【Regular Seminar】Non-Gaussian Stochastic Dynamical Systems

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Non-Gaussian Stochastic Dynamical Systems Non-Gaussian stochastic dynamical systems arise as models for various real-world phenomena and have attracted a lot of attention in mathematics, physics ......

Non-Gaussian Stochastic Dynamical Systems

Non-Gaussian stochastic dynamical systems arise as models for various real-world phenomena and have attracted a lot of attention in mathematics, physics, chemistry, biology, geosciences, economics and other disciplines. Professor Jinqiao Duan set up the study group on non-Gaussian stochastic dynamical systems on October 1, 2015. This weekly event focuses on discussing about Lévy-type stochastic integrals, stochastic differential equations driven by Lévy processes and stochastic dynamical systems with non-Gaussion Lévy noise.  
As a significant class of non-Gaussian processes, Lévy processes have independent and stationary increments, whose sample paths are right-continuous with left limits, and thus have a number of (at most countable) random jump discontinuities occurring at random times, on each finite time interval. They include Brownian motion, Poisson process, Cauchy process, and α-stable Lévy processes as special cases. The α-stable Lévy processes (0<α≤2) may be regarded as generalization of the well-known Brownian motion (α=2). 
Some highlights in our study:
1) The important idea of interlacing, whereby the path of a Lévy process is obtained as the almost-sure limit of a sequence of Brownian motions with drift interspersed with jumps of random size appearing at random times.
2) The collection of all infinitely divisible distributions is in one-to-one correspondence with the collection of all Lévy processes, describe all Lévy processes and all additive processes by characterizating all infinitely divisible distributions.
3) A martingale-based proof of the Lévy-Itô decomposition of an arbitrary Lévy process into Brownian (whose sample paths are continuous in time) and Poisson (whose sample paths have jumps quantified by a jump measure) parts. An important by-product of the Lévy-Itô decomposition is the Lévy-Khintchine formula, which says that each Lévy process has a specific form for its characteristic function.
4) Introduce selfsimilarity and other structures, subordination and density transformation, recurrence and transience, potential theory, Wiener-Hopf factorizations, and unimodality and multimodality.
5) Move beyond Lévy processes to study more general Markov processes and their associated semigroups of linear mappings. The structure of Lévy processes is the paradigm case and this is exhibited both through the Courrège theorem for the infinitesimal generator of Feller processes and the Beurling-Deny formula for symmetric Dirichlet forms.
6) Integration with respect to Brownian motion and Poisson integration are unified using the idea of a martingale-valued measure.
7) A number of useful spin-offs from stochastic integration: the Doleans-Dade stochastic exponential, Girsanov's theorem and its application to change of measure, the Cameron-Martin formula and the beginnings of analysis in Wiener space and martingale representation theorem.
8) Ito's formula for Lévy-type stochastic integrals and stochastic differential equations driven by Lévy processes. Under general conditions, the solutions of these SDEs are Feller processes, solutions also give rise to stochastic flows and hence generate random dynamical systems.
9) Establish an effective framework for describing geometric structures of stochastic dynamical systems with Lévy noise, provide a decomposition of the state space by invariant manifolds.
10) Quantify solution orbits in dynamical system driven by Lévy noise, and study pathwise properties of solution orbits.

(Reported by Yuan Shenglan )