Section: New Results

A class of self-similar processes with stationary increments in higher order Wiener chaoses.

Participant : Benjamin Arras.

Self similar processes with stationary increments (SSSI processes) have been studied for a long time due to their importance both in theory and in practice. Such processes appear as limits in various renormalisation procedures [69] . In applications, they occur in various fields such as hydrology, biomedicine and image processing. The simplest SSSI processes are simply Brownian motion and, more generally, Lévy stable motions. Apart from these cases, the best known such process is probably fractional Brownian motion (fBm). A construction of SSSI processes that generalizes fBm to higher order Wiener chaoses was proposed in [73] . These processes read

t + X t = d h t H (x 1 ,...,x d )dB x 1 ...dB x d

where h t H verifies:

  1. h t H L ^ 2 ( d ),

  2. c>0,h ct H (cx 1 ,...,cx d )=c H-d 2 h t H (x 1 ,...,x d ),

  3. ρ0,h t+ρ H (x 1 ,...,x d )-h t H (x 1 ,...,x d )=h ρ H (x 1 -t,...,x d -t).

In [41] , we define a class of such processes by the following multiple Wiener-Itô integral representation:

X t α = d ||𝐭 * -𝐱|| 2 H-d 2 -||𝐱|| 2 H-d 2 dB x 1 ...dB x d (16)

where t[0,1], 𝐭 * =(t,...,t) and α=H-1+d 2. When d=1, this is just fBm. In order to study the local regularity of this class of processes as well as the asymptotic behaviour at infinity, we use wavelet's methods. More precisely, following ideas from [46] , we obtain the following wavelet-like expansion:

Almost surely,

t[0,1]X t α = j 𝐤 d ϵE 2 -jH I α+1 (ψ (ϵ) )(2 j 𝐭 * -𝐤)-I α+1 (ψ (ϵ) )(-𝐤)I d (ψ j,𝐤 (ϵ) ).

From this representation, we get several results about this class of processes. Namely:

  • There exists a strictly positive random variable A d of finite moments of any order and a constant, b d >1, such that:

    ωΩ * sup (s,t)[0,1]|X t α (ω)-X s α (ω)| |t-s| H (log(b d +|t-s| -1 )) d 2 A d (ω).
  • There exists a strictly positive random variable B d of finite moments of any order and a constant c d >3, such that:

    ωΩ * sup t + |X t α (ω)| (1+|t|) H (loglog(c d +|t|)) d 2 B d (ω).

Using an estimate from [54] , we compute the uniform almost sure pointwise Hölder exponent of X α defined by:

γ X α (t)=sup{γ>0:lim sup ρ0|X t+ρ α -X t α | |ρ| γ <+}.

We get the following result:

Almost surely,

t(0,1),γ X α (t)=H.

In the last part of [41] , we give general bounds on the Hausdorff dimension of the range and graphs of multidimensional anisotropic SSSI processes defined by multiple Wiener integrals. Let Y t H =γ(H,d)I d (h t H ) where γ(H,d) is a normalizing positive constant such that 𝔼[|Y 1 H | 2 ]=1. Let 1 2<H 1 ...H N <1. Let {𝕐 t H } be the multidimensional process defined by:

{𝕐 t H }={(Y t H 1 ,...,Y t H N ):t + }

where the coordinates are independent copies of the process Y t H . Following classical ideas from [78] and using again the estimate from [54] , we obtain:

Almost surely,

dim R E (𝕐 H )minN;dim E+ j=1 k (H k -H j ) d H k ,k=1,...,N,

dim Gr E (𝕐 H )mindim E+ j=1 k (H k -H j ) d H k ,k=1,...,N,dim E+ i=1 N (1-H i ) d.


dim R E (𝕐 H )minN;dim E+ j=1 k (H k -H j ) H k ,k=1,...,N,

dim Gr E (𝕐 H )mindim E+ j=1 k (H k -H j ) H k ,k=1,...,N;dim E+ i=1 N (1-H i ).

where E + .