Lihu Xu (徐禮虎)

I am now an associate professor in the Department of Mathematics at the University of Macau, I am also affiliate to the Center for Applied Mathematics and the Center for Artificial Intelligence and Robotics at UM. My google scholar page is here.

I received my BSc at Shandong University in 07/2001, MSc at Peking University in 07/2004, and PhD at Imperial College in 01/2008. I did postdoctoral research at Bonn University (09/2007-08/2008), Technology University of Eindhoven (09/2008-08/2010), and Technology University of Berlin (10/2010-08/2011). From 09/2011 to 11/2013, I was a lecturer (permanent position, equal to assistant professor) at Brunel University, meanwhile, I visited Boston College (10/2013-12/2013). I was an assistant professor from 01/2014 and an associate professor from 08/2019 at UM. 

Teaching at UM

I was awarded the Teaching Excellence Award of Faculty of Science and Technology in 2018/2019, and nominated for this award in 2021/2022.  

  • 2022/2023-1: Computational Statistics (UG)
  • 2021/2022-2: Introduction to Stochastic Calculus (UG)
  • 2021/2022-1: Topics in Probability and Statistics: High Dimensional Probability (PG)
  • 2021/2022-1:  Data-driven Sampling (UG)
  • 2020/2021-2: Introduction to Stochastic Calculus (UG)
  • 2020/2021-2: Computational Statistics (UG)
  • 2020/2021-1: Topics in Probability and Statistics: High Dimensional Probability (PG)
  • 2019/2020-2:  Introduction to Stochastic Differential Equations (PG)
  • 2019/2020-1:  Introduction to Stochastic Calculus (UG)
  • 2019/2020-1:  Probability (UG)
  • 2018/2019-2:  Applied Statistics (UG)
  • 2018/2019-2:  Introduction to Stochastic Differential Equations (PG)
  • 2018/2019-1:  Introduction to Stochastic Calculus (UG)
  • 2017/2018-2:  Introduction to Stochastic Differential Equations (PG)
  • 2017/2018-1:  Introduction to Stochastic Calculus (UG)
  • 2017/2018-1:  Probability (UG)
  • 2016/2017-2:  Introduction to Real Analysis and Hilbert Space(UG)
  • 2016/2017-1:  Introduction to Stochastic Calculus (UG)
  • 2016/2017-1:  Probability (UG)
  • 2015/2016-2:  Advanced Probability (PG)
  • 2015/2016-1:  Introduction to Stochastic Calculus (UG)
  • 2015/2016-1:  Probability (UG)
  • 2014/2015-2:  Probability and Statistics for Engineering (UG)
  • 2014/2015-1:  Introduction to Stochastic Calculus (UG)
  • 2014/2015-1:  Probability (UG)
  • 2013/2014-2:  Probability and Statistics for Engineering (UG) 

Teaching at Brunel

  • 2012/2013-2:  Probability (UG)
  • 2012/2013-1:  Analysis II (UG) 
  • 2011/2012-2:  Analysis I (UG) 
  • 2011/2012-1: Financial Mathematics (UG, Final Year Project Course)

Teaching at TU Berlin

  • 2010/2011-2: Introduction to Stochastic Partial Differential Equations (PhD)

If you are interested in my research or being a PhD student under my supervision, please feel free to contact me by email:, You may find the information of PhD program and financial support from here


1. Dr. Huiming Zhang (2020–2022): Tenure track Associate Professor at Beihang University (北京航空航天大學)

PhD Students

  1. Dr. Fang Xie(2015–2018): L1 penalized methods in high-dimensional regressions and its theoretical properties,   Assistant Professor at UIC of BNU-HKBU
  2. Dr. Peng Chen (2017–2021): Probability approximations with their applications to limit theorems and stochastic algorithms,  Assistant Professor at NanHang University
  3. Dr. Xinghu Jin (2017–2021): Approximation to steady states of M/Ph/M+n queques,    Assistant Professor at Hefei Industry University
  4. Dr. Xin Xu (2017–2021): Aggregation in several chemotaxis models (Joint with Prof. Xuefeng Wang and Prof. Linlin Su at SUSTech),     Postdoc at Nankai University
  5. Dr. Jianya Lu (2018–2022): Limit theorems of time series and stochastic algorithms,     Lecturer  (Assistant Professor)  in Statistics at University of Essex, see the news of UM. 
  6. Ms. Qiuran Yao (2019–): Joint with Prof. Lianjie SHU in FBA at UM
  7. Mr. Xiang Li (2020–):
  8. Mr. Yu Wang (2021–):
  9. Mr. Yingjun Mo (2022–):

Master Students

  1. Ms. Fang Xie (2013—2015): High dimensional lasso regressions
  2. Mr. Libang Deng (2013—2015): Asset pricing models and an SPDE
  3. Mr. Zhilei Fan (2015—2017): False discovery rate and self-normalized Cramer type moderate deviations
  4. Ms. Qiuran Yao (2016–2019): Lasso and estimation consistency of Tweedie’s compound Poisson model with elastic net
  5. Mr. Xiang Li (2018–2020): L1-regression of a truncated estimator of the mean and a stable approximation
  6. Ms. Jinjin Yu (2018–2021): Variable selection with square root loss function based on kernel methods
  7. Mr.  Man-Nok Lou (2019-2021): Applications of linear regressions
  8. Mr. Qiyang Pei (2021–): Gaussian processes and their applications to data science

Research Interests

My research interests include several topics in the field of probability, statistics and data science: stochastic (partial) differential equations, stochastic processes, long term behavior of stochastic systems, large and moderate deviations, normal and nonnormal approximations by Stein’s method, with their applications to high dimensional statistics and stochastic algorithms in machine learning. I am also interested in applying probability to machine learning and mathematical modellings. I was awarded the Research Excellence Award of Faculty of Science and Technology in 2016/2017. 

Research Grants

I have received 7 external grants from FDCT of S.A.R. Macau and NSFC of China. Among them, two general project grants from NSFC were very prestigious and competitive.   


  1. Peng Chen, Qi-Man Shao and Lihu Xu: A probability approximation framework: Markov process approach,  Annals of Applied Probability (to appear).
  2. Peng Chen, Jianya Lu and Lihu Xu: Approximation to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations,  Applied Mathematics & Optimizations (to appear).
  3. Changsong Deng, Rene Schilling and Lihu Xu: Singular integrals of subordinators with applications to structure properties of SPDEs,  
    Transactions of the American Mathematical Society  (to appear).
  4. Peng Chen, Ivan Nourdin, Lihu Xu, Xiaochuan Yang and Rui Zhang: Non-integrable stable approximation by Stein’s method, Journal of Theoretical Probability, 2, 1137-1186, 2022.
  5. Jianya Lu, Yuzhen Tan and Lihu Xu: Central limit theorem and self-normalized Cramer type moderate deviation for Euler-Maruyama scheme,  Bernoulli, 28 (2), 937-964, 2022. 
  6. Peng Chen, Xinghu Jin, Xiang Li and Lihu Xu: A generalized Catoni’s M-estimator under finite \alpha-th moment assumption with \alpha \in (1,2),  Electronic Journal of Statistics, 15 (2), 5523-5544, 2021.
  7. Amarjit Budhiraja, Yong Chen and Lihu Xu: Large deviations of the entropy production rate for a class of Gaussian processes,  
    Journal of Mathematical Physics, 62 (5), 052702, 2021. 
  8. Hao Shen, Jian Song, Rongfeng Sun and Lihu Xu: Scaling limit of a directed polymer among a Poisson field of independent walks, Journal of Functional Analysis, 281 (5), 109066, 2021.
  9. Peng Chen; Ivan Nourdin and Lihu Xu: Stein’s method for asymmetric alpha-stable distribution, with applications to stable CLT,  Journal of Theoretical Probability, 34 (3), 1382-1407, 2021. 
  10. Xiaobin Sun, Ran Wang, Lihu Xu and Xue Yang: Large deviation for two-time-scale stochastic Burgers equation, Stochastic and Dynamics, no. 5, Paper No. 2150023, 37 pp, 2021.
  11. Ran Wang, Jie Xiong and Lihu Xu: Large deviation principle of occupation measures for non-linear monotone SPDEs, Science China Mathematics, 64 (4), 799-822, 2021.
  12. Zhao Dong; Feng-Yu Wang and Lihu Xu: Irreducibility and asymptotics of stochastic Burgers equation driven by alpha-stable processes, Potential Analysis, 52, 371-392, 2020.
  13. Jinzhu Jia, Fang Xie and Lihu Xu: Sparse Poisson regression with penalized weighted score function. Electronic Journal of Statistics, 13(2), 2898–2920, 2019.
  14. Peng Chen and Lihu Xu: Approximation to stable law by the Lindeberg principle. Journal of Mathematical Analysis and Applications, 480(2), 123338, 28 pp, 2019.
  15. Xiaobin Sun, Yingchao Xie and Lihu Xu: Exponential mixing for SPDEs driven by highly degenerate Lévy noises. Illinois Journal of Mathematics, 64(1), 75-102, 2019.
  16. Lihu Xu: Approximation of stable law in Wasserstein-1 distance by Stein’s method, Annals of Applied Probability, 29(1), 458-504, 2019.

    This paper breaks new grounds on the Stein’s famous method and also opens up many new interesting research problems’ (from a 5 pages’ referee’s report).

  17. Xiao Fang; Qi-Man Shao and Lihu Xu: Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula, Probability Theory and Related Fields, 174(3,4), 945–979, 2019.
  18. Lihu Xu: Singular integrals of stable subordinator, Statistics & Probability Letters, Vol. 139, 115-118, 2018.
  19. Xiaobin Sun, Yimin Xiao, Lihu Xu and Jianliang Zhai: Uniform dimension results for a family of Markov processes, Bernoulli, Vol. 24 4B, 3924-3951, 2018.
  20. Ran Wang and Lihu Xu: Asymptotics for stochastic reaction-diffusion equation driven by subordinate Brownian motions, Stochastic Processes and Their Applications, Vol. 128, 5, 1772-1796, 2018.
  21. Fang Xie, Lihu Xu and Youcai Yang: Lasso for sparse linear regression with exponentially β-mixing errors. Statistics and Probability Letters, 125 (2017)64–70.
  22. Ran Wang, Jie Xiong and Lihu Xu: Irreducibility of stochastic real Ginzburg-Landau equation driven by alpha-stable noises and applications, Bernoulli, 23 (2017), no. 2, 1179–1201.
  23. Lihu Xu, Wen Yue and Tusheng Zhang: Smooth densities of the laws of perturbed diffusion processes, Statistics and Probability Letters, 119 (2016)55–62.
  24. Yong Chen, Hao Ge, Jie Xiong and Lihu Xu: The large deviation principle and steady-state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields, Journal of Mathematical Physics, 57 (2016), no. 7, 073302, 16 pp.
  25. Xiao-Hong Chen, Qi-Man Shao, Wei Biao Wu and Lihu Xu: Self-normalized Cramer Type Moderate Deviations under Dependence, Annals of Statistics, 44 (2016), no. 4, 1593–1617.
  26. Feng-Yu Wang, Jie Xiong and Lihu Xu: Asymptotics of sample entropy production rate for stochastic differential equations, Journal of Statistical Physics, 163 (2016), no. 5, 1211–1234.
  27. Zhao Dong, Symon Peszat and Lihu Xu: On some  smoothening effects of the transition semigroup  of  a  Levy process, Journal of Mathematical Analysis and Applications434 (2016), no. 2, 1566–1580.
  28. Matthias Winter, Lihu Xu, Jianliang Zhai and Tusheng Zhang: The dynamics of the stochastic shadow Gierer-Meinhardt system, Journal of Differential Equations 260 (2016), no. 1, 84–114.
  29. Fang Li and Lihu Xu: Finite time blowup of the stochastic shadow Gierer-Meinhardt System, Electronic Communication on Probability, 20 (2015).
  30. Feng-Yu Wang, Lihu Xu and Xicheng Zhang: Gradient estimates for SDEs driven by multiplicative Levy noise, Journal of Functional Analysis, 269 (2015), 3195–3219.
  31. Ran Wang and Lihu Xu: Asymptotics of the Entropy Production Rate for d-Dimensional Ornstein–Uhlenbeck Processes. Journal of Statistical Physics, 160 (2015), no. 5, 1336–1353.
  32. Feng-Yu Wang and Lihu Xu: Log-Harnack inequality for Grushin type semigroup, Revista Mathematica Iberoamericna, 30 (2014), no. 2, 405-418.
  33. Lihu Xu: Exponential mixing for 2D SDEs forced by degenerate Levy noises, Journal of Evolution Equations14 (2014), no. 2, 249-272.
  34. Zhao Dong, Lihu Xu and Xicheng Zhang: Exponential ergodicity of stochastic Burgers equations forced by alpha-stable processes, Journal of Statistical Physics, (2014), no. 4, 929-949.
  35. Lihu Xu: Ergodicity of stochastic real Ginzburg-Landau equations forced by alpha-stable noises, Stochastic Processes and their Applications, 123, no 10, (2013), 3710-3736.
  36. Feng-Yu Wang and Lihu Xu: Derivative Formula and Applications for Hyperdissipative Stochastic Navier-Stokes/Burgers Equations, Infinite Dimensional Analysis Quantum Probability and Related Topics, Vol. 15, No. 3 (2012), pp 19.
  37. Sergio Albeverio, Arnaud Debussche and Lihu Xu: Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises, Applied Mathematics & Optimizations, 66, 2 (2012), 273-308.
  38. Enrico Priola, Armen Shirikyan, Lihu Xu and Jerzy Zabczyk: Exponential ergodicity and regularity for equations with Levy noise, Stochastic Processes and their Applications, 122, 1 (2012), 106-133.
  39. Zhao Dong, Lihu Xu and Xicheng Zhang: Invariance measures of stochastic 2D Navier-Stokes equations driven by alpha-stable processes, Electronic Communication on Probability, Vol. 16 (2011), 678-688.
  40. Lihu Xu: A modified log-Harnack inequality and asymptotically strong Feller property, Journal of Evolution Equations, 11 (2011), 925-942.
  41. Feng-Yu Wang, Jiang-Lun Wu and Lihu Xu: Log-Harnack inequality of stochastic Burgers equations, Journal of Mathematical Analysis and Applications, 384 (2011), 151–159.
  42. Enrico Priola, Lihu Xu and J. Zabczyk: Exponential mixing for SPDEs with Levy noises, Stochastic and Dynamics, 11 (2011), 521-534.
  43. Marco Romito and Lihu Xu: Ergodicity of 3D stochastic Navier-Stokes equations driven by mildly degenerate noises, Stochastic Processes and their Applications, 121, (2011), no. 4, 673-700.
  44. Lihu Xu and Boguslaw Zegarlinski: Existence and exponential mixing of the alpha-stable systems with unbounded interactions, Electronic Journal of Probability, Vol. 15 (2010), 1994-2018.
  45. Lihu Xu and Boguslaw Zegarlinski: Ergodicity of finite and infinite dimensional alpha-stable systems, Stochastic Analysis and its Applications, 27 (2009), no. 4, 797–824.
  46. Robert Olkiewicz, Lihu Xu and Boguslaw Zegarlinski: Nonlinear problems in infinite interacting particle systems, Infinite Dimensional Analysis Quantum Probability and Related Topics, 11 (2008), no. 2, 179–211.
NameTime Affiliation
Ran Wang09/2014-12/2014, 05/2018-08/2018 Wuhan University, Associate Professor
Longjie Xie05/2015-08/2015Jiangsu Normal University, Professor
Xiaobin Sun05/2016-08/2016. 05/2018-08/2018 Jiangsu Normal University, Associate Professor
 Gaofeng Zong02/2019-05/2019Shandong University of Finance and Economics, Associate Professor
Yuzhen Tan 02/2019-09/2019 Shandong Normal University, Assistant Professor
Hui Jiang04/2019-07/2019 Nanhang University, Professor 
Xuhui Peng07/2021-07/2022 Hunan Normal University, Associate Professor
  1.  Dr. Huiming Zhang will join Beihang University (北京航空航天大學) as a tenure track Associate Professor.
  2. On 22/08/2022, Mr. Yingjun Mo, a new PhD student, joined our research group. 
  3. On 15/06/2022, Dr. Jianya Lu passed his PhD oral defense. He will join University of Essex in UK as a Lecturer (equivalent to Assistant Professor), see the news of UM. 
  4. On 17/08/2021, Dr. Huiming Zhang received ‘青年科學基金’ from NSFC (國家自然科學基金).   
  5. On 08/07/2021, Dr. Xinghu Jin and Dr. Xin Xu passed their PhD oral defenses. Dr. Jin will join Hefei Industry University (合肥工業大學) as an assistant professor and Dr. Xin Xu will join Nankai University (南開大學) as a postdoc in this fall. 
  6. On 14/05/2021, Dr. Peng Chen passed his PhD oral defense. He will join NanHang University (南京航空航天大學) as an assistant professor in this fall.  
  7. Dr. Lihu Xu will give an invited talk at 7th National Annual Conference of Probability Theory (20/08/2021-22/08/2021, 4 plenary talks, 9 invited talks, 22 sessions) . He was also invited to organize a session . 
  8. On 13/03/2021, our research group members climbed a small mountain at 九澳. 
  9. On 27/02/2021. our research group members climbed Mazu mountain. 
  10. On 20/10/2020, Dr. Huiming Zhang joined our group as a postdoc.