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.
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.
