no. 2, 1052-1077.

• Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition
  (with Le Chen, David Nualart, and Fei Pu) J. Functional Analysis 282(2) Paper No. 109290.

• Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension d≥1
  (with David Nualart and Fei Pu) SIAM J. Math. Analysis 53(2) 2084-2133.

• Spatial ergodicity for SPDEs via Poincaré-type inequalities
  (with Le Chen, David Nualart, and Fei Pu) Electr. J. Probab. 26 1-37.

• A CLT for dependent random variables, with an application to an infinite system of interacting diffusion processes
  (with Le Chen, David Nualart and Fei Pu) Proceedings of the Amer. Math. Soc. 149(12) 5367-5384.

1. Analysis of a stratified Kraichnan model
   (with Jingyu Huang) Electronic J. Probab. 25 Article no. 122, 1-67.

2. Dissipation in parabolic SPDEs
   (with Kunwoo Kim, Carl Mueller, and Shang-Yuan Shiu) J. Statist. Phys. 179 502-534.

• Dense blowup for parabolic SPDEs
  (with Le Chen, Jingyu Huang, and Kunwoo Kim) Electronic J. Probab. Paper No. 118, 33 pp.

• Talagrand concentration inequalities for stochastic partial differential equations (with Andrey Sarantsev) Stochastic and Partial Differential Equations: Analysis & Computation 7(4) 679-698.

• Global solutions to reaction-diffusion equations with super-linear drift and multiplicative noise
  (with Robert C. Dalang and Tusheng Zhang) Ann. Probab. 47(1) 519-559.

• The dimension of the range of a random walk
  (with Nicos Georgiou, Kunwoo Kim, and Alex D. Ramos) Electr. J. Probab. 23(83) 1-31.

• A macroscopic multifractal analysis of parabolic stochastic PDEs
  (Online access; with Kunwoo Kim and Yimin Xiao) Comm. Math. Phys. 360, 307-346.

• On the multifractal local behavior of parabolic stochastic PDEs
  (with Jingyu Huang) Electr. Comm. Probab. 49 11 pp. 

• Intermittency and multifractality: A case study via stochastic PDEs
  (with Kunwoo Kim and Yimin Xiao) Ann. Probab., Vol. 45, No. 6A, 3697-3751.

• A conversation with Mu-Fa Chen
  (with Ed Waymire) Notices of the AMS; Vol. 64, No. 6, 616-619.

• On the macroscopic fractal geometry of some random sets
  (with Yimin Xiao) In: Stochastic Analysis and Related Topics, Progress in Probability, Vol. 72, 179-206, Birkhäuser, Springer International AG 2017 (Fabrice Baudoin and Jonathan Peterson, ed.s)

• A boundedness trichotomy for the stochastic heat equation
  (with Le Chen and Kunwoo Kim) Annales de l'Inst. Henri Poincaré Vol. 53, No. 4, 1991-2004.

• Strong invariance and noise comparison principles for some parabolic SPDEs
  (with Mathew Joseph and Carl Mueller) Ann. Probab. Vol. 45, No. 1, 377-403.

• Dissipation and high disorder
  (with Le Chen, Michael Cranston, and Kunwoo Kim) Ann. Probab. Vol. 45, No. 1, 82-99.

• Decorrelation of total mass via energy
  (with Le Chen and Kunwoo Kim) Potential Analysis 45 157-166.

• Semi-discrete semi-linear parabolic SPDEs
  (with Nicos Georgiou, Mathew Joseph, and Shang-Yuan Shiu) Ann. Applied Probab. 25(5) 2959-3006.

• Non-linear noise excitation and intermittency under high disorder
  (with Kunwoo Kim) Proc. Amer. Math. Soc. 143(9) 4073-4083.

• Non-linear noise excitation of intermittent stochastic PDEs and the topology of LCA groups
  (with Kunwoo Kim) Ann. Probab. 43(4) 1944-1991.

• Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion
  (with Mohammud Foondun and Pejman Mahboubi) Stoch. Partial Differ. Equ.: Anal. Comput. 3(2) 133-158.

• Brownian motion and thermal capacity
  (with Yimin Xiao) Ann. Probab. 43(1) 405-434.

• Initial measures for the stochastic heat equation
  (with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu) Ann. Inst. H. Poincaré Probab. Statist. 50(1) 136-153.

• Parabolic SPDEs and intermittency
  Markov Processes Relat. Fields 20 45-80.

• Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimention \(k\ge1\)
  (with Robert C. Dalang and Eulalia Nualart) SPDEs: Analysis and Computations 1 94-151.

• Intermittency and chaos for a non-linear stochastic wave equation in dimension 1
  (with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu) In: Malliavin Calculus and Stochastic Analysis, Springer Proceedings in Mathematics and Statistics, Vol. 34, 251-279.

• On the chaotic character of the stochastic heat equation, before the onset of intermittency
  (with D. Conus and M. Joseph) Ann. Probab. 41(3B) 2225-2260

• On the chaotic character of the stochastic heat equation, II
  (with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu) Probab. Theory Rel. Fields 156 483-533.

• On the stochastic heat equation with spatially-colored random forcing
  (with M. Foondun) Trans. Amer. Math. Soc. 365(1) 409-458.

• Weak nonmild solutions to some SPDEs
  (with Daniel Conus) Ill. J. Math. 54(4) (2010) 1329-1341 [published in 2012].

• Correlation-length bounds, and estimates for intermittent islands in parabolic SPDEs
  (with Daniel Conus and Mathew Joseph) Electr. J. Probab. Vol. 17, Paper 102 (15 pp).

• Packing dimension profiles and Lévy processes
  (with René Schilling and Yimin Xiao) Bull. London Math. Soc. 44 931-943.

• An asymptotic theory for randomly-forced discrete nonlinear heat equations
  (with Mohammud Foondun) Bernoulli 18(3) 1042-1060.

• On the existence and position of the farthest peaks of a family of stochastic heat and wave equations
  (with Daniel Conus) Probab. Theory Rel. Fields 152(3-4) 681-701.

• Critical Brownian sheet does not have double points
  (with R.C. Dalang, E. Nualart, D. Wu, and Y. Xiao) Ann. Probab. 40 (4) 1829-1859.

• Charged polymers in the attractive regime: a first order transition from Brownian scaling to four-point localization
  (with Y. Hu and M. Wouts) J. Statist. Physics, 144(5) (2011), pp. 948--977.

• Dynkin's isomorphism theorem and the stochastic heat equation
  (with N. Eisenbaum and M. Foondun) Potential Analysis, 34 (2011), no. 3, pp. 243-260.

• A local-time correspondence for stochastic partial differential equations
  (with Mohammud Foondun and Eulalia Nualart) The Transactions of the AMS, Vol. 363 (5) 2481-2515.

• On the global maximum of the solution to the stochastic heat equation with compact-support initial data
  (with Mohammud Foondun) Ann. Inst. Henri Poincaré Probab. Stat. 46(4) 895-907.

• Strong approximations in a charged-polymer model
  (with Yueyun Hu) Periodica Mathematica Hungarica, Vol. 61(1-2) 213-224. 

• Zeros of a two-parameter random walk
  (with P. Révész) In "Dependence in Probability, Analysis, and Number Theory," 265-278 (I. Berkes, R. C. Bradley, H. Dehling, M. Peligrad, and R. Tichy, editors) Kendrick Press, Heber City.

• Charged polymers and random walk in random scenery
  (with Xia Chen) In: Optimality: The Third Erich L. Lehmann Symposium, IMS Lecture Notes-Monograph Series, Vol. 57 (Javier Rojo, Editor), 237-251, IMS, Hayward, California.

• Harmonic analysis of additive Lévy processes
  (with Yimin Xiao) Probability Theory and Related Fields 145, 459-515.

• From fractals and probability to Lévy processes and stochastic PDEs
  Progress in Probability, Vol. 61, 111-141 (mostly a Survey).

• Hitting probabilities for systems of nonlinear stochastic heat equations with multiplicative noise
  (with Robert C. Dalang and Eulalia Nualart) Probability Theory and Related Fields 177, 371-427.

• Intermittence and nonlinear stochastic partial differential equations
  (with Mohammud Foondun) Electronic J. Probab. Vol. 14, Paper no. 21, 548-568.

• Capacities in Wiener space, quasi-sure lower functions, and Kolmogorov's ε-entropy
  (with David A. Levin and Pedro J. Méndez-Hernández) Stochastic Proc. Appl.

• 118 no. 10, 1723-1737.Level sets of the stochastic wave equation driven by a symmetric Lévy noise.
  (with Eulalia Nualart) Bernoulli 14, 899-925.

• Packing-dimension profiles and fractional Brownian motion. (with Yimin Xiao)
  The Math. Proc. Cambridge Phil. Soc. 145, no. 1, 205-213.

• Packing dimension of the range of a Lévy process. (with Yimin Xiao)
  The Proceedings of the AMS, Vol. 136, No. 7, 2597-2607.

• Slices of the Brownian sheet: New results and open problems.
  Stochastic Analysis, Random Fields, and Applications V. Progress in Probability 59, 135-174, Birkhäuser Verlag 2007, Basel.

• Dynamical percolation on general trees.
  Probability Theory and Related Fields 140(1-2), 129-167.

• Hausdorff dimension of the contours of symmetric additive Lévy processes (with Narn-Rueih Shieh and Yimin Xiao).Probability Theory and Related Fields 140(1-2), 169-193.

• Hitting probabilities for systems of nonlinear stochastic heat equations with additive noise (with Robert C. Dalang and Eulalia Nualart) ALEA, Vol. III, 231-371.

• Normal numbers are normal (Expository).
  Clay Mathematics Institute Annual Report 2006, p. 15 (continued on pp. 27-31).

• Images of the Brownian sheet (with Yimin Xiao).
  The Transactions of the AMS, Vol. 359, No. 7, 3125-3151.

• Convex rearrangements, generalized Lorenz curves, and correlated Gaussian data (with Y. Davydov, Z. Shi, and R. Zitikis). The J. of Statistical Planning and Inference 137, 915-934.

• Sectorial local non-determinism and the geometry of the Brownian sheet (with Dongsheng Wu and Yimin Xiao). Electr. J. Probab., Vol. 11, Paper 33, pp. 817-843.

• A note on a.s. finiteness of perpetual integral functionals of diffusions (with Paavo Salminen and Marc Yor), Electr. Comm. Probab., Vol. 11, Paper 11, pp. 108-117.

• Exceptional times and invariance for dynamical random walks (with David Asher Levin and Pedro J. Méndez-Hernández). Probab. Th. Rel. Fields 134(3), 383-416. (Earlier preprint)

• An extreme-value analysis of the LIL for Brownian motion (with David A. Levin and Zhan Shi) Electr. Comm. Probab.,Vol. 10, Paper 20, pp. 196-206.

• On dynamical Gaussian random walks (with David Asher Levin and Pedro J. Méndez-Hernández) Ann. Probab. 33(4), 1452-1478.

• Lévy processes: capacity, and Hausdorff dimension (With Yimin Xiao)
  Ann. Probab. 33(3), 841-878.

• Level crossings of a two-parameter random walk, (With P. Révész, and Z. Shi)
  Stoch. Proc. Their Appl. 115, 359-380.

• On the explosion of local times of Brownian sheet along lines, (with P. Révész and Z. Shi)
  Ann. IHP. 40(1), 1-24.

• Brownian sheet and quasi-sure analysis,
  Fields Inst. Comm. Vol. 44, 25-47. (Survey)

• Recurrent lines in two-parameter isotropic stable stable sheets, (with R. C. Dalang)
  Stoch. Proc. Appl. 114, 81-107.

• Additive Lévy processes: Capacity and dimension, (with Yimin Xiao)
  Progress in Probab. Vol. 57, 151-170. (Survey)

• Optimal reward on a sparse tree with random edge-weights, (with T. M. Lewis)
  J. of Applied Probab., Vol. 40(4), 926-945.

• Measuring the range of an additive Lévy process (or "Local Times of Additive Lévy Processes, II: Existence;" w/ Y. Xiao and Y. Zhong, Ann. Probab. 31(2), 1097-1141.

• Intersections of Brownian motions,
  Expos. Math., 21(2), 97-114.

• Local times of additive Lévy processes (with Y. Xiao and Y. Zhong)
  Stoch. Proc. Their Appl., 104, 193-216.

• Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes. (with Y. Xiao) The Proceedings of the AMS, 131(8), 2611-2616.

• The codimension of the zeros of a stable process in random scenery. Sém. de Probab. XXXVII Lecture Notes in Math.1832, 236-245.

• On the most visited sites of symmetric Markov processes. (w/ N. Eisenbaum)
  Stoch. Proc. and Their Appl., 101, 241-256.

• Bounds on gambler's ruin probabilities in terms of moments (w/ S. N. Ethier)
  Methodology and Computing in Applied Probab., 4(1), 55-68.

• Level sets of additive Lévy processes (w/ Y. Xiao)
  Ann. Probab., 30(1), 62-100.

• Images and level sets of additive random walks (w/ Y. Xiao), In: High-dimensional Probab.II, 329-346, Birkhaüser, Boston, (Seattle, 1999; Ed's: E. Giné, D. M. Mason, J. A. Wellner).

• Limsup random fractals (w/ Y. Peres and Y. Xiao)
  Elec. J. Probab., 5(4), 1-24.

• Sojourn times of Brownian sheet (w/ R. Pemantle)
  Periodica Math. Hung., 41(1-2), 187-194.

• On sums of iid random variables indexed by N parameters
  Séminaire de Probab. XXXIV, Lec. Notes in Math., 151-156.

• Fast sets and points for fractional Brownian motion (w/ Z. Shi)
  Séminaire de Probab. XXXIV, Lec. Notes in Math., 393-416.

• Boundary crossings and the distribution function of the maximum of Brownian sheet (w/ E. Csáki and Z. Shi), Stoch. Proc. Their Appl., 90, 1-18.

• Brownian sheet images and Bessel-Riesz capacity
  Trans. Amer. Math. Soc. 351(7), 2607-2622.

• Iterated Brownian motion and its intrinsic skeletal structure (w/ T. M. Lewis)
  Progress in Probab.,45, 201-210, Birkhäuser-Verlag, Basel.

• Capacity estimates, boundary crossings and the Ornstein-Uhlenbeck process in Wiener space (w/ E. Csáki and Z. Shi), Elec. Comm. Probab., 4(13), 103-109.

• Stochastic calculus for Brownian motion on a Brownian fracture (w/ T. M. Lewis)
  Ann. Applied Probab., 9, 629-667.

• Brownian sheet and capacity (w/ Z. Shi)
  Ann. Probab., 27, 1135-1159.

• A law of the iterated logarithm for stable processes in random scenary (w/ T. M. Lewis)
  Stoch. Proc. Their Appl., 74, 89-121.

• Brownian motion in a Brownian crack (w/ K. Burdzy)
  Ann. Applied Probab., 8(3), 708-748.

• Chung's law for integrated Brownian motion (w/ Z. Shi)
  Trans. Amer. Math. Soc., 350(10), 4253-4246.

• Logarithmic averages of stable random variables are asymptotically normal (w/ I. Berkes and L. Horváth), Stoch. Proc. Their Appl., 77, 35-51.

• Gaussian measure of a small ball and capacity in Wiener space (w/ Z. Shi), Asymptotic Methods in Probab. Statist., 453-465, Elsevier Science, Amsterdam, (ed: B. Szyszkowicz).

• Some polar sets for the Brownian sheet
  Séminaire de Probab., XXXI, Lec. Notes in Math. 1655, 190-197.

• Escape rates for Lévy processes
  Stud. Sci. Math. Hung., 33, 177-183.

• The uniform modulus of iterated Brownian motion (w/ T. M. Lewis)
  J. Theoret. Probab., 9(2), 317-333.

• Chung's law of the iterated logarithm for iterated Brownian motion (w/ T. M. Lewis)
  Ann. Inst. de l' IHP, 32(3), 349-359.

• A strong approximations for logarithmic averages (w/ L. Horváth)
  Stud. Sci. Math. Hungar., 31, 187-196.

• On a problem of Erdös and Taylor (w/ T. M. Lewis and Z. Shi)
  Ann. Probab., 23(3), 761-787.

• Lévy classes and self-normalization
  Elec. J. Probab., 1(1), 1-18.

• Deviation inequalities for continuous martingales
  Stoch. Proc. Their Appl., 65, 17-30.

• The gap between the past supremum and the future infimum of a transient Bessel process
  Séminaire de Probab. XXIX, Lec. Notes in Math. 1613, 220-230.

• The distribution of bubbles of Brownian sheet
  Ann. Probab.,23(2), 786-805.

• The favorite point of a Poisson process (w/ T. M. Lewis)
  Stoch. Proc. Their Appl., 57, 19-38.

• Weight functions and pathwise local central limit theorems (w/ L. Horváth)
  Stoch. Proc. Their Appl., 59, 105-123.

• The level sets of iterated Brownian motion (w/ K. Burdzy)
  Séminaire de Probab. XXIX, Lec. Notes in Math. 1613, 231-236.

• A discrete fractal in Z1
  Proc. Amer. Math. Soc., 120(2), 577-584.

• Intersection local time for points of infinite multiplicity. (w/ R. F. Bass and K. Burdzy)
  Ann. Probab., 22(2), 566-626.

• Exact rates of convergence to Brownian local times
  Ann. Probab., 22(3), 1295-1330.

• Law of the iterated logarithm for local times of the empirical process. (w/ R. F. Bass)
  Ann. Probab., 32(1), 388-399.

• On the future infima of some transient processes. (w/ T. M. Lewis and W. V. Li)
  Probab. Theory Related Fields, 99, 337-360.

• An embedding of compensated compound Poisson processes with applications to local time
  Ann. Probab., 21(1), 340-361.

• Intersection local times and Tanaka formulas, (w/ R. F. Bass)
  Ann. Inst. de l' IHP, 29(3), 419-451.

• Strong approximations to Brownian local time (w/ R. F. Bass)
  Sem. in Stoch. Proc., 1992, 43-65, Birkhaüser, (E. Çinlar, K. L. Chung, M. J. Sharp, R. F. Bass and K. Burdzy, Ed.'s).

• Rates of convergence to Brownian local time (w/ R. F. Bass)
  Stoch. Proc. Their Appl., 47, 197-213.

• Level crossings of empirical processes
  Stoch. Proc. Their Appl., 43, 331-343.

• Moment inequalities for functionals of Brownian convex hull
  Ann. Probab., 20(2), 627-630.

• Local asymptotic laws for Brownian convex hull
  Probab. Theory Related Fields, 93, 377-392.

• Stochastic calculus and continuity of local times of Lévy processes (w/ R. F. Bass)
  Séminaire de Probab. , XXVI, Lec. Notes in Math., 1526, 1-11.

• Local times on curves and uniform invariance principles (w/ R. F. Bass)
  Probab. Theory Related Fields, 92, 465-492.