Passive tracer in non-Markovian, Gaussian velocity field

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Model 3G

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Passive tracer in non-Markovian, Gaussian velocity field Tymoteusz Chojecki Institute of Mathematics, UMCS, pl. Marii Curie-Skłodowskiej 1, 20-031, Lublin, Poland

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Article history: Received 2 February 2018 Received in revised form 3 August 2018 Accepted 5 August 2018 Available online xxxx

We consider the trajectory of a tracer that is the solution of an ordinary differential equation X˙ (t) = V (t , X (t)), X (0) = 0, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see Fannjiang and √ Komorowski (1999), Carmona and Xu (1996) and Komorowski et al. (2012), that X (t)/ t converges in law, as t → +∞, to a normal, zero mean vector, provided that the field V (t , x) is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function. © 2018 Published by Elsevier B.V.

Keywords: Passive tracer Central limit theorem

1. Introduction and some assumptions

1

In this paper we would like to show the central limit theorem for a passive tracer model, when the velocity field is non-Markovian but Gaussian and exponentially mixing in time. Passive tracer model is given by the following equation,

⎧ ⎪ ⎨

dX (t)

⎪ ⎩

X (0) = 0,

dt

= V (t , X (t)) ,

2 3 4

t > 0, (1.1)

∑d

where V : R1+d × Ω → Rd is a real, d−dimensional, incompressible i.e. p=1 ∂xp Vp (t , x) ≡ 0, zero mean, Gaussian random vector field over a probability space (Ω , F , P). This model describes a trajectory of particle motion in an incompressible, disordered flow and has applications e.g. in turbulent diffusion and stochastic homogenization, see Majda and Kramer (1999), Kraichnan (1970), Warhaft (0000) and Sreenivasan and Schumacher (2010). Some basic problems concerning the asymptotic behavior of the tracer are: the law of large numbers (LLN) i.e. whether X (t)/t converges to √a constant vector v∗ (called the Stokes drift), as t → +∞ and the central limit theorem (CLT), i.e. whether (X (t) − v∗ t)/ t is convergent in law to a normal vector N(0, κ ). The covariance matrix κ = [κij ]i,j=1,...,d is called turbulent diffusivity of the tracer. It is generally believed that, if the field is stationary and sufficiently strongly mixing, then the central limit theorem holds, see e.g. Arnold (1964), Kraichnan (1970) and Taylor (1923). It is expected, see Majda and Kramer (1999), Arnold (1964) and Kraichnan (1970), that both the LLN, with v∗ = 0, and the CLT for the tracer trajectory hold when the velocity field is zero mean, Gaussian, incompressible and its covariance matrix R(t , x) = [Rpq (t , x)]p,q=1,...,d , given by Rpq (t , x) := E[Vp (t , x)Vq (0, 0)],

p, q = 1, . . . , d, (t , x) ∈ R

1+d

,

E-mail address: [email protected]. https://doi.org/10.1016/j.spl.2018.08.002 0167-7152/© 2018 Published by Elsevier B.V.

Please cite this article in press as: Chojecki T., Passive tracer in non-Markovian, Gaussian velocity field. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.08.002.

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6 7 8 9 10 11 12 13 14 15 16 17

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exponentially decays in time, i.e. there exists C > 0 such, that d ∑

|Rpq (t , x)| ≤ Ce−|t |/C , for all (t , x) ∈ R1+d .

(1.2)

p,q=1 3 4 5 6 7

8

9 10 11

12

13 14 15

16

17 18 19 20 21 22 23

24

25 26 27

28

29

30

The CLT has been established in Komorowski and Papanicolaou (1997), in the case of T −dependent fields, i.e. those for which exists T > 0 such that R(t , x) = 0, |t | > T , x ∈ Rd . In case when the vector field V (t , x) is Markovian (not necessarily Gaussian) and satisfies the spectral gap condition, the CLT has been established in Fannjiang and Komorowski (1999), Theorem A, see also Komorowski et al. (2012), Carmona and Xu (1996) and Koralov (1999). In the Gaussian case when the covariance matrix is of the form Rpq (t , x) =

∫ Rd

eix·ξ −γ (ξ )|t | Rˆ pq (dξ ),

p, q = 1, . . . , d, (t , x) ∈ R1+d ,

(1.3)

ˆ ·) = [Rˆ pq (·)] are even (because the field is real), then where both γ (·) and non-negative Hermitian matrix valued measure R( the field is Markovian. It can be shown, see Chapter 12 of Komorowski et al. (2012), that the spectral gap condition holds, provided there exists γ0 > 0 such that γ (ξ ) ≥ γ0 ,

ξ ∈ Rd .

(1.4)

Until now the CLT has not been proved in such a generality for fields satisfying (1.2). We would like to perform a step in this direction. In this paper we consider non-Markovian fields, where the exponential factor in (1.3) is replaced by a function h : [0, +∞) × Rd → R i.e. the covariance matrix is defined as follows Rpq (t , x) =

∫ Rd

eix·ξ h(|t |, ξ )Rˆ pq (ξ )dξ ,

p, q = 1, . . . , d, (t , x) ∈ R1+d ,

(1.5)

where Rˆ pq (ξ ) is a density of Rˆ pq (·). We show in Proposition 2.1 that the function h is non-negative definite iff R(t , x) = [Rpq (t , x)] is non-negative definite. Therefore, the largest (in the sense of inclusion) set of functions h in (1.5) which can be examined is the class of non-negative definite functions. In this paper, we study a smaller family of functions, namely we assume that h is completely monotone in the sense of Bernstein, see (1.7). Let us denote by rˆ the power–energy spectrum. It is a scalar non-negative, integrable function given by formula

ˆ ξ ), rˆ (ξ ) := trR(

ξ ∈ Rd ,

(1.6)

where tr is the trace. Let r(dξ ) := rˆ (ξ )dξ . The main result of the paper, see Theorem 2.1, is the CLT for the trajectory of a tracer moving in a field whose covariance matrices are given by (1.5), where the function h is completely monotone i.e. h ∈ C ∞ (0, +∞) ∩ C [0, +∞),

(−1)n h(n) (t , ξ ) ≥ 0, t > 0, r a.e. ξ , n = 0, 1, . . .

and satisfies (1.8). From the Bernstein Theorem (Lax, 2002 Theorem 3., p. 138) we know that h(t , ξ ) =

+∞

∫

e−λt µ(ξ , dλ),

t ∈ [0, +∞), r a.e. ξ ,

(1.7)

0 31 32 33

34

35 36 37 38 39 40 41 42 43

where µ(ξ , ·) is a non-negative, finite Borel measure on [0, +∞) for r a.e. ξ . For example from Schilling et al. (0000), Lemma 4.5, we know, that all completely monotone functions are non-negative definite. We assume that there exists λ0 > 0 such that supp µ(ξ , ·) ⊂ [λ0 , +∞),

r a.e. ξ .

(1.8)

Observe that this assumption implies (1.2). Note also that when µ(ξ , dλ) = δγ (ξ ) (dλ), we recover the Markovian case, see (1.3). In this way we generalize results from Fannjiang and Komorowski (1999), Koralov (1999) and Carmona and Xu (1996). In Section 3.1 we show (see (3.7)) an example of a covariance matrix which is of the form (1.5) but not of the form (1.3). We prove Theorem 2.1 in Section 3 by embedding the field V (t , x) into a larger space where we add one dimension and one argument i.e. a space of d + 1 dimensional fields V˜ (t , x, y). We define a field V˜ (t , x, y) in such a way that the field (V˜ (t , x, 0)) has the same distribution as the field (V (t , x), 0). The process V˜ (t , ·, ·) has the Markov property in appropriate function space. This process also has the spectral gap property. At the end we use Theorem 12.13 from Komorowski et al. (2012). In Section 4 we show the proof of Proposition 2.1. Please cite this article in press as: Chojecki T., Passive tracer in non-Markovian, Gaussian velocity field. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.08.002.

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3

2. Preliminaries and the statement of the main result

1

ˆ ξ ) = [Rˆ pq (ξ )]p,q=1,...,d , which guarantee that function First, we present some assumptions on matrix valued function R( R(t , x) (defined in (1.5)) is non-negative definite: Rˆ pq (ξ ) = Rˆ ∗qp (ξ ),

ˆ ξ )η · η ≥ 0, R(

p, q = 1, . . . , d, r a.e. ξ ,

η ∈ Cd , r a.e. ξ .

4

(2.2)

5

Proposition 2.1. For any N ≥ 1, α1 , . . . , αN ∈ C and t1 , . . . , tN ∈ R we have h(|tp − tq |, ξ )αp α¯ q ≥ 0,

for r a.e. ξ ,

3

(2.1)

Now we present the result which explains why we need to deal with functions h, which are non-negative definite in t for r a.e. ξ .

N ∑

2

6 7

8

(2.3)

9

p,q=1

iff the matrix valued function R given by (1.5) is non-negative definite.

10

The proof of this proposition is presented in Section 4. Since E|V (0, 0)|2 < +∞, we have

∫

h(0, ξ )rˆ (ξ )dξ < +∞.

Rd

11 12

(2.4)

Recall that we have assumed that h is of the form (1.7). To fulfill Assumption (2.4) we require that esssup µ(ξ , [0, +∞)) < +∞.

14

(2.5)

ξ

It implies that |h| is bounded. Above esssup is the essential supremum with respect to r. Then (2.4) is implied by

∫

rˆ (ξ )dξ < +∞.

Rd

(2.6)

• For r a.e. ξ we have

17

18 19 20 21

22

Rˆ pq (ξ ) = Rˆ qp (−ξ ),

p, q = 1, . . . , d.

(2.7)

• For any t ∈ R

23

24

h(|t |, ξ ) = h(|t |, −ξ ),

r a.e. ξ .

(2.8)

Assumption (2.8) implies (see (1.7))

µ(ξ , ·) = µ(−ξ , ·),

25

26

r a.e. ξ .

(2.9)

To sum up, in this paper we consider the model in which a d-dimensional random vector field V (t , x) has the covariance ˆ ξ ) and h(|t |, ξ ) satisfy Assumptions (2.1)–(2.9). matrix given by (1.5), where R( To be able to solve Eq. (1.1) we need to assume an appropriate regularity of the field. Namely, we assume that there exists the second derivative in x of the field V (t , x) and it is continuous. This implies (see (1.5)) Rd

15

16

Assumptions (2.1)–(2.6) imply that the matrix valued function R(·, ·) is non-negative definite. From Ibragimov and Rozanov (1978), Section I.3, we know that there exits a unique (in the sense of law) stationary Gaussian random vector field V (t , x) such that R(t , x) is its covariance matrix. We want to deal with a real field, so we need to assume, see Roznaov (1967) Theorem 4.2., p. 18, that

∫

13

(1 + |ξ |4 )rˆ (ξ )dξ < +∞.

(2.10)

Moreover, we assume that the field V (t , x) is incompressible, i.e.

27 28 29 30 31

32

33

d

∇x · V (t , x) =

∑

∂xp Vp (t , x) ≡ 0,

(t , x) ∈ R1+d .

34

p=1

We can rewrite this using (1.5) as follows d ∑

ξp Rˆ pq (ξ ) = 0,

q = 1, . . . , d, r a.e. ξ .

35

(2.11)

p=1

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From the above equality we obtain d ∑

2

∂xp Rpq (t , x) = 0,

(t , x) ∈ R1+d , q = 1, . . . , d.

(2.12)

p=1 3 4 5 6

7 8 9

Formula (1.5) and condition (2.10) imply in particular (see Theorem 3.1 from Qualls and Watanabe, 1973) that there exists a version of V (t , x) with the realizations which grows slower than linearly in (t , x) a.s., therefore Eq. (1.1) can be solved globally in t, and the process X (t) is defined for all t ∈ [0, +∞). Now we can present our main result of this paper. Theorem 2.1. Assume that V (t , x) is a zero-mean Gaussian random vector field whose covariance matrix √ R(t , x) = [Rpq (t , x)]p,q=1,...,d is given by (1.5). In addition suppose that (2.1)–(2.11) hold. Then the random variables X (t)/ t, where X (t) is defined in (1.1), converge weakly to the normal vector N(0, κ ), where the limit covariance matrix κ = [κpq ] satisfies: 1

κpq = lim

10

t →+∞

11

12

t

E[Xp (t)Xq (t)] p, q = 1, . . . , d.

3. Proof of Theorem 2.1 Consider a d + 1−dimensional field ˜ V : R2+d × Ω → R1+d with the covariance matrix

˜ Rpq (t , x, y) =

13

∫ ∫ Rd

R 14

∫

dξ

m(A × B) =

{∫

A

µ ˜ (ξ , dλ) :=

21 22

1 2

A ∈ B(Rd ), B ∈ B(R),

(3.2)

B

1(0,+∞) (λ)µ(ξ , dλ) +

∫ ∫ ∫

1 ˜ Rpq (t , x, y) = π

1 2

1(−∞,0] (λ)µ(ξ , −dλ),

R

R

Rd

λ eiτ t +ix·ξ +iλy Rˆ pq (ξ )µ ˜ (ξ , dλ)dτ dξ . λ2 + τ 2

This equality together with Assumptions (2.1)–(2.11) implies that ˜ Rpq (t , x, y) is the covariance matrix of some zero mean, Gaussian, stationary in time and space, random, real valued vector field ˜ V (t , x, y) (Ibragimov and Rozanov, 1978, Section I.3). Observe that for y = 0 we have

˜ Rpq (t , x, 0) =

∫ ∫ R

23

+ 24

} µ ˜ (ξ , d λ ) ,

recall that µ is given by (1.7). The covariance matrix in full Fourier transform form is given by

19

20

(3.1)

where µ ˜ is defined as follows

17

18

t , y ∈ R, x ∈ Rd ,

for p, q = 1, . . . , d, ˜ Rpq = 0, if p or q = d + 1. Above m(dξ , dλ) := µ ˜ (ξ , dλ)dξ would be the measure given by

15

16

e−|λt | eiξ ·x+iλy Rˆ pq (ξ )m(dξ , dλ),

1 2

∫

∫

0

e Rd

Rd

λ|t | iξ ·x

e

1

e−|λt | eiξ ·x µ ˜ (ξ , dλ)Rˆ pq (ξ )dξ =

∫

2

µ(ξ , −dλ)Rˆ pq (ξ )dξ = Rpq (t , x),

∫ Rd

+∞

e−λ|t | eiξ ·x µ(ξ , dλ)Rˆ pq (ξ )dξ

0

(3.3)

p, q = 1, . . . , d, (t , x) ∈ R

1+d

,

−∞

where matrix Rpq (t , x) is given by (1.5). Observe that ˜ Vd+1 (t , x, y) ≡ 0, because ˜ Rd+1,d+1 ≡ 0. So the field satisfies

{ } d { } ˜ V (t , x, 0), (t , x) ∈ R1+d = (V (t , x), 0) , (t , x) ∈ R1+d ,

25

(3.4)

d

26 27 28 29 30

where = denotes equality in the law. Let r˜ be a measure r˜ (dξ , dλ) := µ ˜ (ξ , dλ)rˆ (ξ )dξ . From (1.8) we know that |λ| ≥ λ0 , r˜ a.e. ξ . Let ρ > (d + 1)/2. Denote by E the Hilbert space that is the completion of the space of functions v = (v1 , . . . , vd+1 ) : Rd+1 → Rd+1 with components in Cc∞ (Rd+1 ) (the space of infinitely differentiable compactly supported functions), under the norm

∥v∥2E :=

31

32

33

∫ Rd+1

( ) |v (z)|2 + |∇v (z)|2 (1 + |z |2 )−ρ dz .

Here

|v (z)|2 :=

d+1 ∑ i=1

vi2 (z)

and

|∇v (z)|2 :=

d+1 ∑ (

)2 ∂zj vi (z),

z ∈ Rd+1 .

i,j=1

Please cite this article in press as: Chojecki T., Passive tracer in non-Markovian, Gaussian velocity field. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.08.002.

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Consider the process

1

V˜ t := V˜ (t , ·, ·), t ≥ 0.

2

From Komorowski et al. (2012) Chapter 12, we know that the process takes values in space E . This space contains the realization of the field V˜ (0, ·, ·). This is a separable Hilbert space which turns out to be a subspace of the space of C 1 maps v : R1+d → R1+d . It can be shown that process {V˜ t , t ≥ 0} is stationary. On the space E we introduce a measure π as a distribution of V˜ (0, ·, ·). Process {V˜ t , t ≥ 0} has the Markov property with the corresponding transition semigroup (Pt )t ≥0 i.e.

[

]

E F (V˜ t +h )|Vt = Ph F (V˜ t ),

h ≥ 0, t ≥ 0, F ∈ Bb (E ),

(3.5)

where Vt := σ (V˜ s , s ≤ t) is a natural filtration of the process {V˜ t , t ≥ 0}. (Pt )t ≥0 is a semigroup of Markov contractions on L2 (π ). Its generator L : D(L) → L2 (π ) has the spectral gap property (see Komorowski et al., 2012, Corollary 12.15) i.e.

− ⟨LF , F ⟩L2 (π ) ≥ γ0 ∥F ∥2L2 (π ) ,

for F ∈ D(L), such that

∫

Fdπ = 0,

3 4 5 6

7

8 9

10

E

where γ0 was introduced in (1.4). We consider the following equation

11 12

⎧ ˜ dXp (t) ⎪ ⎪ =˜ Vp (t , X˜ (t), Y (t)), p = 1, . . . , d, t ≥ 0, ⎪ ⎪ dt ⎪ ⎪ ⎨ dY (t) =˜ Vd+1 (t , X˜ (t), Y (t)), dt ⎪ ⎪ ⎪ ⎪ X˜ (0) = 0, ⎪ ⎪ ⎩ Y (0) = 0.

(3.6)

13

The existence and uniqueness of the solution (3.6) comes from the form of the covariance of the field V˜ (·). Because

14

d ˜ Vd+1 (t , x, y) ≡ 0 we have Y (t) = 0 for t ≥ 0. This and (3.4) imply {X˜ (t), t ≥ 0} = {X (t), t ≥ 0}, where X (t) was defined in

15

(1.1). So we will omit writing the symbol tilde over X (t). Now we introduce the so called environment process Ut := ˜ V (t , X (t) + ·, Y (t) + ·) ∈ E ,

16 17

P a.e. t ≥ 0.

18

Observe that

19

t

∫

F (Us )ds,

X (t) =

t ≥ 0,

20

0

where F : E → Rd is given by F (ω) = (ω1 (0, 0), . . . , ωd (0, 0)), ω ∈ E . Process {Ut , t ≥ 0} is also Markovian, see Komorowski et al. (2012) Proposition 12.19 (i). Observe that

∑

Vd+1 (t , x, y) = Vi (t , x, y) + ∂y˜ ∂xi ˜

∑d

∑

Vi (t , x, y), ∂xi ˜

t , y ∈ R, x ∈ Rd .

23

i=1

i=1

Let Z (t , x, y) :=

∂ ˜ , x, y). Observe that

i=1 xi Vi (t

EZ 2 (t , x, y) = EZ 2 (0, 0, 0) =

d ∑ i,j=1

∂x2i xj˜ Rij (0, 0, 0) =

24

d ∑

∂xj

{ d ∑

j=1

} ∂xi Rij (0, 0) = 0,

25

i=1

where the last two equalities above come respectively from (3.3) and (2.12). Since ∇ · ˜ V = 0 we know that π is an invariant measure for the process {Ut , t ≥ 0}, see Komorowski et al. (2012) Proposition 12.19 (ii). Moreover, the transition semigroup (Rt )t ≥0 for {Ut , t ≥ 0}, on L2 (π ), with generator L : D(L) → L2 (π ), has the spectral gap property (Komorowski et al., 2012 Corollary 12.22) i.e.

− ⟨LF , F ⟩L2 (π ) ≥ γ0 ∥F ∥2L2 (π ) ,

for F ∈ D(L), such that

∫

Fdπ = 0.

e Rd

28 29

Rˆ pq (dξ ), p, q = 1, . . . , d, 0 < α ≤ 1, (t , x) ∈ R1+d ,

31 32

33

Observe that Theorem 2.1 can be applied to a field whose covariance matrix is given by Rpq (t , x) =

27

30

3.1. Example of non-Markovian field which satisfies the assumption of Theorem 2.1

ix·ξ −γ (ξ )|t |−γ˜ (ξ )|t |α

26

E

The CLT for the process {X (t), t ≥ 0} (which is defined in (1.1)) comes directly from Theorem 12.13 of Komorowski et al. (2012). □

∫

22

d

d

∇ ·˜ V (t , x, y) =

21

34

(3.7)

Please cite this article in press as: Chojecki T., Passive tracer in non-Markovian, Gaussian velocity field. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.08.002.

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where functions γ˜ , γ : Rd → R satisfy (1.4). Observe that the function h(t , ξ ) =

+∞

∫

e−t

αλ

µ(ξ , dλ),

t ≥ 0, r a.e.ξ ,

(3.8)

0 3

4

where α ∈ (0, 1] is completely monotone in t. Indeed, we know that H(t , ξ ) =

+∞

∫

e−t λ µ(ξ , dλ),

t ≥ 0, r a.e. ξ

0 5 6 7 8

is completely monotone. However, φ (t) := t α can be represented by an integral from ψ (t) := α t α−1 , which is completely monotone on (0, +∞), if α ∈ (0, 1]. Therefore, from Theorem 8 of Schoenberg (1938) we have that h(t , ξ ) = H(φ (t), ξ ) is completely monotone on [0, +∞). Now we will check condition (1.8). Observe that we can write for µ(ξ , dλ) := δγ˜ (ξ ) (dλ), t ≥ 0, +∞

∫

e−t

9

αλ

α

µ(ξ , dλ) = e−γ˜ (ξ )t =

0 10 11

12

+∞

∫

e−γ˜ (ξ )λt g(λ)dλ,

where g is a smooth density of T (1), where {T (t), t ≥ 0} is a α -stable subordinator (Zolotarev, 1983, p. 201, formula (2.10.7)). Observe that for t ≥ 0 e

−γ (ξ )t −γ˜ (ξ )t α

+∞

∫

−(γ˜ (ξ )λ+γ (ξ ))t

e

=

g(λ)dλ =

14

+∞

∫

˜ ξ , λ)dλ, r a.e. ξ , e−λt g(

where

) ( ⎧ ⎨γ˜ (ξ )−1 g λ − γ (ξ ) , ˜ ξ , λ) = γ˜ (ξ ) g( ⎩ 0,

λ > γ (ξ ) , λ ≤ γ (ξ ).

15

˜ ξ , λ)dλ fulfills (1.8). We can see that the last expression in (3.9) is of the form (1.7) and the measure µ ˜ (ξ , dλ) := g(

16

4. Proof of Proposition 2.1

17 18 19

20

Denote by A(ξ ) some non-negative Hermitian, matrix valued function such that

ˆ ξ ) = A(ξ )rˆ (ξ ), R(

ξ ∈ Rd ,

where trA(ξ ) ≡ 1

(4.1)

21

for r a.e. ξ . Choose a function φˆ ∈ S (R ), where S (R ) is the class of C valued Schwartz functions. Then

22

⏐2 ⏐ ⎞ ⎛ ⏐ ⏐∑ ∫ ∫ N ∑ ⏐ ⏐ N 1 ⎝ αp V (tp , x) · φˆ (x)dx⏐⏐ = αp α¯ q h(tp − tq , ξ )⎠ A(ξ )φ (ξ ) · φ (ξ )rˆ (ξ )dξ , 0 ≤ E⏐⏐ (2π )d Rd Rd ⏐ p=1 ⏐ p,q=1

23 24

d

d

d

where φ (ξ ) is the inverse Fourier transform of a function φˆ (x). If we choose φˆ j (x), such that φj (ξ ) = ej ψ (ξ ), where ψ (ξ ) is a scalar function, ej = (0, . . . , 1, . . . , 0) and summing by j = 1, . . . , d, we obtain



25

0≤

1 (2π )d



jth position

⎛

∫ Rd

N ∑

⎝ p,q=1

 ⎞

αp α¯ q h(tp − tq , ξ )⎠ |ψ (ξ )|2

d ∑

A(ξ )ej · ej rˆ (ξ )dξ .

j=1



 =1



26

Considering (4.1) and the fact that ψ was arbitrary, we obtain (2.3). □

27

Uncited references

28

29

30 31

(3.9)

0

0 13

t ≥ 0, r a.e. ξ ,

0

Rosenblatt (1971) Acknowledgments I would like to express my deep gratitude to Professor Tomasz Komorowski whose help and comments have been invaluable. I acknowledge the support of the National Science Centre: NCN grant 2016/23/B/ST1/00492. Please cite this article in press as: Chojecki T., Passive tracer in non-Markovian, Gaussian velocity field. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.08.002.

STAPRO: 8306

T. Chojecki / Statistics and Probability Letters xx (xxxx) xxx–xxx

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References Arnold, V.I., 1964. Instability of dynamical systems with several degrees of freedom. Soviet Math. 5, 581–585. Carmona, R., Xu, L., 1996. Homogenization for time dependent 2-D incompressible Gaussian flows, Preprint. Fannjiang, A., Komorowski, T., 1999. Turbulent diffusion in Markovian flows. Ann. Appl. Probab. 9, 591–610. Ibragimov, I.A., Rozanov, Y.A., 1978. Gaussian Random Processes. Springer, New York. Komorowski, T., Landim, C., Olla, S., 2012. Fluctuations in Markov Processes. Springer. Komorowski, T., Papanicolaou, G., 1997. Motion in a Gaussian, incompresible flow. Ann. Appl. Probab. 229–264. Koralov, L., 1999. Transport by time dependent stationary flow. Comm. Math. Phys. 199 (3), 649–681. Kraichnan, R., 1970. Diffusion in a random velocity field. Phys. Fluids 13, 22–31. Lax, P., 2002. Functional Analysis. Wiley-Interscience, New York. Majda, A.J., Kramer, P.R., 1999. Simplified models for turbulent diffusion: Theory, numerical modeling, and physical phenomena. Phys. Rep. 314, 237–574. Qualls, C., Watanabe, H., 1973. Asymptotic properties of Gaussian random fields. T.A.M.S. 177, 155–171. Rosenblatt, M., 1971. Markov Processes. Structure and Asymptoti Behavior. Springer- Verlag. Roznaov, Yu. A., 1967. Stationary Random Processes. Holden-Day, San Francisco. Schilling, R.L., Song, R., Vondracek, Z., Bernstein Functions Theory and Applications, De Gruyter. Schoenberg, I.J., 1938. Metric spaces and completely monotone functions. Ann. of Math. (2) 39 (4), 811–841. Sreenivasan, K.R., Schumacher, J., 2010. Lagrangian views on turbulent mixing of passive scalars. Phil. Trans. R. Soc. A (368), 1561–1577. Taylor, G.I., 1923. Diffusions By Continuous Movements. Proc. London Math. Soc. (2) 20, 196–211. Warhaft, Z., Passive scalars in turbulent flows, Annu. Rev. Fluid Mech. 32, 203–240. Zolotarev, V.M., 1983. Odnomernye Ustojchivye Raspredeleniya. Nauka.

Please cite this article in press as: Chojecki T., Passive tracer in non-Markovian, Gaussian velocity field. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.08.002.

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