Abstract
We consider stochastic perturbations of geodesic flow for left-invariant metrics on finite-dimensional Lie groups and study the Hörmander condition and some properties of the solutions of the corresponding Fokker–Planck equations.
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Notes
We can of course go back and forth between TG and \(T^*G\) with the help of the metric.
We use the usual identifications: if f, g are two co-vectors with coordinates \(f_i, g_j\), respectively, then the two-form \(f\wedge g\) is identified with the antisymmetric matrix \(\omega _{ij}=f_ig_j-f_jg_i\) and \((f\wedge g)(\xi ,\eta )= \omega _{ij}\xi ^i\eta ^j\) for any two vectors \(\xi ,\eta \).
In particular, when working on manifolds, one often has to distinguish carefully between the Itô and Stratonovich integrals. In the stochastic processes we will use here, this issue mostly does not come up. A typical case where it does come up is, for example, the formal equation for the Brownian motion on G in our setting: \(a^{-1}\dot{a}= \sigma \dot{w}\), or \(\mathrm{d}a=a\circ \mathrm{d}w\). In this case, the equation should be interpreted in the sense of Stratonovich. See, for example, Birrell et al. (2017) for a discussion of related topics.
Recall that a group is unimodular of the notions of left-invariant and right-invariant Haar measures coincide. This is the same as demanding that the maps \(y\rightarrow \mathrm{Ad}\,a^* y\) preserve the volume in \(\mathfrak g^*\), i.e., have determinant 1.
Note that with Itô integration of the particle trajectories might not stay in M.
Here p stands for parabolic, as the definition is tied to the parabolic Hörmander condition.
Here and below this is of course meant only in the context of the example we are considering in this subsection.
References
Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Addison-Wesley Publishing Company Inc., Redwood City (1987)
Agrachev, A.A., Sarychev, A.V.: Navier–Stokes equations: controllability by means of low modes forcing. J. Math. Fluid Mech. 7(1), 108–152 (2005)
Agrachev, A.A., Sarychev, A.V.: Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing. Commun. Math. Phys. 265(3), 673–697 (2006)
Arnaudon, M., Chen, X., Cruzeiro, A.B.: Stochastic Euler–Poincaré reduction. J. Math. Phys. 55(8), 081507 (2014)
Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16(1), 319–361 (1966)
Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Springer, Berlin (1998)
Birrell, J., Hottovy, S., Volpe, G., Wehr, J.: Small mass limit of a Langevin equation on a manifold. Ann. Henri Poincaré 18(2), 707–755 (2017)
Bismut, J.-M.: Mecanique Aléatoire, Lecture Notes in Math, vol. 866. Springer, Berlin (1981)
Bismut, J.-M.: The hypoelliptic Laplacian on the cotangent bundle. J. Am. Math. Soc. 18(2), 379–476 (2005)
Einstein, A.: Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 322(8), 549–560 (1905)
Földes, J., Glatt-Holtz, N., Richards, G., Thomann, E.: Ergodic and mixing properties of the Boussinesq equations with a degenerate random forcing. J. Funct. Anal. 269(8), 2427–2504 (2015)
Freidlin, M.I., Wentzell, A.D.: On the Neumann problem for PDEs with a small parameter and the corresponding diffusion processes. Probab. Theory Relat. Fields 152(12), 101–140 (2012)
Glatt-Holtz, N.E., Herzog, D.P., Mattingly, J.C.: Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations. arXiv:1706.01997 (2017)
Hairer, M.: Ergodic Theory for Stochastic PDEs, Lecture Notes from the LAM-EPSRC Short Course Held in July 2008
Hairer, M.: On Malliavin’s proof of Hörmander’s theorem. Bull. Sci. Math. 135(6–7), 650–666 (2011)
Hairer, M., Mattingly, J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. (2) 164(3), 993–1032 (2006)
Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. Academic Press, New York (1980)
Herzog, D.P., Mattingly, J.C.: A practical criterion for positivity of transition densities. Nonlinearity 28, 2823–2845 (2015)
Hinch, E.J.: Application of the Langevin equation to fluid suspensions. J. Fluid Mech. 72(3), 499–511 (1975)
Hochgerner, S., Ratiu, T.: Geometry of Non-holonomic Diffusion. arXiv:1204.6438 (2012)
Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147–171 (1967)
Jurdjevic, V.: Geometric Control Theory, Cambridge Studies in Advanced Mathematics, vol. 52 (1997)
Khasminskii, R.: Stochastic Stability of Differential Equations. Stochastic Modelling and Applied Probability, vol. 66, 2nd edn. Springer, Berlin (2012)
Kraichnan, R.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–23 (1967)
Kuksin, S.B.: Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2006)
Lázaro-Camí, J.-A., Ortega, J.-P.: Stochastic Hamiltonian dynamical systems. Rep. Math. Phys. 61(1), 65122 (2008)
Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems. Texts in Applied Mathematics, vol. 17, 2nd edn. Springer, New York (1999)
Marsden, J., Weinstein, A.: Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Order in chaos (Los Alamos, N.M., 1982). Phys. D 7(1–3), 305–323 (1983)
Romito, M.: Ergodicity of the finite dimensional approximation of the 3D Navier–Stokes equations forced by a degenerate noise. J. Stat. Phys. 114(1–2), 155–177 (2004)
Roux, J.-N.: Brownian particles at different times scales: a new derivation of the Smoluchowski equation. Phys. A Stat. Mech. Appl. 188(4), 526–552 (1992)
Shirikyan, A.: Exact controllability in projections for three-dimensional Navier–Stokes equations. Ann. Inst. H. Poincar Anal. Non Linaire 24(4), 521–537 (2007)
Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (University of California, Berkeley, California, 1970/1971), vol. 3. Probability Theory, pp. 333–359 (1972)
Villani, C.: Hypocoercivity. Mem. Am. Math. Soc. 202(950), iv+141. https://doi.org/10.1090/S0065-9266-09-00567-5 (2009)
Acknowledgements
We thank Jonathan Mattingly for an illuminating discussion. We also thank the referees for their very helpful comments, which were important for improving the original version of the article. The research was supported in part by Grants DMS 1362467 and DMS 1159376 from the National Science Foundation.
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Communicated by Alex Kiselev.
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Hu, W., Šverák, V. Dynamics of Geodesic Flows with Random Forcing on Lie Groups with Left-Invariant Metrics. J Nonlinear Sci 28, 2249–2274 (2018). https://doi.org/10.1007/s00332-018-9446-1
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DOI: https://doi.org/10.1007/s00332-018-9446-1