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Dr Jiang-Lun Wu

Reader

Specialist Subjects: Stochastic partial differential equations and random fields, non- standard analysis, stochastic analysis, Levy-type processes and their generators, probability theory on algebraic structures, infinite dimensional analysis, mathematical physics, quantum field theory, financial mathematics

Jiang-Lun Wu was born in 1963 in Shaanxi Provenice, P R China. He graduated from Northwest University in Xi´an in 1984 and he received his Ph D in 1991 from the Chinese Academy of Sciences, Beijing, where he was appointed as associate professor in 1993. He was awarded A v Humboldt Research Fellow at Ruhr-University Bocchum, Germany and then DFG research fellow from 1993 to 2000. He was appointed as Lecturer at Swansea University in 2000 and then promoted to Senior Lecturer and Reader In 2005 and 2007, respectively.

His research interests cover a wide range of topics in analysis and probability, including stochastic analysis, nonstandard analysis, analysis over infinite dimensional spaces. His research has sometimes been motivated by problems of mathematical physics, specifically constructive quantum fields and statistical mechanics, with a feature of linking problems of existence of probability measures on infinite dimensional spaces with functional analytic questions to probability theory. His early research resulted in nonstandard analysis approach to stochastic calculus and stochastic differential equations in infinite dimensional spaces, hyperfinite flat integrals for Euclidean random fields and large deviation, all rooted in stochastic quantization. He has experience in cooperation with a number of mathematicians in different areas. His substantive joint work with S. Albeverio and H. Gottschalk (both Bonn) on quantum field theory has led a remarkable and surprising result on constructing new interesting models with indefinite metric and nontrivial S-matrix. This is achieved in a tour de force of analysis and probability by replacing Gaussian processes for Nelson's free field with Lévy processes. The resulting theory has a nontrivial S-matrix and satisfies the Gårding-Wightman axioms. Thus the axioms of quantum mechanics are reconciled with special relativity, albeit in a simple physical model. Our result has been noticed remarkably by the mathematical physics community. A recent Harvard Ph D dissertation by W. G. Ritter (under the supervision of the world leading mathematical physicist A Jaffe) was inspired by their work. His joint work with S. Albeverio and B. Ruediger (Koblenz) on functional analytic and probabilistic aspects of Lévy type generators is based on N. Jacob's pioneering work on pseudo-differential operators and Lévy type stochastic processes. Further in this direction, with N. Jacob and A. Potrykus (both Swansea), he is investigating stochastic nonlinear pseudo-differential equations and the associated critical nonlinearity exponents.

With A. Truman (Swansea), he examined the Cauchy problem for stochastic Burgers equations with Lévy noise and for fractal Burgers equations forced by Lévy space-time white noise. These are the first two works of our series of investigations on stochastic fluid dynamics with non Gaussian white noise force. Their recent paper in Journal of Functional Analysis devoted to investigating noise perturbation of integro-differential conservation laws which produced a series results in understanding the behaviour of noise perturbation of non-local conservation laws in fluid dynamics.

With S. Albeverio and T. Zhang (Manchester), and with D. Applebaum (Sheffield), he initiated to study parabolic stochastic partial differential equations (SPDEs) driven by Lévy white noise, which led a very active research area and stimulated further very interesting investigations, e.g. in mathematical finance by B. Oksendal's Oslo school, and further in SPDEs by L. Mytnik (Haifa), N. Fournier (Nancy), E. Hausenblas (Salzburg), A. Lasota and T. Szarek (both Katowice), and S. Peszat (Krakow) and J. Zabczyk (Warsaw), M. R�ckner (Bielefeld), etc., just to mention a few.

With S. Albeverio and Y. Sun (Singapore), and with J. Berger, H. Osswald (both Munich) and Y. Sun, he has published two papers on the nonstandard analysis framework of joint measurability and duality, establishing a positive framework for Doob's famous 1937 joint measurability conjecture. These papers were noticed by world leading nonstandard analysts H. J. Keisler (Wisconsin) and P. A. Loeb (Illinois) in their mathematical research on equilibrium economics, as well as cited by P. J. Hammond and by D. Duffie (both Stanford) in their recent research in economics and finance, respectively. Along with this framework, with Osswald, he published a paper devoted to analyse the continuity of solutions to infinite dimensional stochastic differential equations of Langevin type. More recently he has published a paper to establish a general flat integral formula for generalized random fields via Loeb's rich measure structure.

With J. Bennett (his former PhD student at Swansea), he have written three papers devoted to stochastic optimization problems associated with stochastic differential equations driven by Lévy type processes arising from mathematical finance. Further related to his previous work on constructive quantum field theory, with Albeverio and Gottschalk, he has presented a notion of partly divisible probability measures on locally compact Abelian groups, which has a link to free random variables in non commutative probability theory. Along this line, with N Privault (La Rochelle), he has published a paper on establishing quantum Poisson stochastic calculus. Recently with D. Parashar (Warwick), he is studying deformed Burgers equations perturbed by quantum Lévy noise (which consists of a quantum Gaussian noise and an independent Poisson noise).

 

Recent Publications

 

  1. An optimal control problem associated with SDEs driven by Lévy-type processes (with J. Bennett), Stoch. Anal. Appl., to appear. 60H30,60J75, 49L25,93E20.
  2. Explicit construction of SDEs associated with polar-decomposed Lévy measures and application to stochastic optimization (with J. Bennett), Frontiers of Mathematics in China , 2 (2007), 539--558. 60H30, 60G52, 49J22.
  3. A hyperfinite flat integral for generalized random fields, J. Math. Anal. Appl., 330 (2007), 133--143. 60G60, 28E05 (03H05).
  4. Martingale property for empirical processes (with S. Albeverio, Yeneng Sun), Transactions of American Mathematical Society, 359 (2007), 517--527. 60G42, 60G44 (03H05).
  5. On a stochastic nonlinear equation arising from 1-D integro-differential scalar conservation laws (with A. Truman), Journal of Functional Analysis, 238 (2006), 612--635. 60H15, 35R60.
  6. Fractal Burgers equation driven by Lévy noise (with A. Truman), pp 295-310 in Stochastic Partial Differential Equations and Applications (eds. G. Da Prato, L. Tubaro), Lecture Notes in Pure and Applied Mathematics, 245 , Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton, London, New York, 2006. 60H15, 35R60.
  7. On the continuity of pathwise solutions to Langevin equations in infinite dimensions (with H. Osswald), Acta Appl. Math., 83 (2004), 289--312. 60G17, 60G15, 60H05 (03H05).
  8. Stochastic Burgers equation with Lévy space-time white noise (with A. Truman), pp 298-323 in Probabilistic Methods in Fluids (eds. I.M. Davies et al), World Scientific, Singapore, 2003. 60H15, 35R60
  9. On nonstandard product measure spaces (with Josef Berger, Horst Osswald, Yeneng Sun). Illinois Journal of Mathematics, 46 (2002), no. 1, 319--330. 03H05, 28A35, 28E05 (26E35, 60A10)
  10. SPDEs leading to local, relativistic quantum vector fields with indefinite metric and nontrivial S-matrix (with S. Albeverio, H. Gottschalk), pp 21--38 in Stochastic Partial Differential Equations and Applications (eds. G. Da Prato, L. Tubaro), Lecture Notes in Pure and Applied Mathematics, Volume 227 , Marcel Dekker, Inc., New York/Basel, 2002.
  11. Analytic and probabilistic aspects of Lévy processes and fields in quantum theory (with S. Albeverio, B. R�diger), pp 187--224 in Lévy Processes: Theory and Applications (eds. O. Barndorff-Nielsen, T. Mikosch, S. Resnick), Birkhäuser, Boston, 2001. 81Sxx (60G51, 60K40, 81Txx)
  12. On nonstandard construction of stable type Euclidean random field measures and large deviation (with S. Albeverio), pp 1--18 in Reuniting the Antipodes: Constructive and Nonstandard Views of the Continuum (eds. U. Berger, H. Osswald, P. Schuster), Kluwer Academic Publishers, Boston/Dordrecht/London, 2001.

PhD

 

Swansea University

Tel: 01792 602757

Fax: 01792 295843

Email: J.L.Wu@swansea.ac.uk