Random integral operators related to the point processes

arXiv:1707.09922v1 [math.PR] 31 Jul 2017

Random integral operators related to the point processes A. A. Dorogovtsev∗ and Ia. A. Korenovska† August 1, 2017

Abstract In the article we study properties of the random integral operator in L2 (R) whose kernel is obtained as a convolution of Gaussian density with a stationary point process.

1

Introduction

Let Θ be a stationary point process on the real line [1]. In this paper we consider integral operators in L2 (R) with the kernel X p(u − θ)p(v − θ), (1) k(u, v) = θ∈Θ

where p is some square-integrable function. The necessity in the investigation of such random kernels arises in the theory of stochastic flows. Namely, in the articles [2, 3] the strong random operators related to an Arratia flow [4] were introduced. If {x(u, t), u ∈ R, t ≥ 0} is an Arratia flow then for every f ∈ L2 (R) and t > 0 Tt f (u) = f (x(u, t)), u ∈ R, is a random element in L2 (R). It was proved in [2] that Tt is a strong random operator in Skorokhod sense [5] but it is not a bounded random operator [3]. Since it is known that the map x(·, t) : R → R is a step function with probability one then for any function f with a bounded support L2 (R)-norm of Tt f equals to zero with positive probability. To avoid such situation one can consider ∗

Institute of Mathematics, National Academy of Sciences of Ukraine Tereshchenkivska Str. 3, Kiev 01601, Ukraine, [email protected] † Institute of Mathematics, National Academy of Sciences of Ukraine Tereshchenkivska Str. 3, Kiev 01601, Ukraine, [email protected]

1

f ∗ pε , where pε is a density of normal distribution with zero mean and variance ε. Then, due to the change of variable formula for an Arratia flow [3], one can obtain Z Z Z X 2 Tt (f ∗ pε )(u) du = ∆y(θ, t) pε (v1 − θ)pε (v2 − θ)f (v1 )f (v2 )dv1 dv2 , R

R

θ: ∆y(θ,t)>0

R

(2) where {y(u, s), u ∈ R, s ∈ [0; t]} is a conjugated Arratia flow [4]. In the right part of (2) one may see the quadratic form of the operator similar to (1). Hence, the knowledge of the properties of (1) can help us in the investigation of random operators constructed from the stochastic flows. The article continues studying of characteristics of random operators from [6, 7].

2

Shifts of Gaussian density along a point process

We will start with the following statement. Theorem 2.1. Let Θ be a stationary ergodic point process on R [1] and E|Θ ∩ [ 0; 1) | < +∞. Then there exists an event Ω0 of probability one such that for each ω ∈ Ω0 a linear combinanations of the functions {pε (· − θ(ω)); θ(ω) ∈ Θ(ω)} are dense in L2 (R). Proof. Lets break the proof into steps: Lemma 2.1. Let Θ be a stationary ergodic point process on R with E|Θ ∩ [ 0; 1) | < +∞. Then with probability one X 1 = +∞. |θ| θ∈Θ Proof. Its sufficient to prove that X

θ∈Θ∩[ 1;+∞)

1 = +∞ a.s. θ

(3)

P P 1 Since θ∈Θ∩[ 1;+∞) 1θ ≥ ∞ + 1) |, then it is enough to show that n=1 n+1 |Θ ∩ [ n; nP 1 for a sequence ξn = |Θ ∩ [ n; n + 1) | the series ∞ n=1 n ξn diverges almost surely. One ∞ may note that {ξn }P n=0 is a stationary, ergodic, and E|ξ0 | < ∞. Hence, due to ergodic theorem, for Sn = nk=0 ξn the following convergence holds 1 Sn → Eξ0 , n → ∞ a.s. n

2

(4)

Thus, with probability one C = supn∈N n1 Sn < +∞, and there exists N ∈ N such that for any n ≥ N Eξ0 1 Sn ≥ . (5) n 2 Using this one can check that for any m ∈ N m X 1

k

k=2

ξk =

m  X Sk k=2

Hence, by (5), the series

Sk−1 − k k−1



P+∞

Sk−1 k=2 k(k−1)

+

m X k=2

m X Sk−1 Sk−1 ≥ 2C + . k(k − 1) k(k − 1) k=2

(6)

diverges, which, by (6), proves the statement.

Corollary 2.1. Using Lemma 2.1 and Muntz theorem one may check that there exists Ω0 of probability one such that for any ω ∈ Ω0 and 0 < a < b a linear combinations θ(ω) of the functions u , θ(ω) ∈ Θ(ω) are dense in L2 ([a; b]).

Corollary 2.2. There exists Ω0 of probability one such that for  any ω ∈ Ω0 and a < b a linear combinations of the functions eθ(ω)u , θ(ω) ∈ Θ(ω) are dense in L2 ([a; b]).  Proof. Denote by LS fk , k = 1, n the linear span of f1 , . . . , fn . Lets notice that for any a < b and f ∈ L2 ([a; b]) the following relations hold 
2 d f, LS eθu , θ ∈ Θ L2 ([a;b]) = inf cθ

= inf cθ

Z

eb

ea

f (ln u) −

X θ∈Θ

cθ u

θ

!2

Z

a

b

f (u) −

X θ∈Θ

θu

cθ e

!2

du =

  θ 2 du −a ˜ , ≤ e d f , LS v , θ ∈ Θ L2 ([ea ;eb ]) u

where the function f˜(u) = f (ln u) from L2 ([ea ; eb ]). Thus, due to Corollary 2.1, with probability one for any a < b and f ∈ L2 ([a; b]) 
d f, LS eθu , θ ∈ Θ L2 ([a;b]) = 0. Corollary 2.3. There exists Ω0 of probability one such that for any ω ∈ Ω0 and a < b a linear combinations of the functions {pε (u − θ(ω)), θ(ω) ∈ Θ(ω)} are dense in L2 ([a; b]).

3

Proof. To prove this statement lets consider a fixed point θ˜ ∈ Θ, and a linear bounded operator B in L2 ([a; b]) such that (Bf ) (u) = f (u)h(u), where h(u) = √

1 − u2 − θ˜2 e 2ε e 2ε . 2πε

Then for any a < b and f ∈ L2 ([a; b]) d (f, LS {pε (u − θ), θ ∈ Θ})2L2 ([a;b]) =  n  θ˜2 −θ2 θu  o2   √ u2 θ˜2 = d B f (u) 2πεe 2ε e 2ε , LS B e− 2ε e ε , θ ∈ Θ

L2 ([a;b])

 n θu o2 = d B f˜(u), LS Be ε , θ ∈ Θ

L2 ([a;b])

=

,

√ u2 θ˜2 where f˜(u) = f (u) 2πεe 2ε e 2ε . Since B is a bounded linear operator in L2 ([a; b]) then  n θu o2 o2  n θu ≤ kBk2 d f˜(u), LS e ε , θ ∈ Θ , d B f˜(u), LS Be ε , θ ∈ Θ L2 ([a;b])

L2 ([a;b])

which, due to Corollary 2.2, equals to 0. To end the proof of the theorem it is enough, by Corollary 2.3, to note that for any f ∈ L2 (R) d (f (u), LS {pε (u − θ), θ ∈ Θ})2L2 (R) =
2 = lim d f (u)1I[−m;m] (u), LS {pε (u − θ), θ ∈ Θ} L2 ([−m;m] . m→∞

Consequently, with probability one the linear span of the functions {pε (· − θ); θ ∈ Θ} is dense in L2 (R). The theorem is proved.

3

Properties of the integral random operator

Now let us turn to the integral operator with the kernel (1). Let pε be the same as before. Lemma 3.1. For any f ∈ L2 (R) and a stationary point process Θ with E|Θ∩[0; 1]| < +∞ 2 X Z f (u)pε (u − θ)du < +∞ a.s. θ∈Θ

R

4

Proof. Using Campbell’s formula [1] one can check that for every f ∈ L2 (R) E

X Z θ∈Θ

R

f (u)pε (u − θ)du =

Z Z R

=C

R

=C

Z Z R

R

≤E

XZ Z

R

R

θ∈Θ

|f (u)||f (v)|E

Z Z R

2

|f (u)||f (v)|

X θ∈Θ

Z

R

|f (u)||f (v)|pε(u − θ)pε (v − θ)dudv =

R

pε (u − θ)pε (v − θ)dudv =

pε (u − t)pε (v − t)dtdudv =

|f (u)||f (v)|p2ε(u − v)dudv = C

Z

2

−ελ2

h (λ)e

R

dλ ≤ C

Z

R

|f (u)|2du,

where C = E|Θ ∩ [0; 1]|, and h is the Fourier transform of f ∈ L2 (R). Remark 3.1. It follows from the proof of Lemma 3.1 that the following integral operator XZ f (u)pε (u − θ)du · pε (v − θ) Af (v) = θ∈Θ

R

is well-defined and is a strong random operator in Skorokhod sense [5].

Next lemma shows that A is not a bounded random operator in most interesting cases. Lemma 3.2. Let Θ be an ergodic stationary point process such that essup|Θ∩[0; 1]| = +∞. Then A is not a bounded random operator. Proof. It can be checked that under the condition on the process Θ with probability one there exists an increasing sequence of natural numbers {nk ; k ≥ 1} such that sup |Θ ∩ [ nk ; nk + 1) | = +∞. k≥1

Consider the following sequence of functions from L2 (R) fk = 1I[ nk ;nk +1) ,

k ≥ 1.

Then 2

kAfk k ≥

X

θ∈Θ∩[ nk ;nk +1)

Z

1

pε (v)dv 0

2

pε (1)2 .

Hence, supk kAfk k = +∞, and lemma is proved. For a fixed interval [a; b] lets denote by Qa,b the projection in L2 (R) onto L2 ([a; b]). 5

Remark 3.2. For any a, b ∈ R the random operators AQa,b , Qa,b A are bounded. Proof. One can check, by H¨older inequality, that for any f, g ∈ L2 (R) Z b XZ g(v)pε (v − θ) f (u)pε (u − θ)du ≤ (AQa,b f, g) = θ∈Θ

a

R

1

1

≤ 2− 4 (b − a) 2 kgkL2 (R) kf kL2 (R)

X θ∈Θ

max pε (u − θ).

u∈[a;b]

By Campbell’s formula [1], Z X E max pε (u − θ) = max pε (u − r)dr < +∞. θ∈Θ

Thus,

P

θ∈Θ

u∈[a;b]

R u∈[a;b]

maxu∈[a;b] pε (u − θ) < +∞ a.s., which proves the statement.

Lemma 3.3. For any a, b ∈ R with probability one the random operator AQa,b = Qa,b AQa,b is a nuclear. Proof. To prove the statement lets estimate the nuclear norm of Qa,b AQa,b . For any θ ∈ Θ denote by eθ the function eθ = Qa,b pε (· − θ). Evidently, the operator eθ ⊗ eθ is a nuclear, and its nuclear norm equals to keθ k2 . Lets notice that X XZ b 2 E keθ k = E pε (u − θ)2 du = θ∈Θ

=C

θ∈Θ

Z bZ a

R

a

pε (u − v)2 dvdu < +∞,

where, as before, C = E|Θ ∩ [0; 1]|. Its enough to note that X Qa,b AQa,b = eθ ⊗ eθ .

(7)

θ∈Θ

Lemma is proved. Due to the previous statement, the image K of the unit ball in L2 ([a; b]) under the operator AQa,b is a compact set with probability one. We obtain the following statement about asymptotic behavior of Kolmogorov width for the compact set K. Theorem 3.1. Let Θ be an ergodic stationary point process. Then with probability one there exists C > 0 such that   (Cn−b)2 (Cn+a)2 − − ε ε dn (K) = O e ∨e , n → ∞. 6

Proof. The representation (7) allows to estimate Kolmogorov widths of K. Lets denote by Nx , x > 0, the number of elements in the set Θ ∩ [−x; x], and by dn the n-th Kolmogorov width of K. It follows from (7) that X keθ k2 . (8) d Nx ≤ θ∈Θ\[−x;x]

Due to ergodic theorem, Nx ∼ 2Cx when x → +∞. To estimate the right part of (8) suppose that x > max{−a, b}, and consider the sum X X (b − a)pε (θ − b)2 . keθ k2 ≤ θ∈Θ,θ>x

θ∈Θ,θ>x

Denote by ξn = |Θ ∩ [ n; n + 1) |. Then {ξn ; n ≥ 1} is a stationary ergodic sequence. For a natural x +∞ X X pε (θ − b)2 ≤ pε (x − b)2 ξk . θ∈Θ,θ>x

k=x

Pk

For any k ≥ 1 let Sk = j=1 ξj . Since Sk ∼ Ck, k → ∞ a.s., then, by Abel transform, one can check that +∞ X k=x

2

pε (k − b) ξk = −pε (x − b) Sx−1 + ∼C

Lets notice that +∞ X k=x

2

+∞ X k=x

+∞ X k=x

Sk (pε (k − b)2 − pε (k + 1 − b)2 ) ∼

(pε (k − b)2 − pε (k + 1 − b)2 )k, x → +∞.

+∞

(pε (k − b)2 − pε (k + 1 − b)2 )k =

(2k+1−2b) (k−b)2 1 X 1 − (x−b)2 (1 − e− ε )e− ε k ∼ e ε , 2πε k=x 4π

and the statement is proved. For any interval [a; b] AQa,b is a bounded (nuclear) random operator. Despite this, when [a; b] increases to R, AQa,b must converge to unbounded random operator A. Consequently, one can expect that the operator norm kAQa,b k will increase to infinity when [a; b] increases to R. Using the arguments from the proof of Lemma 3.2 one can prove the following statement. Theorem 3.2. Let Θ be a Poisson point process with intensity 1. Then ln ln n kAQ−n,n k → +∞, n → ∞ a.s. ln n 7

Proof. It follows from the proof of Lemma 3.2 that kAQ−n,n k ≥ C max ξk , 1,n

where the random variables {ξn ; n ≥ 1} were introduced before. Now {ξn ; n ≥ 1} are independent random variables with poissonian distribution with intensity 1. Consequently, e−1 P {ξ1 ≥ m} ∼ , m → +∞. m! For any R > 0 P {maxk=1,n ξk ≤ mn R} = (1 −P {ξ1 > mn R})n . Thus, for mn = lnlnlnnn maxk=1,n ξk → +∞, n → ∞ a.s., mn and the theorem is proved.

Acknowledgment. The first author acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) within the project ”Stochastic Calculus and Geometry of Stochastic Flows with Singular Interaction” for initiation of international collaboration between the Institute of Mathematics of the FriedrichSchiller University Jena (Germany) and the Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev.

References [1] G. Last, M. Penrose. Lectures on the Poisson Processes. Cambridge University Press, 2016 (to appear). [2] A. A. Dorogovtsev. Krylov-Veretennikov expansion for coalescing stochastic flows. // Commun. Stoch. Anal. — 2012. — 6, No 3. — P. 421-435. [3] Ia. A. Korenovska. Properties of strong random operators generated by an Arratia flow (in Russian). // Ukr. Mat. Zh. - 2017. - 69, 2. - pp. 157-172 [4] R. Arratia. Coalescing Brownian motions on the line. PhD Thesis, University of Wisconsin, Madison, 1979. [5] A.V.Skorokhod. Random linear operators. D.Reidel Publishing Company, Dordrecht, Holland, 1983, 198 p. [6] A. A. Dorogovtsev. Semigroups of finite-dimensional random projections. Lithuanian Mathematical Journal, v.51 (2011), No.3, p.p.330-341. [7] I. A. Korenovska. Random maps and Kolmogorov widths. // Theory of Stochastic Processes. — 2015. — 20(36), No 1. — P. 78-83. 8