## Thesis etd-09052018-182135 |

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Thesis type

Tesi di laurea magistrale

Author

MASSA, DENISE

URN

etd-09052018-182135

Thesis title

Reguralisation by noise and notions of irregularity of a path

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Romito, Marco

**controrelatore**Trevisan, Dario

Keywords

- irregularity
- regularisation by noise
- roughness
- SPcondition

Graduation session start date

21/09/2018

Availability

Withheld

Release date

21/09/2088

Summary

The purpose of this thesis is to investigate some properties that a path w may have in order to say that it regularises an ordinary differential equation (ODE)

d x_t=b(t,x_t) d t +d w_t,

in a sense that the same equation without w can show non-uniqueness.

In general, the regularisation by noise deals with the study of equations which may be not well-posed, in particular they can lack of uniqueness, but they become well-posed adding an arbitrary continuous path w, which usually is some kind of noise. A goal would be being able to quantify which and how the properties of w influence this solution,

in particular in terms of point-wise and uniform regularity.

After a detailed presentation of the problem and an overview of the most important results already known on this topic, we focus on the study of the averaging operator T^w f which

is intended to average a function f by means of the path w, which can be sampled from the law of a stochastic process. For our purposes, we concentrate on the particular case of a function w sampled according to the law of the fractional Brownian motion of Hurst index H in (0,1).

This choice has the advantage of being a simple process for which many other results about existence and uniqueness of associated SDE are available. Besides, the freedom in the choice of the Hurst parameter $H$ gives us the possibility of explore different degrees of

irregularity of the perturbation on the regularisation phenomenon.

Through the notion of (\rho,\gamma)-irregularity, we start our survey observing that the regularising properties of the ODE are strictly related to the ones of the operator T^w, which is defined in both contexts of Fourier-Lebesgue spaces and Besov spaces.

In the first case, we prove the boundedness of T^w, but in the second case that remains an open problem and we study the image of T^wf for fixed f. It turns out that the properties of T^w are strongly dependent on some quantities that involve the path w and are

sufficient to have good L^1-estimations and prove uniqueness of the solution of the ODE in the Young integral formulation.

In the second part of this thesis, inspired by this notion of (\rho,\gamma)-irregularity, we present several notions of irregularity of a path, some of which acquired from analysis and the theory of rough paths, and some others arisen in our investigation of path-by-path regularity. Thus, we introduce the SP-condition which turns out to be a more practical and

general concept because in a sense it can summarise some of the other notions, for particular values of the parameters involved.

The dissertations proceeds studying the basic properties of the following notions: (\rho,\gamma)-irregularity, scaling property, \theta-roughness, anti Hölder irregularity and SP-condition. The examination of their connected implications seems to put

the SP-condition in a privileged position among them. This could suggest that we may concentrate on a deep understanding of this condition if we want to explicate which properties a generic path w must have in order to have a good regularising effect on

an ODE.

d x_t=b(t,x_t) d t +d w_t,

in a sense that the same equation without w can show non-uniqueness.

In general, the regularisation by noise deals with the study of equations which may be not well-posed, in particular they can lack of uniqueness, but they become well-posed adding an arbitrary continuous path w, which usually is some kind of noise. A goal would be being able to quantify which and how the properties of w influence this solution,

in particular in terms of point-wise and uniform regularity.

After a detailed presentation of the problem and an overview of the most important results already known on this topic, we focus on the study of the averaging operator T^w f which

is intended to average a function f by means of the path w, which can be sampled from the law of a stochastic process. For our purposes, we concentrate on the particular case of a function w sampled according to the law of the fractional Brownian motion of Hurst index H in (0,1).

This choice has the advantage of being a simple process for which many other results about existence and uniqueness of associated SDE are available. Besides, the freedom in the choice of the Hurst parameter $H$ gives us the possibility of explore different degrees of

irregularity of the perturbation on the regularisation phenomenon.

Through the notion of (\rho,\gamma)-irregularity, we start our survey observing that the regularising properties of the ODE are strictly related to the ones of the operator T^w, which is defined in both contexts of Fourier-Lebesgue spaces and Besov spaces.

In the first case, we prove the boundedness of T^w, but in the second case that remains an open problem and we study the image of T^wf for fixed f. It turns out that the properties of T^w are strongly dependent on some quantities that involve the path w and are

sufficient to have good L^1-estimations and prove uniqueness of the solution of the ODE in the Young integral formulation.

In the second part of this thesis, inspired by this notion of (\rho,\gamma)-irregularity, we present several notions of irregularity of a path, some of which acquired from analysis and the theory of rough paths, and some others arisen in our investigation of path-by-path regularity. Thus, we introduce the SP-condition which turns out to be a more practical and

general concept because in a sense it can summarise some of the other notions, for particular values of the parameters involved.

The dissertations proceeds studying the basic properties of the following notions: (\rho,\gamma)-irregularity, scaling property, \theta-roughness, anti Hölder irregularity and SP-condition. The examination of their connected implications seems to put

the SP-condition in a privileged position among them. This could suggest that we may concentrate on a deep understanding of this condition if we want to explicate which properties a generic path w must have in order to have a good regularising effect on

an ODE.

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