Diffusion Processes on Manifolds
Fabrice Debbasch, Claire Chevalier
Abstract
This is an informal introduction to stochastic analysis on both Riemannian and Lorentzian manifolds. We review the basics underlying the construction of diffusions on manifolds, highlighting the important differences between the Riemannian and Lorentzian cases. We also discuss a few recent applications which range from biophysics to cosmology.
Full-text: Open access
Permanent link to this document: http://projecteuclid.org/euclid.imsc/1233152936
Digital Object Identifier: doi:10.1214/074921708000000318
References
[1] Angst, J. and Franchi, J. (2007). Central limit theorem for a class of relativistic diffusions. J. Math. Phys. 48 (8).
[2] Barbachoux, C., Debbasch, F. and Rivet, J. P. (2001). Covariant Kolmogorov equation and entropy current for the relativistic Ornstein–Uhlenbeck process. Eur. Phys. J. B 23 487.
[3] Barbachoux, C., Debbasch, F. and Rivet, J. P. (2001). The spatially one-dimensional relativistic Ornstein–Uhlenbeck process in an arbitrary inertial frame. Eur. Phys. J. B 19 37.
[4] Braga, J., Desterro, J. M. P. and Carmo-Fonseca, M. (2004). The spatially one-dimensional relativistic Ornstein–Uhlenbeck process in an arbitrary inertial frame. Mol. Bio. Cell 15 4749–4760.
[5] Chevalier, C. and Debbasch, F. (2007). Fluctuation-dissipation theorems in an expanding universe. J. Math. Phys. 48 023304.
[6] Chevalier, C. and Debbasch, F. (2008). Is Brownian motion sensitive to geometry fluctuations? J. Stat. Phys. 131 (4) 717–731.
[7] Chevalier, C. and Debbasch, F. (2007). Relativistic transport and fluctuation-dissipation theorem. In 17th International Symposium on Trans- port Phenomena, Toyama, Japan, 2006.
[8] Chevalier, C. and Debbasch, F. (2008). Relativistic diffusions: A unifying approach. J. Math. Phys. 49 043303.
[9] Chevalier, C. and Debbasch, F. (2008). A unifying approach to relativistic diffusions and H-theorems. Mod. Phys. Lett. B 22 (6) 383–392.
[10] Chevalier, C. and Debbasch, F. (2008). Brownian motion and entropy growth on irregular surfaces. International Journal of Numerical Analysis and Modelling, to appear.
[11] Christensen, M. (2004). How to simulate anisotropic diffusion processes on curved surfaces. J. Comput. Phys. 201 421–435.
[12] Debbasch, F. (2004). A diffusion process in curved space-time. J. Math. Phys. 45 (7) 2744.
[13] Debbasch, F. and Chevalier, C. (2007). Relativistic stochastic processes: A review. In Proceedings of Medyfinol 2006, XV Conference on Nonequilibrium Statistical Mechanics and Nonlinear Physics, Mar del Plata, Argentina, Dec. 4–8 2006 (O. Descalzi, O. A. Rosso and H. A. Larrondo, eds.) A.I.P. Conference Proceedings 913. American Institute of Physics, Melville, NY.
[14] Debbasch, F., Mallick, K. and Rivet, J. P. (1997). Relativistic Ornstein–Uhlenbeck process. J. Stat. Phys. 88 945.
[15] Debbasch, F. and Moreau, M. (2004). Diffusion on an interface: A geometrical approach. Physica A 343 81–104.
[16] Debbasch, F. and Rivet, J. P. (1998). A diffusion equation from the relativistic Ornstein–Uhlenbeck process. J. Stat. Phys. 90 1179.
[17] Dowker, F., Henson, J. and Sorkin, R. F. (2004). Quantum gravity phenomenology, Lorentz invariance and discreteness. Mod. Phys. Lett. A 19 1829–1840.
[18] Dubrovin, B. A., Noviko, S. P. and Fomenko, A. T. (1984). Modern Geometry — Methods and Applications. Springer-Verlag, New York.
[19] Dudley, R. M. (1965). Lorentz-invariant Markov processes in relativistic phase space. Arkiv f. Mat. 6 (14) 241–268.
[20] Dunkel, J. and Hänggi, P. (2005). Theory of relativistic Brownian motion: The (1+1)-dimensional case. Phys. Rev. E 71 016124.
[21] Dunkel, J. and Hänggi, P. (2005). Theory of relativistic Brownian motion: The (1+3)-dimensional case. Phys. Rev. E 72 036106.
[22] Emery, M. (1989). Stochastic Calculus in Manifolds. Springer-Verlag.
[23] Franchi, J. (2006). Relativistic diffusion in Gödel’s universe. arXiv math.PR/0612020.
[24] Franchi, J. and le Jan, Y. (2007). Relativistic diffusions and Schwarzschild geometry. Comm. Pure Appl. Math. 60 (2) 187–251.
[25] K. Gödel (1949). An example of a new type of cosmological solution of Einstein’s field equations of gravitation. Rev. Mod. Phys. 21 447–450.
[26] Hawking, S. W. and Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge.
[27] Ikeda, I. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, second edition. North-Holland Mathematical Library.
[28] Israel, W. (1987). Covariant fluid mechanics and thermodynamics: An introduction. In Relativistic Fluid Dynamics (A. Anile and Y. Choquet-Bruhat, eds.) Lecture Notes in Mathematics 1385. Springer-Verlag, Berlin.
[29] Itô, K. (1950). On stochastic differential equations on a differentiable manifold i. Nagoya Math. J. 1 35–47.
[30] Itô, K. (1953). On stochastic differential equations on a differentiable manifold. II. Mem. Coll. Sci. Univ. Kyoto. Ser. A 28 82–85.
[31] Kolb, E. W. and Turner, M. S. (1990). The Early Universe. Frontiers in Physics. Addison-Wesley Publishing Company, Redwood City.
[32] Lamperti, J. (1977). Stochastic Processes. Applied Mathematical Sciences 23. Springer-Verlag, New-York.
[33] McKean, H. P. (1969). Stochastic integrals. Academic Press, New York and London.
[34] Nehls, S. et al (2000). Dynamics and retention of misfolded proteines in native ER membranes. Nat. Cell. Bio. 2 288–295.
[35] Øksendal, B. (1998). Stochastic Differential Equations, fifth edition. Universitext. Springer-Verlag, Berlin.
[36] Peebles, P. J. E. (1980). The Large-Scale Structure of the Universe. Princeton University Press, Princeton.
[37] Rigotti, M. and Debbasch, F. (2005). A H-theorem for the Relativistic-Ornstein–Uhlenbeck process in curved space-time. J. Math. Phys. 46 103303.
[38] Sbalzarini, I. F. et al. (2006). Simulations of (an)isotropic diffusion on curved biological interfaces. Biophysical J 90 (3) 878–885.
[39] Wald, R. M. (1984). General Relativity. The University of Chicago Press, Chicago.
Institute of Mathematical Statistics Collections
