Isotropic Stochastic Flows

Baxendale, Peter; Harris, T. E.

doi:10.1214/aop/1176992360

Cited by 91 publications

(120 citation statements)

References 17 publications

Supporting

Mentioning

118

Contrasting

Unclassified

Order By: Relevance

“…We note that in the case where inertia of the particles can be neglected, our results reduce to calculating the Lyapunov exponent for a spatially correlated Brownian motion, which was discussed from a mathematical point of view by Le Jan (1985) and Baxendale & Harris (1986).…”

Section: Illustration and Contextmentioning

confidence: 89%

Aggregation of inertial particles in random flows

et al. 2005

View full text Add to dashboard Cite

We describe a criterion for particles suspended in a randomly moving fluid to aggregate. Aggregation occurs when the expectation value of a random variable is negative. This variable evolves under a stochastic differential equation. We analyse this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short, such that the stochastic differential equation is a Langevin equation.

show abstract

Section: Illustration and Contextmentioning

confidence: 89%

Aggregation of inertial particles in random flows

et al. 2005

View full text Add to dashboard Cite

show abstract

“…Its positivity w as established in 8 for velocity elds that are a nite-dimensional Ornstein-Uhlenbeck process. The Jacobian and the evolution of curves in isotropic stochastic ows with zero drift is studied in 4 . The trajectories of 1.1 behave v ery di erently from mean zero time dependent ows.…”

Section: Evolution Of the Jacobian Matrixmentioning

confidence: 99%

“…In particular we should have that E ffYt; yg E ffSt; yg We note further that the generator 10.2 is formally of the form L J f = c J ; J @ 2 f @J@J : 10.6 Such di usion processes have t ypically positive non-zero Lyapunov exponents, that is, the limit = lim T!1 lnjjJtjj T exists with probability one, and 0. The computation of for some time dependent o ws was done in 4,8,31,19,20 . We shall show in Section 11 that in our case = 0.…”

Section: Evolution Of the Jacobian Matrixmentioning

confidence: 99%

Evolution of trajectory correlations in steady random flows

Fannjiang¹,

Ryzhik²,

Papanicolaou³

1997

Recent Advances in Partial Differential Equations, Venice 1996

View full text Add to dashboard Cite

Abstract. W e analyze the behavior of the correlation for two nearby trajectories of motion in a random incompressible o w with nonzero mean and small uctuations. W e show that the Fourier transform of the Richardson function of a passive scalar advected by the ow satis es, under certain conditions, a radiative transport equation. We also study the stretching of curves advected by the ow and show that their length grows algebraically in time, and not exponentially as it does for time dependent, zero mean random ows. Contents

show abstract

“…Although extensive literature is available for stochastic flows driven by standard Brownian motion (see [5,13]), very little is known when the driver of the flow is changed to a nonMarkovian, non-diffusive process, such as fractional Brownian motion. For instance, some of the very basic results concerning the tangent flow are yet to be unearthed in the case when the flow is driven by fractional Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

“…To this end, note first that from (14) we can bound the second integrand by |D 1−α t− B H γ,t− (r)| ≤ k 1 (α, β) B H γ β,T (t − r) α+β−1 . Now using (5) and Assumptions (A2)-(A4), the first integrand can be bounded by…”

mentioning

confidence: 99%