Noise-intensity bifurcations of transition paths by Morse index formula

Zhihao Zhao; Jinqiao Duan

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arXiv · 2508.01954

Noise-intensity bifurcations of transition paths by Morse index formula

Zhihao Zhao·Jinqiao Duan

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Abstract

We study most probable transition paths (MPTPs) of a gradient diffusion system in $\mathbb{R}^n$ with fixed endpoints, in the framework of Onsager--Machlup (OM) theory. The existence theory for such paths is developed within Lagrangian variational theory: above the Mañé critical value $c_u(L)$ the energy-penalized action admits global minimizers, while below the endpoint-dependent critical value $k_0(L;x_\pm)$ only time-truncated minimizers exist; at regular levels between them, free-time extremals exist. The main result is a $\{0,1\}$-index theorem: at a nondegenerate free-time extremal, the Morse index of the full Hessian equals the fixed-time Morse index plus a correction $ν\in\{0,1\}$, equal to $1$ exactly when the reduced (minimal fixed-time) action satisfies $S''(T)<0$. The loss of local minimality along a branch is thereby split into two mechanisms: a conjugate-point mechanism and a duration mechanism. Combined with a Hamiltonian spectral-flow formula, this yields a bifurcation criterion in the noise intensity $σ$ and a sharp local stability criterion expressed in terms of the crossing instants. As an example, the one-dimensional quartic double well is analyzed in detail.

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BibTeX RIS

Zhihao Zhao, Jinqiao Duan. 2026-09-20. Noise-intensity bifurcations of transition paths by Morse index formula. https://arxiv.org/abs/2508.01954

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