##
**Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths.**
*(English)*
Zbl 1288.60068

The object of study in this work is a {rough stochastic differential equation} of the form
\[
dZ_t = V_0(Z_t)dt + \sum_{i=1}^d V_i(Z_t)dX_t, \quad Z_0=z\in \mathbb{R}^d,
\]
where the vector fields \(V_0, V_i\in \mathbb{R}^d\) are smooth, and \(X\) is a (random) {rough path}, Hölder continuous of index \(\gamma>\frac{1}{3}\) [T. J. Lyons, Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)]. In particular, they focus on driving paths \(X\) that stem from a two-sided {fractional Brownian motion} (fBm) with Hurst parameter \(H\in(\frac{1}{3},\frac{1}{2})\). Under the celebrated {Hörmander condition} on the Lie algebra generated by the vector fields they establish the regularity of laws and ergodicity of the solution process. In other words, they answer two questions. Whether the transition densities are “smooth”, i.e., whether they admit a density with respect to the \(n\)-dimensional Lebesgue measure, and whether such a system admits a “unique invariant measure”.

The key point in establishing the smoothness of transitions densities with probabilistic methods, that is via the Malliavin calculus of variations, is to guarantee the invertability of the Malliavin matrix.

To this aim the authors introduce a {modulus of \(\theta\)-Hölder roughness}. Basically they ask for a constant \(L_\theta(X)\), \(\theta\in(0,1)\) such that for any test vector \(\varphi\in\mathbb{R}^n\) and small \(\varepsilon>0\) there exists a time \(t\in[0,T]\) such that for every \(s\) with \(|t-s|\leq\varepsilon\) \[ |\langle \varphi,\delta X_{s,t}\rangle|>L_\theta(X)\varepsilon^\theta. \] In particular, they show that an fBm of Hurst parameter \(H\leq \frac{1}{2}\) is almost surely \(\theta\)-Hölder rough for any \(\theta>H\).

The solution \(Z\) is a controlled rough path [M. Gubinelli, J. Funct. Anal. 216, No. 1, 86–140 (2004; Zbl 1058.60037)] with a “derivative process” \(Z'\) defined by the increment equality \[ \delta Z_{s,t} = Z'\delta X_{s,t} + R^Z_{s,t} \quad \text{with} \quad \|R^Z_{s,t}\|_{2\gamma}<\infty. \] The main technical result is now an estimate of \(Z'\), in terms of \(\|Z\|_\infty\) in Proposition 1, essentially they show \[ \|Z'\|_\infty \lesssim \frac{\|Z\|_\infty}{L_\theta(X)} \;. \] The authors interpret this estimate as a deterministic version of Norris’s lemma, the probabilistic counterpart in Malliavin’s proof.

The approach to the Malliavin calculus of variation presented here relies on the Mandelbrot-van Nesse representation of fBm as a “functional of the past” and a \(\frac{1}{2}-H\) fractional integral of a one-sided Brownian path. The work now consists in establishing the relation between the Cameron-Martin space of the underlying Wiener space and a “Cameron-Martin”-type analogue for fBm conditioned on its past. This way they can also relate the Mallavin matrix to the underlying Wiener space.

Similar techniques are then applied to establish the Feller property and to obtain the ergodicity result.

The article closes with two examples: The {hypoelliptic Ornstein-Uhlenbeck process} in \(n\) dimensions, driven by an \(m\)-dimensional fBM for \(H>\frac{1}{3}\), and multidimensional {linear Stratonovich equations} driven by fBm.

The presented work covers four areas each of great technical complexity of its own. The area of rough paths, fractional calculus, the Malliavin calculus of variations and ergodicity of non-Markovian systems. Still, the language is clear and a lot of explanations guide through the calculations. This allows also readers that are not so experienced in the field to follow the beautiful arguments.

The key point in establishing the smoothness of transitions densities with probabilistic methods, that is via the Malliavin calculus of variations, is to guarantee the invertability of the Malliavin matrix.

To this aim the authors introduce a {modulus of \(\theta\)-Hölder roughness}. Basically they ask for a constant \(L_\theta(X)\), \(\theta\in(0,1)\) such that for any test vector \(\varphi\in\mathbb{R}^n\) and small \(\varepsilon>0\) there exists a time \(t\in[0,T]\) such that for every \(s\) with \(|t-s|\leq\varepsilon\) \[ |\langle \varphi,\delta X_{s,t}\rangle|>L_\theta(X)\varepsilon^\theta. \] In particular, they show that an fBm of Hurst parameter \(H\leq \frac{1}{2}\) is almost surely \(\theta\)-Hölder rough for any \(\theta>H\).

The solution \(Z\) is a controlled rough path [M. Gubinelli, J. Funct. Anal. 216, No. 1, 86–140 (2004; Zbl 1058.60037)] with a “derivative process” \(Z'\) defined by the increment equality \[ \delta Z_{s,t} = Z'\delta X_{s,t} + R^Z_{s,t} \quad \text{with} \quad \|R^Z_{s,t}\|_{2\gamma}<\infty. \] The main technical result is now an estimate of \(Z'\), in terms of \(\|Z\|_\infty\) in Proposition 1, essentially they show \[ \|Z'\|_\infty \lesssim \frac{\|Z\|_\infty}{L_\theta(X)} \;. \] The authors interpret this estimate as a deterministic version of Norris’s lemma, the probabilistic counterpart in Malliavin’s proof.

The approach to the Malliavin calculus of variation presented here relies on the Mandelbrot-van Nesse representation of fBm as a “functional of the past” and a \(\frac{1}{2}-H\) fractional integral of a one-sided Brownian path. The work now consists in establishing the relation between the Cameron-Martin space of the underlying Wiener space and a “Cameron-Martin”-type analogue for fBm conditioned on its past. This way they can also relate the Mallavin matrix to the underlying Wiener space.

Similar techniques are then applied to establish the Feller property and to obtain the ergodicity result.

The article closes with two examples: The {hypoelliptic Ornstein-Uhlenbeck process} in \(n\) dimensions, driven by an \(m\)-dimensional fBM for \(H>\frac{1}{3}\), and multidimensional {linear Stratonovich equations} driven by fBm.

The presented work covers four areas each of great technical complexity of its own. The area of rough paths, fractional calculus, the Malliavin calculus of variations and ergodicity of non-Markovian systems. Still, the language is clear and a lot of explanations guide through the calculations. This allows also readers that are not so experienced in the field to follow the beautiful arguments.

Reviewer: Jan Gairing (Berlin)

### MSC:

60H07 | Stochastic calculus of variations and the Malliavin calculus |

26A33 | Fractional derivatives and integrals |

35B65 | Smoothness and regularity of solutions to PDEs |

60G10 | Stationary stochastic processes |

60G22 | Fractional processes, including fractional Brownian motion |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

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\textit{M. Hairer} and \textit{N. S. Pillai}, Ann. Probab. 41, No. 4, 2544--2598 (2013; Zbl 1288.60068)

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