Probability and Statistics Seminar

Sequential Monte Carlo (SMC) algorithms find applications across a wide range of areas, most notably in non-linear filtering and smoothing for state space models. The performance of SMC is limited, particularly in smoothing applications, by the phenomenon of ancestral degeneracy, an inevitable consequence of the resampling step. We study the genealogies induced by resampling, showing that the genealogies of many popular SMC algorithms converge to Kingman’s n-coalescent. The tractability of this limiting process allows ancestral degeneracy to be quantified and thus hopefully mitigated, by informing the choice of algorithm and tuning parameters. Joint work with Paul Jenkins, Adam Johansen and Jere Koskela.

We consider the (1+1)-dimensional stochastic heat equation (SHE) with multiplicative white noise and the Cole-Hopf solution of the Kardar-Parisi-Zhang (KPZ) equation. We show an exact way of computing all the positive real Lyapunov exponents of the SHE for a large class of initial data which includes any bounded deterministic positive initial data and the stationary initial data. As a consequence, we derive exact formulas for the upper tail large deviation rate functions of the KPZ equation for general initial data. This is a joint work with Promit Ghosal.