Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields


Whenever one uses translation invariant mean Green’s functions to describe the behavior in the mean and to estimate dispersion coefficients for diffusion in random velocity fields, the spatial homogeneity of the transition probability of the transport process is implicitly assumed. This property can be proved for deterministic initial conditions if, in addition to the statistical homogeneity of the space-random velocity field, the existence of unique classical solutions of the transport equations is ensured.

When uniqueness condition fails and translation invariance of the mean Green’s function cannot be assumed, as in the case of nonsmooth samples of random velocity fields with exponential correlations, asymptotic dispersion coefficients can still be estimated within an alternative approach using the Itô equation.

Numerical simulations confirm the predicted asymptotic behavior of the coefficients, but they also show their dependence on initial conditions at early times, a signature of inhomogeneous transition probabilities.

Such memory effects are even more relevant for random initial conditions, which are a result of the past evolution of the process of diffusion in correlated velocity fields, and they persist indefinitely in case of power law correlations.

It was found that the transition probabilities for successive times can be spatially homogeneous only if a long-time normal diffusion limit exits. Moreover, when transition probabilities, for either deterministic or random initial states, are spatially homogeneous, they can be explicitly written as Gaussian distributions.


N. Suciu
-Chair for Applied Mathematics I, Friedrich-Alexander University, Erlangen-Nuremberg, Germany
-Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj Napoca, Romania


Cite this paper as:

N. Suciu, Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields, Phys. Rev. E, 81 (2010), 056301,
doi: 10.1103/PhysRevE.81.056301


see the expansion block below.


About this paper


Phys. Rev. E

Publisher Name
Print ISSN


Online ISSN


Google Scholar Profile

google scholar link

[1] R. Phythian and W. D. Curtis, The effective long-time diffusivity for a passive scalar in a Gaussian model fluid flow, J. Fluid Mech. 89, 241 (1978).
CrossRef (DOI)

[2] M. Dentz and D. M. Tartakovsky, Self-consistent four-point closure for transport in steady random flows, Phys. Rev. E 77, 066307 (2008).
CrossRef (DOI)

[3] M. Dentz and B. Berkowitz, Exact effective transport dynamics in a one-dimensional random environment, Phys. Rev. E 72, 031110 (2005).
CrossRef (DOI)

[4] N. Suciu, C. Vamoş, H. Vereecken, K. Sabelfeld, and P. Knabner, Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media, Water Resour. Res. 44, W08501 (2008).
CrossRef (DOI)

[5] N. Suciu, C. Vamoş, F.A. Radu, H. Vereecken, and P. Knabner, Persistent memory of diffusing particles, Phys. Rev. E 80, 061134 (2009).
CrossRef (DOI)

[6] J. Honkonen, Stochastic processes with stable distributions in random environments, Phys. Rev. E 53, 327 (1996).
CrossRef (DOI)

[7] J. L. Doob, Stochastic Processes (Wiley, New York, 1990).

[8] P. E. Kloeden and E. Platen, Numerical Solutions of Stochastic Differential Equations (Springer, Berlin, 1999).

[9] Y. Aït-Sahalia, Telling from Discrete Data Whether the Underlying Continuous-Time Model Is a Diffusion, J. Finance 57, 2075 (2002).
CrossRef (DOI)

[10] J. G. Conlon and A. Naddaf, N.Y. J. Math. 6, 153 (2000).

[11] T. Delmotte and J.-D. Deuschel, On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to ϕ interface model, Probab. Theory Relat. Fields 133, 358 (2005).
CrossRef (DOI)

[12] C. L. Zirbel, Lagrangian observations of homogeneous random environments, Adv. Appl. Probab. 33, 810 (2001).
CrossRef (DOI)

[13] A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions (Springer, New York, 1987), Vol. I.

[14] A. Fannjiang and T. Komorowski, Diffusive and Nondiffusive Limits of Transport in Nonmixing Flows, SIAM J. Appl. Math. 62, 909 (2002).
CrossRef (DOI)

[15] A. J. Majda and P. R. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Phys. Rep. 314, 237 (1999).
CrossRef (DOI)

[16] N. Suciu, C. Vamoş, and J. Eberhard, Evaluation of the first-order approximations for transport in heterogeneous media Water Resour. Res. 42, W11504 (2006).
CrossRef (DOI)

[17] H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys. 65, 97 (1979).
CrossRef (DOI)

[18] M. H. A. Davis, Functionals of diffusion processes as stochastic integrals, Math. Proc. Cambridge Philos. Soc. 87, 157 (1980).
CrossREf (DOI)

[19] C. Vamoş, N. Suciu, and H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comput. Phys. 186, 527 (2003).
CrossRef (DOI)

[20] G. Sposito and G. Dagan, Predicting solute plume evolution in heterogeneous porous formations, Water Resour. Res. 30, 585 (1994).
CrossRef (DOI)

[21] C. Nicolis and G. Nicolis, Propagation of extremes in space, Phys. Rev. E 80, 061119 (2009).
CrossRef (DOI)

[22] F. N. C. Paraan, M. P. Solon, and J. P. Esguerra, Brownian motion of a charged particle driven internally by correlated noise, Phys. Rev. E 77, 022101 (2008)
CrossRf (DOI)

[23] M. A. Despósito and A. D. Vinales, Memory effects in the asymptotic diffusive behavior of a classical oscillator described by a generalized Langevin equation, ibid. 77, 031123 (2008);
CrossRef (DOI)

[24] R. L. S. Farias, R. O. Ramos, and L. A. da Silva, Stochastic Langevin equations: Markovian and non-Markovian dynamics, ibid. 80, 031143 (2009).
CrossRef (DOI)

[23] V. Ilyin, I. Procaccia, and A. Zagorodny, Stochastic processes crossing from ballistic to fractional diffusion with memory: Exact results, Phys. Rev. E 81, 030105(R) (2010).
CrossRef (DOI)

[24] B. Dybiec and E. Gudowska-Nowak, Discriminating between normal and anomalous random walks,Phys. Rev. E 80, 061122 (2009).
CrossRef (DOI)

[25] B. B. Mandelbrot and J. W. van Ness, Fractional Brownian Motions, Fractional Noises and Applications, SIAM Rev. 10, 422 (1968).
CrossRef (DOI)

[26] T. Marquardt, Fractional Lévy processes with an application to long memory moving average processes, Bernoulli 12, 1099 (2006).
CrossRef (DOI)

[27] S. N. Majumdar, Persistence of a particle in the Matheron-de Marsily velocity field, Phys. Rev. E 68, 050101(R) (2003).
CrossRef (DOI)

[28] R. García-García, A. Rosso, and G. Schehr, Longest excursion of fractional Brownian motion: Numerical evidence of non-Markovian effects, Phys. Rev. E 81, 010102(R) (2010).
CrossRef (DOI)

[29] G. Matheron and G. de Marsily,   Water Resour. Res. 16, 901 (1980).
CrossRef (DOI)


Related Posts