Flatness of invariant manifolds for stochastic partial differential equations driven by Lévy processes

TL;DR

This work shows that for semilinear SPDEs driven by Lévy processes with small jumps, any invariant finite-dimensional manifold must possess a baseline flatness at least equal to the number of small-jump driving sources, captured by the dimension of the span of the jump directions $\\gamma^k(h)$. The authors develop local and global flatness theorems showing that invariance enforces tangency of these directions to the manifold, leading to a local direct-sum decomposition into a linear subspace and a transverse manifold; under connectivity, these local structures extend globally. The results generalize prior HJM findings to a broad SPDE framework and are illustrated by a Lévy-driven Hull-White extension of the Vasiček model, where invariant manifolds are necessarily foliations generated by a one-dimensional span of an exponential kernel. This provides a structural understanding of invariant manifolds in infinite-dimensional stochastic systems and informs the design of finite-dimensional realizations in finance and related applications.

Abstract

The purpose of this note is to prove that the flatness of an invariant manifold for a semilinear stochastic partial differential equation driven by Lévy processes is at least equal to the number of driving sources with small jumps. We illustrate our findings by means of an example.

Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Example 2.4
• Proposition 2.5
• proof
• Theorem 2.6
• proof
• Theorem 2.7
• proof