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Weekly seminar links: Probability

Probability

The probability group consists of three professors (Dayue Chen, Yong Liu and Yanxia Ren) and two associate professors (Daquan Jiang and Fuxi Zhang). They have a routine seminar on Monday, and often hold workshops and conferences. Their research fields include stochastic analysis, measure-valued Markov processes, approximation theory, random walks, interacting particle systems, statistical physics, as well as their applications. Some achievements in recent years are listed as follows. 


1. Measure-valued Markov Processes (REN Yanxia)


The measure-valued process is an important current branch of probability. The field has roots in diverse areas of pure and applied science, including branching processes, population genetics models, interacting particle systems and stochastic partial differential equations. In 1951, Feller observed that the size of a large population can be modeled by a diffusion obtained from the Galton-Watson process by scaling and passing to the limit. The Feller diffusion approximation is now a key tool in mathematical population genetics. Superprocesses arise as an extension of this idea to models that record not only the size of the population, but also the location of individuals within it. For example, the location of an individual could be a spatial position inRd, say, or her genetic type. In the genetics setting, the ‘spatial motion’ of individuals is a model of mutation between types. The process used to describe the spatial motion or genetic type of an individual (called the underlying process) can be any Levy process, or more generally, any Markov processes (satisfying some technical conditions, for example, a Hunt process).

Measure-valued processes are used to model population growth in biological systems, so it is no surprise that limit theorems for Measure-valued processes have attracted attention of many researchers during the past two decades.


1) In [Ren-Song-Zhang,J. Funct. Anal.(2014)] Yanxia Ren et al. established spatial central limit theorems for a large class of supercritical branching symmetric Markov processes with general spatial-dependent branching mechanisms. The central limit theorems established in this paper are more satisfactory in the sense that the normal random variables in our theorems are non-degenerate. Recently, [Ren-Song-Zhang,Ann. Probab.(2017)] established a spatial central limit theorem for a large class of supercritical branching, not necessarily symmetric, Markov processes with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently for supercritical branching symmetric Markov processes [Ren-Song-Zhang,J. Funct. Anal.(2014)]. The spectral theory of non symmetric strongly continuous semigroups developed to prove the central limit theorem is of independent interest.


2) In [Kyprianou-Liu-Murillo Salas-Ren,Ann. Inst. Henri Poincaré Probab. Stat.(2012)], Ren et al. offered a probabilistic treatment of the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. A new path-wise version of Evans's immortal particle decomposition (called a spine decomposition) for super-Brownian motion, a very important technique, was given and used. This paper also offered an exactX(logX)2moment dichotomy for the almost sure convergence of the so-called derivative martingale at its critical parameter to a non-trivial limit. This differs to the case of branching Brownian motion, and branching random walk, where a moment ‘gap’ appears in the necessary and sufficient conditions.


3) In [Kyprianou-Ren,Stoch. Proc. Appl.(2014)], Ren and her coauthor showed a strong law of large numbers for a class of supercritical super-stable processes. Earlier results in this direction have focused on supercritical super-Brownian motion, which has compact support property. But super-stable process does not have compact support property and the path of stable process is not continuous, so the generalization to stable motion processes is new and substantive. This paper used a new topology, the shallow topology, and identified the limit. The definition of shallow convergence uses a set of test functions which are "swiftly decreasing'' and leads to a notion of convergence that is stronger than the vague topology but weaker than the weak topology. The proofs use Fourier analysis to give a "frequency domain'' result that is then converted via stochastic estimates back to an SLLN.


4) In [Chu-Li-Ren,Bernoulli(2014)], Ren and her coauthor considered supercritical branching processes with immigration. The small value probabilities of a martingale limit W were studied. It was proved that the small value probability results pointing at new phase transitions in the case of immigration. The proof is based on the so-calledbranching tree heuristic method[Mörters-Ortgiese,Bernoulli(2008)], which roughly speaking is a probabilistic approach to obtain the limit random variable W (or some of its properties) instead of the classical integral and complex analysis technique.


2. Stochastic Analysis and Stochastic Partial Differential Equations (LIU Yong)


1) Time regularity: Time regularity is a fundamental problem in the theory of stochastic processes. The time regularity of Ornstein-Uhlenbeck SPDE driven by Levy processes “is of prime interest in study of non-linear stochastic PDEs” [Brzeniak et al.,C.R. Acad. Sci. Paris Ser. I(2010)]

In [Liu-Zhai,C. R. Acad. Sci. Paris Ser. I(2012)], Yang Liu and hia coauthor gave a necessary and sufficient condition of càdlàg modification of stochastic processes in Hilbert spaces, and then obtained a necessary and sufficient condition of càdlàg modification of Ornstein- Uhlenbeck process with cylindrical stable noise in a Hilbert space. This result and its corollary denied a conjecture in [Priola-Zabczyk,PTRF(2011)]; gave a negative answer to Problem 1, a positive answer to Problem 4 and partly answer Problem 3 in [Brzeniak et al.,C. R. Acad. Sci. Paris Ser. I(2010)].


2) Complex Wiener-Ito Chaos decomposition: Wiener-Ito Chaos decomposition and its representation of stochastic multiple integrals is an elegant theorem and a deep analytic tool in stochastic analysis. This theorem gives the representation of Ito stochastic multiple integrals for direct sum decomposition of square integrable functional on Wiener spaces.  The deep relation of direct sum decomposition, the eigenfunctions of Ornstein-Uhlenbeck operator (Hermite polynomial), Ito multiple integrals and Malliavin calculus is given through this theory. In 1951, Ito showed the connection between stochastic multiple integrals and chaos decomposition. In 1953, he introduced the complex Hermite polynomial and established the theory of complex stochastic multiple integrals.

In [Chen-Liu,Kyoto J. Math.(2014)], Chen and Liu proved that the complex Hermite polynomial is the eigenfunction of complex Ornstein–Uhlenbeck operator; and then in [Chen-Liu,Infin. Dimens. Anal. Quantum Probab. Relat. Top.(2017)], Chen and Liu obtained a new direct sum decomposition of complex square integrable functionals on complex Gaussian-Hilbert spaces, and gave a representation of real stochastic multiple integrals for the real part and the imaginary part of complex chaos decomposition. As its application, combining central limit theorem of real stochastic multiple integrals (4th moment theorem) [Nualart-Pecciti,Ann. Probab.(2005)], Chen and Liu relatively easily proved central limit theorem of complex stochastic multiple integrals.


3) In [Gong-Liu-Liu-Luo,J. Funct. Anal.(2014)], Liu et al. extend the spectral gap comparison theorem of Andrews and Clutterbuck [Andrews-Clutterbuck, JAMS (2011)] to the abstract Wiener spaces.


3. Applied Probability (JIANG Daquan, ZHANG Fuxi)


1) Jiang focused his research interest on the rigorous mathematical theory of nonequilibrium statistical physics and its applications. He and collaborators proved several important equalities describing the statistical and dynamic properties of nonequilibrium systems, including Jarzynski’s equality and various fluctuation theorems. The discovery of these equalities is one of the most important progresses in recent two decades in the field of nonequilibrium statistical physics.

In [Jia-Jiang-Qian,Ann. Appl. Probab.(2016)], Jiang and his colleagues established some deep properties of taboo probabilities and used them to prove that for a discrete-time or continuous-time Markov chain, the earliest forming time of a family of cycles passing through the same set of states is independent of which one of these cycles is firstly formed. Exploiting these properties, they proved that the sample circulations along a family of cycles passing through a common state satisfy a large deviation principle and its rate function satisfies the Gallavotti-Cohen type fluctuation theorem.


2) In [Xia-Zhang,Stoch. Proc. Appl.(2012)], Fuxi Zhang and her coauthor defined a family of polynomial birth-death point processes (PBDP), and proposed a general sketch of the PBDP approximation. Namely, they devised in an evolution with the PBD as the invariant measure, by regarding the PBD random variable as the cardinality of particles in some system. An analogue for PBDP was devised. They constructed a coupling of processes whose initial configurations have only one or two different points. Consequently, the upper bounds of Stein's factors were estimated. They considered locally dependent point processes as approximated objectors, which usually describe the space and time of rare random events. Finally, they “give substantially improved bounds when replacing approximating Poisson or Compound Poisson processes” by our new processes.

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Faculty:

CHEN Dayue ProfessorDiscrete-time Markov chains in particular, interacting particle systems, random walk on trees and in a random environment, percolation.dayue@math.pku.edu.cn
JIANG Daquan Associate ProfessorProbability theory and stochastic processes, Dynamical systems and ergodic theoryjiangdq@math.pku.edu.cn
LIU Yong ProfessorStochastic Analysis on infinite dimensional spaces, SPDEs and interacting diffusions and their applications to mathematical physics liuyong@math.pku.edu.cn
REN Yanxia ProfessorSuperprocesses (Measure-valued Markov Processes)yxren@math.pku.edu.cn
ZHANG Yuan Assistant ProfessorInteracting particle systems and their applications, random walk, random interlacementzhangyuan@math.pku.edu.cn
ZHANG Fuxi Associate ProfessorProbability: Interacting particle systems, random walk, Gaussian fields, Stein’s method, Nonequilibrium steady state zhangfxi@math.pku.edu.cn