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

Perruchaud, Pierre

Electron. J. Probab. 25, Paper No. 39, 26 p. (2020).

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.

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

Keywords:

homogenization; mixing; riemannian manifolds; diffusion processes; rough paths theory

Citations:

Zbl 1329.60274

References:

[1]	J. Angst, I. Bailleul, and C. Tardif, Kinetic Brownian motion on Riemannian manifolds, Electron. J. Probab. 20 (2015), no. 110, 40. · Zbl 1329.60274
[2]	D. Applebaum, Probability on compact Lie groups, Probability Theory and Stochastic Modelling, vol. 70, Springer, Cham, 2014, With a foreword by Herbert Heyer. · Zbl 1302.60007
[3]	M. Arnaudon, K. A. Coulibaly, and A. Thalmaier, Brownian motion with respect to a metric depending on time: definition, existence and applications to Ricci flow, C. R. Math. Acad. Sci. Paris 346 (2008), no. 13-14, 773-778. · Zbl 1144.58019
[4]	I. Bailleul, A stochastic approach to relativistic diffusions, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 3, 760-795. · Zbl 1206.60053
[5]	I. Bailleul, Flows driven by rough paths, Rev. Mat. Iberoam. 31 (2015), no. 3, 901-934. · Zbl 1352.34089
[6]	J. Birrell, S. Hottovy, G. Volpe, and J. Wehr, Small mass limit of a Langevin equation on a manifold, Ann. Henri Poincaré 18 (2017), no. 2, 707-755. · Zbl 1361.82027
[7]	J.-M. Bismut, Hypoelliptic Laplacian and probability, J. Math. Soc. Japan 67 (2015), no. 4, 1317-1357. · Zbl 1334.35019
[8]	E. Breuillard, P. Friz, and M. Huesmann, From random walks to rough paths, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3487-3496. · Zbl 1179.60017
[9]	I. Chevyrev, Random walks and Lévy processes as rough paths, Probab. Theory Related Fields 170 (2018), no. 3-4, 891-932. · Zbl 1436.60087
[10]	I. Chevyrev, P. K. Friz, A. Korepanov, I. Melbourne, and H. Zhang, Multiscale systems, homogenization, and rough paths, Probability and analysis in interacting physical systems, Springer Proc. Math. Stat., vol. 283, Springer, Cham, 2019, pp. 17-48. · Zbl 1443.60060
[11]	M. Christensen and J. B. Pedersen, Diffusion in inhomogeneous and anisotropic media, The Journal of Chemical Physics 119 (2003), 5171-5175.
[12]	K. A. Coulibaly-Pasquier, Brownian motion with respect to time-changing Riemannian metrics, applications to Ricci flow, Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011), no. 2, 515-538. · Zbl 1222.58030
[13]	A. M. Davie, Differential equations driven by rough paths: An approach via discrete approximation., AMRX, Appl. Math. Res. Express 2007 (2008), 40 (English).
[14]	P. Friz, P. Gassiat, and T. Lyons, Physical Brownian motion in a magnetic field as a rough path, Trans. Amer. Math. Soc. 367 (2015), no. 11, 7939-7955. · Zbl 1390.60257
[15]	P. K. Friz and M. Hairer, A course on rough paths, Universitext, Springer, Cham, 2014, With an introduction to regularity structures. · Zbl 1327.60013
[16]	P. K. Friz and N. B. Victoir, Multidimensional stochastic processes as rough paths, Cambridge Studies in Advanced Mathematics, vol. 120, Cambridge University Press, Cambridge, 2010, Theory and applications. · Zbl 1193.60053
[17]	L. Gross, Abstract Wiener spaces, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, Univ. California Press, Berkeley, Calif., 1967, pp. 31-42.
[18]	L. Gross, Abstract Wiener measure and infinite dimensional potential theory, Lectures in Modern Analysis and Applications, II, Lecture Notes in Mathematics, Vol. 140. Springer, Berlin, 1970, pp. 84-116. · Zbl 1316.00048
[19]	M. Heidernätsch, M. S. Bauer, and G. Radons, Characterizing n-dimensional anisotropic brownian motion by the distribution of diffusivities., The Journal of Chemical Physics 139 (2013), no. 18, 184105.
[20]	D. P. Herzog, S. Hottovy, and G. Volpe, The small-mass limit for Langevin dynamics with unbounded coefficients and positive friction, J. Stat. Phys. 163 (2016), no. 3, 659-673. · Zbl 1346.82027
[21]	E. P. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics, vol. 38, American Mathematical Society, Providence, RI, 2002. · Zbl 0994.58019
[22]	S. Ishiwata, H. Kawabi, and R. Namba, Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part i, 2018. · Zbl 1476.60067
[23]	S. Ishiwata, H. Kawabi, and R. Namba, Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part ii, 2018. · Zbl 1476.60067
[24]	N. van Kampen, Diffusion in inhomogeneous media, Journal of Physics and Chemistry of Solids 49 (1988), 673-677.
[25]	K. Kuwada, Convergence of time-inhomogeneous geodesic random walks and its application to coupling methods, Ann. Probab. 40 (2012), no. 5, 1945-1979. · Zbl 1266.60060
[26]	X.-M. Li, Effective diffusions with intertwined structures, arXiv:1204.3250 (2012), 1-33.
[27]	X.-M. Li, Limits of random differential equations on manifolds, Probab. Theory Related Fields 166 (2016), no. 3-4, 659-712. · Zbl 1356.60056
[28]	X.-M. Li, Random perturbation to the geodesic equation, Ann. Probab. 44 (2016), no. 1, 544-566. · Zbl 1372.60083
[29]	M. Liao, Lévy processes in Lie groups, Cambridge Tracts in Mathematics, vol. 162, Cambridge University Press, Cambridge, 2004. · Zbl 1076.60004
[30]	O. Lopusanschi and D. Simon, Area anomaly in the rough path brownian scaling limit of hidden Markov walks, 2017. · Zbl 1482.60129
[31]	O. Lopusanschi and D. Simon, Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs, Stochastic Process. Appl. 128 (2018), no. 7, 2404-2426. · Zbl 1396.60076
[32]	K. V. Mardia and P. E. Jupp, Directional statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2000. · Zbl 0935.62065
[33]	M. A. Pinsky, Isotropic transport process on a Riemannian manifold, Trans. Amer. Math. Soc. 218 (1976), 353-360. · Zbl 0341.60041
[34]	D. · Zbl 0925.60004