Persistence of a particle in the Matheron-de Marsily velocity field

We show that the longitudinal position x(t) of a particle in a (d+1)-dimensional layered random velocity field (the Matheron-de Marsily model) can be identified as a fractional Brownian motion (fBm) characterized by a variable Hurst exponent H(d)=1-d/4 for d<2 and H(d)=1/2 for d>2. The fBm becomes marginal at d=2. Moreover, using the known first-passage properties of fBm we prove analytically that the disorder averaged persistence [the probability of no zero crossing of the process x(t) up to time t], has a power-law decay for large t with an exponent theta=d/4 for d<2 and theta=1/2 for dgreater than or equal to2 (with logarithmic correction at d=2), results that were earlier derived by Redner based on heuristic arguments and supported by numerical simulations [S. Redner, Phys. Rev. E 56, 4967 (1997)].