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Differentiating unstable diffusion

Angxiu Ni

TL;DR

This work addresses the problem of computing the linear response of stochastic differential equations under perturbations to initial conditions, drift, and diffusion, especially in unstable or chaotic regimes. It introduces a path-kernel formula that temper’s the path-perturbation via a schedule $α_t$, shifting part of the perturbation to the probability kernel and yielding a robust expression for ${δE[Φ(X^γ_T)]}$ that combines a pathwise term and a stochastic kernel term: ${δE[Φ(X^γ_T)] = E[ dΦ(X_T) v_T + Φ(X_T) ∫_0^T (α_t v_t / σ(x_t))·dB_t ]}$. The paper provides a discrete-time derivation, continuous-time and ergodic extensions, and a detailed discussion of how the method degenerates to pure path-perturbation, pure kernel-differentiation, and the BEL formula under suitable limits; it also presents an algorithm and demonstrates the approach on the Lorenz-96 model with noise, showing improved stability and accuracy over existing methods. The results offer a practical path to stable sensitivity analysis for high-dimensional, unstable stochastic systems with diffusion-perturbation components, supported by a concrete numerical package. Overall, the path-kernel framework unifies several linear-response techniques and provides actionable procedures for tempering gradient growth in chaotic stochastic dynamics.

Abstract

We derive a path-kernel formula for the linear response of SDEs, where the perturbation applies to initial conditions, drift coefficients, and diffusion coefficients. It tempers the unstableness by gradually moving the path-perturbation to hit the probability kernel. Then we derive a pathwise sampling algorithm and demonstrate it on the Lorenz 96 system with noise.

Differentiating unstable diffusion

TL;DR

This work addresses the problem of computing the linear response of stochastic differential equations under perturbations to initial conditions, drift, and diffusion, especially in unstable or chaotic regimes. It introduces a path-kernel formula that temper’s the path-perturbation via a schedule , shifting part of the perturbation to the probability kernel and yielding a robust expression for that combines a pathwise term and a stochastic kernel term: . The paper provides a discrete-time derivation, continuous-time and ergodic extensions, and a detailed discussion of how the method degenerates to pure path-perturbation, pure kernel-differentiation, and the BEL formula under suitable limits; it also presents an algorithm and demonstrates the approach on the Lorenz-96 model with noise, showing improved stability and accuracy over existing methods. The results offer a practical path to stable sensitivity analysis for high-dimensional, unstable stochastic systems with diffusion-perturbation components, supported by a concrete numerical package. Overall, the path-kernel framework unifies several linear-response techniques and provides actionable procedures for tempering gradient growth in chaotic stochastic dynamics.

Abstract

We derive a path-kernel formula for the linear response of SDEs, where the perturbation applies to initial conditions, drift coefficients, and diffusion coefficients. It tempers the unstableness by gradually moving the path-perturbation to hit the probability kernel. Then we derive a pathwise sampling algorithm and demonstrate it on the Lorenz 96 system with noise.

Paper Structure

This paper contains 19 sections, 5 theorems, 62 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Main results and structure of paper
  4. Deriving the formula
  5. Notations
  6. Intuition
  7. Derivation
  8. Changing variables
  9. Taking Derivative
  10. Continuous-time and ergodic statements
  11. How to use in unstable cases, and shortcomings
  12. Degeneration to some well-known cases
  13. Degeneracy to pure path-perturbation
  14. Degeneracy to pure kernel-differentiation
  15. Degeneracy to a big Gaussian
  16. ...and 4 more sections

Key Result

theorem 1

Fix any $x_0$, $v_0$, and any $\alpha_t$ (called a 'schedule') a scalar process adapted to $\EuScript{F}_t$ and independent of $\gamma$. Consider the Ito SDE, Its linear response has the expression Here $v_t$ is, starting from $v_0$, the solution of the damped path-perturbation equation

Figures (6)

  • Figure 1: Intuition of the derivative. For different $\gamma$, we compare the blue bundle with the red bundle.
  • Figure 2: Plot of $x^0_t, x^1_t$ from a typical orbit of length $T=2$.
  • Figure 3: $\Phi^{avg}_T$ and $\delta \Phi^{avg}_T$ for different $\gamma^1$. The dots are $\Phi^{avg}_T$, and the short lines are $\delta \Phi^{avg}_T$ computed by the kernel-differentiation algorithm; they are computed from the same orbit.
  • Figure 4: $\Phi^{avg}_T$ and $\delta \Phi^{avg}_T$ for different $\gamma^0$ and $\gamma^2$.
  • Figure 5: $\Phi^{avg}$ and $\delta \Phi^{avg}$ of the physical measure for different $\gamma^0$. The red triangles are $\Phi^{avg}$ for the Lorenz 96 system without noise.
  • ...and 1 more figures

Theorems & Definitions (6)

  • theorem 1: path-kernel formula for differentiating SDEs
  • lemma 1: discrete-time differentiation
  • proof
  • theorem 1: path-kernel formula for differentiating SDEs
  • corollary 1: centralized path-kernel formula
  • corollary 2: ergodic path-kernel formula