Stochastic solutions of nonlinear PDE's and an extension of superprocesses

Stochastic solutions of nonlinear pde's: McKean versus superprocesses

Stochastic solutions not only provide new rigorous results for nonlinear pde's but also, through its local non-grid nature, are a natural tool for parallel computation. There are two methods to construct stochastic solutions: the McKean method and superprocesses. Here a comparison is made of these two approaches and their strenghts and limitations are discussed.

New Green's functions for some nonlinear oscillating systems and related PDEs

During the past three decades, the advantageous concept of the Green's function has been extended from linear systems to nonlinear ones. At that, there exist a rigorous and an approximate extensions. The rigorous extension introduces the so-called backward and forward propagators, which play the same role for nonlinear systems as the Green's function plays for linear systems. The approximate extension involves the Green's formula for linear systems with a Green's function satisfying the corresponding nonlinear equation. For the numerical evaluation of nonlinear ordinary differential equations …

Uniqueness for a class of one-dimensional stochastic PDEs using moment duality

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^…

Exact Model Reduction for Damped-Forced Nonlinear Beams: An Infinite-Dimensional Analysis

We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of the linearized beam equation. Reduction of the governing PDE to SSMs provides an explicit low-dimensional model which captures the correct asymptotics of the full, infinite-dimensional dynamics. Our approach is general enough to admit extensions to other types of continuum vibrations. The model-reduction procedure we employ also gives guidelines for a mathem…

一种用于具有滞回的随机振子不变测度的PDE方法

A PDE approach for the invariant measure of stochastic oscillators with hysteresis

本文提出了一种PDE方法用于计算具有滞回的随机振子的不变测度，扩展了弹塑性框架并展示了其在阈值穿越频率计算中的应用。