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Zero noise limit for singular ODE regularized by fractional noise

Łukasz Mądry, Paul Gassiat

Abstract

We consider scalar ODE with a power singularity at the origin, regularized by an additive fractional noise. We show that, as the intensity in front of the noise goes to $0$, the solution converges to the extremal solutions to the ODE (which exit the origin instantly), and we quantify this convergence with subexponential probability estimates. This extends classical results of Bafico and Baldi in the Brownian case. The main difficulty lies in the absence of the Markov property for the system. Our methods combine a dynamical approach due to Delarue and Flandoli, with techniques from the large time analysis of fractional SDE (due in particular to Panloup and Richard).

Zero noise limit for singular ODE regularized by fractional noise

Abstract

We consider scalar ODE with a power singularity at the origin, regularized by an additive fractional noise. We show that, as the intensity in front of the noise goes to , the solution converges to the extremal solutions to the ODE (which exit the origin instantly), and we quantify this convergence with subexponential probability estimates. This extends classical results of Bafico and Baldi in the Brownian case. The main difficulty lies in the absence of the Markov property for the system. Our methods combine a dynamical approach due to Delarue and Flandoli, with techniques from the large time analysis of fractional SDE (due in particular to Panloup and Richard).
Paper Structure (13 sections, 31 theorems, 244 equations)

This paper contains 13 sections, 31 theorems, 244 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main result
  4. Preliminaries
  5. Construction of $\psi_{\alpha,\kappa}$
  6. Stage 1
  7. Stage 2
  8. Choice of $t^*$ and the proof of \ref{['eq:decay_new_past']}
  9. The proof of $\inf_{t \in [1,1+t^*]} X_t > 1$
  10. The proof of \ref{['eq:decay_new_noise']}
  11. Stage 3
  12. Proof of the main theorem
  13. Some technical estimates

Key Result

Theorem 1.1

Let $X^{\varepsilon}$ be as in eq:intro_equation with $W$ a fBm of parameter $H$. Then there exists a random time $\psi_{\varepsilon}$ such that, for suitable $\nu, \alpha>0$,either or In addition, for some $\kappa >0$, it holds, uniformly in $\varepsilon>0$,

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 2.1
  • Proposition 2.2
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 67 more