Integration of Brownian vector fields

Jan, Yves Le; Raimond, Olivier

doi:10.1214/aop/1023481009

Cited by 111 publications

(121 citation statements)

References 17 publications

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“…It is well-known (see, for example, LeJan [11]) that in the physically interesting models, such as Kraichnan velocity [10], the matrix-valued function C = C(x) has the Fourier transformĈ = C(z) given by…”

Section: Is An Isotropic Gaussian Vector Field With Zero Mean and Covmentioning

confidence: 99%

“…, z d ), z is the corresponding column vector, |z| 2 = z * z, I is the identity matrix; α > 0, a ≥ 0, b ≥ 0, A 0 > 0 are real numbers. Similar to [11], we will assume that 0 < α < 2.…”

Section: Is An Isotropic Gaussian Vector Field With Zero Mean and Covmentioning

confidence: 99%

“…For non-smooth vector fields v, analysis of either (1.1) or (1.3) is impossible without further specifying the function v. It is shown in [11] that in many physical models, such as turbulent flows, etc., the function v = (v 1 (t, x), . .…”

Section: Introductionmentioning

confidence: 99%

“…In [11], the authors provide a systematic study of the family of operators (1.6) S s,t : f (x) → E f (X x s,t )|F W s,t…”

Section: Introductionmentioning

confidence: 99%

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Passive scalar equation in a turbulent incompressible Gaussian velocity field

Лотоцкий¹,

Rozovskiĭ²

2004

Russ. Math. Surv.

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Abstract. Time evolution of a passive scalar is considered in a turbulent homogeneous incompressible Gaussian flow. The turbulent nature of the flow results in non-smooth coefficients in the corresponding evolution equation. A strong, in the probabilistic sense, solution of the equation is constructed using Wiener Chaos expansion, and the properties of the solution are studied. Among the results obtained are a certain L p -regularity of the solution and Feynman-Kac-type, or Lagrangian, representation formula. The results apply to both viscous and conservative flows.

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Section: Is An Isotropic Gaussian Vector Field With Zero Mean and Covmentioning

confidence: 99%

Section: Is An Isotropic Gaussian Vector Field With Zero Mean and Covmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In [11], the authors provide a systematic study of the family of operators (1.6) S s,t : f (x) → E f (X x s,t )|F W s,t…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Passive scalar equation in a turbulent incompressible Gaussian velocity field

Лотоцкий¹,

Rozovskiĭ²

2004

Russ. Math. Surv.

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“…Given (1.2) and (1.3) the n-point functions F n of the scalar T obey the so-called Hopf equations (see [16]):…”

Section: Introductionmentioning

confidence: 99%

Passive Advection and the Degenerate Elliptic Operators M n

Hakulinen¹

2003

Communications in Mathematical Physics

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Noise Prevents Collapse of Vlasov‐Poisson Point Charges

Delarue

Flandoli

Vincenzi

2013

Comm Pure Appl Math

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We elucidate the effect of noise on the dynamics of N point charges in a Vlasov-Poisson model driven by a singular bounded interaction force. A too simple noise does not impact the structure inherited from the deterministic case and, in particular, cannot prevent coalescence. Inspired by the theory of random transport in passive scalars, we identify a class of random fields which generate random pulses that are chaotic enough to disorganize the deterministic structure and prevent any collapse of the particles. We obtain strong unique solvability of the stochastic model for any initial configuration of different point charges. In the case where there are exactly two particles, we implement the "vanishing noise method" for determining the continuation of the deterministic model after collapse.

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Integration of Brownian vector fields

Cited by 111 publications

References 17 publications

Passive scalar equation in a turbulent incompressible Gaussian velocity field

Passive scalar equation in a turbulent incompressible Gaussian velocity field

Passive Advection and the Degenerate Elliptic Operators M n

Noise Prevents Collapse of Vlasov‐Poisson Point Charges

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