Interacting particle systems at the edge of multilevel Jack processes

Multilevel Dyson Brownian motions via Jack polynomials

We introduce multilevel versions of Dyson Brownian motions of arbitrary parameter $β>0$, generalizing the interlacing reflected Brownian motions of Warren for $β=2$. Such processes unify $β$ corners processes and Dyson Brownian motions in a single object. Our approach is based on the approximation by certain multilevel discrete Markov chains of independent interest, which are defined by means of Jack symmetric polynomials. In particular, this approach allows to show that the levels in a multilevel Dyson Brownian motion are intertwined (at least for $β\ge 1$) and to give the corresponding link …

Interacting particle systems at the edge of multilevel Dyson Brownian motions

We study the joint asymptotic behavior of spacings between particles at the edge of multilevel Dyson Brownian motions, when the number of levels tends to infinity. Despite the global interactions between particles in multilevel Dyson Brownian motions, we observe a decoupling phenomenon in the limit: the global interactions become negligible and only the local interactions remain. The resulting limiting objects are interacting particle systems which can be described as Brownian versions of certain totally asymmetric exclusion processes. This is the first appearance of a particle system with loc…

Iterative Particle Approximation for McKean-Vlasov SDEs with application to Multilevel Monte Carlo estimation

The mean field limits of systems of interacting diffusions (also called stochastic interacting particle systems (SIPS)) have been intensively studied since McKean \cite{mckean1966class}. The interacting diffusions pave a way to probabilistic representations for many important nonlinear/nonlocal PDEs, but provide a great challenge for Monte Carlo simulations. This is due to the nonlinear dependence of the bias on the statistical error arising through the approximation of the law of the process. This and the fact that particles/diffusions are not independent render classical variance reduction t…

A host-parasite multilevel interacting process and continuous approximations

We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits…

Analysis of Adaptive Multilevel Splitting algorithms in an idealized case

The Adaptive Multilevel Splitting algorithm is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the $k$ less well-adapted particles among $n$ are killed while $k$ new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever $k$. We also obtain a precise asymptotic expansion for the variance of the estimator and the cost …

含时滞的McKean-Vlasov方程与相互作用粒子系统的Milstein格式及对偶多层蒙特卡洛采样

Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean-Vlasov equations and interacting particle systems

针对带点时滞、漂移项可能超线性增长的McKean-Vlasov方程对应的相互作用粒子系统，推导了Milstein格式并证明其一阶强收敛与矩稳定，还提出无需模拟Lévy面积的对偶多层Milstein格式以实现解的期望泛函的最优复杂度估计。