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Regularization by noise for rough differential equations driven by Gaussian rough paths

Rémi Catellier, Romain Duboscq

Abstract

We consider the rough differential equation with drift driven by a Gaussian geometric rough path. Under natural conditions on the rough path, namely non-determinism, and uniform ellipticity conditions on the diffusion coefficient, we prove path-by-path well-posedness of the equation for poorly regular drifts. In the case of the fractional Brownian motion $B^H$ for $H>\frac14$, we prove that the drift may be taken to be $κ>0$ Hölder continuous and bounded for $κ>\frac32 - \frac1{2H}$. A flow transform of the equation and Malliavin calculus for Gaussian rough paths are used to achieve such a result.

Regularization by noise for rough differential equations driven by Gaussian rough paths

Abstract

We consider the rough differential equation with drift driven by a Gaussian geometric rough path. Under natural conditions on the rough path, namely non-determinism, and uniform ellipticity conditions on the diffusion coefficient, we prove path-by-path well-posedness of the equation for poorly regular drifts. In the case of the fractional Brownian motion for , we prove that the drift may be taken to be Hölder continuous and bounded for . A flow transform of the equation and Malliavin calculus for Gaussian rough paths are used to achieve such a result.
Paper Structure (31 sections, 45 theorems, 413 equations)

This paper contains 31 sections, 45 theorems, 413 equations.

Table of Contents

  1. Introduction
  2. Difficulties and extended plan of the paper
  3. Notations and preliminary
  4. Rough differential equation with drift
  5. Rough paths and differential equations in a nutshell
  6. Flow generated by driftless differential equations driven by rough paths
  7. Solution of RDE with drift : flow transform
  8. Rough differential equation with drift : the averaged field
  9. Kolmogorov type theorem for averaged fields
  10. The $p<+\infty$ case.
  11. The $p=+\infty$ case.
  12. The $p<+\infty$ case.
  13. The $p=+\infty$ case.
  14. Malliavin Calculus
  15. Isonormal Gaussian processes
  16. ...and 16 more sections

Key Result

Theorem 1

Let $2<p<4$. Let $\mathbf{W}$ be a $p$-geometric Gaussian rough path such that its first component $(W_t)_{t\in[0,T]}=(\mathbf{W}^1_{0,t})_{t\in[0,T]}$ is $\alpha$-locally non-determinism, namely Assume that $\sigma \in C^\infty_b(\mathbbm{R}^d;(\mathbbm{R}^d)^{\otimes 2})$ is uniformly elliptic. Let $b\in \mathcal{C}^{\kappa}$ with $\kappa + \frac{1}{\alpha} > \frac{3}{2}$ and $\kappa>0$. Then a

Theorems & Definitions (110)

  • Theorem
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • proof
  • Corollary 2.5
  • ...and 100 more