##
**Homogenisation for anisotropic kinetic random motions.**
*(English)*
Zbl 1458.58017

This paper introduces an anisotropic version of kinetic Brownian motion on a Riemannian manifold as follows. Let \(\Sigma\) be a non-degenerate \(d\times d\) covariance matrix, let \(B_t\) be a (non-standard) Brownian motion on \(\mathbb{R}^d\) with this covariance, and let \(\sigma>0\) be a parameter. Then construct the velocity process \(v_t\) on \(\mathbb{S}^{d-1}\) as the solution to the Stratonovich SDE
\[
dv_t=\sigma\Pi_{v_t^{\perp}}\circ dB_t ,\tag{\(*\)}
\]
where \(\Pi_{v_t^{\perp}}\) denotes projection onto the orthogonal complement of \(v_t^{\perp}\). Finally, we obtain a process \(x_t\) on \(\mathbb{R}^d\) by integrating, so that
\[
x_t = x_0 +\int_0^t v_s\, ds .
\]
This gives a diffusion \((x_t,v_t)\) on the unit tangent bundle such that the \(x_t\) component is a \(C^1\) curve with random velocity. For a Riemannian manifold \(\mathcal{M}\), the analogous process on the unit tangent bundle \(T^1\mathcal{M}\) is given by the stochastic development (“rolling without slipping”).

The case when \(\Sigma\) is the identity matrix, called kinetic Brownian motion, was introduced in [X.-M. Li, “Effective diffusions with intertwined structures”, Preprint, arXiv:1204.3250] and studied further in [J. Angst et al., Electron. J. Probab. 20, Paper No. 110, 40 p. (2015; Zbl 1329.60274)]; the main result from those works is that \(x_{\sigma^2 t}\) converges to Brownian motion on \(\mathcal{M}\) as \(\sigma\rightarrow\infty\). The first goal of the present paper is to prove the analogous convergence in the anisotropic case. Rough path techniques are used to prove that, when \(\mathcal{M}=\mathbb{R}^d\), \(x_{\sigma^2 t}\) converges to an anisotropic Brownian motion on \(\mathbb{R}^d\) with covariance given in terms of the invariant measure of \(v_t\). Then define an anisotropic Brownian motion on a general Riemannian manifold \(\mathcal{M}\) to be the stochastic development of a Euclidean anisotropic Brownian motion (because of the anisotropy, this process corresponds to a diffusion on the orthonormal frame bundle but does not project to a diffusion on \(\mathcal{M}\)). Further rough path techniques are then used to show that, for the process \((x_t,v_t)\) on \(T^1\mathcal{M}\), the rescaled process \(x_{\sigma^2 t}\) converges to an anisotropic Brownian motion on the base manifold \(\mathcal{M}\).

The second goal of the paper is to observe that the velocity process \((*)\) can be replaced by a more general class of ergodic Markov processes. In particular, this allows the author to reformulate several more classical approximations to diffusions in this framework and thus reprove the convergence of these processes to the associated diffusions. For example, random flights, random walks on time-varying metrics, and a Langevin process are all treated in this manner.

The case when \(\Sigma\) is the identity matrix, called kinetic Brownian motion, was introduced in [X.-M. Li, “Effective diffusions with intertwined structures”, Preprint, arXiv:1204.3250] and studied further in [J. Angst et al., Electron. J. Probab. 20, Paper No. 110, 40 p. (2015; Zbl 1329.60274)]; the main result from those works is that \(x_{\sigma^2 t}\) converges to Brownian motion on \(\mathcal{M}\) as \(\sigma\rightarrow\infty\). The first goal of the present paper is to prove the analogous convergence in the anisotropic case. Rough path techniques are used to prove that, when \(\mathcal{M}=\mathbb{R}^d\), \(x_{\sigma^2 t}\) converges to an anisotropic Brownian motion on \(\mathbb{R}^d\) with covariance given in terms of the invariant measure of \(v_t\). Then define an anisotropic Brownian motion on a general Riemannian manifold \(\mathcal{M}\) to be the stochastic development of a Euclidean anisotropic Brownian motion (because of the anisotropy, this process corresponds to a diffusion on the orthonormal frame bundle but does not project to a diffusion on \(\mathcal{M}\)). Further rough path techniques are then used to show that, for the process \((x_t,v_t)\) on \(T^1\mathcal{M}\), the rescaled process \(x_{\sigma^2 t}\) converges to an anisotropic Brownian motion on the base manifold \(\mathcal{M}\).

The second goal of the paper is to observe that the velocity process \((*)\) can be replaced by a more general class of ergodic Markov processes. In particular, this allows the author to reformulate several more classical approximations to diffusions in this framework and thus reprove the convergence of these processes to the associated diffusions. For example, random flights, random walks on time-varying metrics, and a Langevin process are all treated in this manner.

Reviewer: Robert Neel (Bethlehem, PA)

### MSC:

58J65 | Diffusion processes and stochastic analysis on manifolds |

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

60G51 | Processes with independent increments; Lévy processes |

### Citations:

Zbl 1329.60274
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\textit{P. Perruchaud}, Electron. J. Probab. 25, Paper No. 39, 26 p. (2020; Zbl 1458.58017)

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