Pathwise methods in regularisation by noise
Pathwise methods in regularisation by noise
Dokument öffnen (2.4MB)Art der Hochschulschrift
DissertationPrüfungsdatum
23.06.2022Datum der Veröffentlichung
18.07.2022Erstgutachter
Gubinelli, MassimilianoZweitgutachter
Flandoli, FrancoMetadaten
Zur Langanzeige
Zitierbare Links 
Inhalt
This thesis concerns the study of regularisation by noise phenomena for ODEs and PDEs. In particular, it focuses on the use of so called pathwise techniques: our aim is to identify analytical properties, satisfied by typical realizations of the noise in consideration, which are sufficient to guarantee a regularising effect. The advantage of this strategy is that, whenever successful, it provides strong probabilistic results (e.g. path-by-path uniqueness) and works for a large class of stochastic processes. Moreover, the theory developed in this way goes beyond the probabilistic framework by studying the regularising effect of generic functions (in the sense of prevalence).
We start by studying abstract nonlinear Young differential equations, needed in later applications; conditions for existence and uniqueness of solutions, as well as regularity of the flow, are provided. Solvability of evolutionary nonlinear Young PDEs, especially of transport and parabolic type, is studied as well.
We then apply the above abstract theory to show the regularising effect of continuous additive perturbations on ODEs. It is shown that there is a non trivial trade-off between the regularity of the parturbation and its effect on the ODE, formalising rigorously the well-known principle "the rougher the noise, the better the regularisation". The results apply for paths sampled as a fractional Brownian motion, as well as for generic continuous functions. We also present a variant of such results, by allowing the presence of an additional multiplicative noise term, again of fractional type.
Third, we analyse the notion of ρ-irregularity of a path, in relation to its regularising effect on ODEs and PDEs; several analytical and geometric properties of ρ-irregular paths are derived, like their Holder roughness and the dimension of their image sets. Useful criteria for stochastic processes to be ρ-irregular are also presented.
Finally, we study the mixing and diffusion enhancing properties of generic shear flows; while this topic does not technically belong to the regularisation by noise context, the techniques developed in previous chapters applies here as well. We obtain a variant of the aforementioned principle that can be summarised as "the rougher the shear flow, the faster the mixing".
We start by studying abstract nonlinear Young differential equations, needed in later applications; conditions for existence and uniqueness of solutions, as well as regularity of the flow, are provided. Solvability of evolutionary nonlinear Young PDEs, especially of transport and parabolic type, is studied as well.
We then apply the above abstract theory to show the regularising effect of continuous additive perturbations on ODEs. It is shown that there is a non trivial trade-off between the regularity of the parturbation and its effect on the ODE, formalising rigorously the well-known principle "the rougher the noise, the better the regularisation". The results apply for paths sampled as a fractional Brownian motion, as well as for generic continuous functions. We also present a variant of such results, by allowing the presence of an additional multiplicative noise term, again of fractional type.
Third, we analyse the notion of ρ-irregularity of a path, in relation to its regularising effect on ODEs and PDEs; several analytical and geometric properties of ρ-irregular paths are derived, like their Holder roughness and the dimension of their image sets. Useful criteria for stochastic processes to be ρ-irregular are also presented.
Finally, we study the mixing and diffusion enhancing properties of generic shear flows; while this topic does not technically belong to the regularisation by noise context, the techniques developed in previous chapters applies here as well. We obtain a variant of the aforementioned principle that can be summarised as "the rougher the shear flow, the faster the mixing".
Schlagwörter
Regularisation by noise, Young integration, Fractional Brownian motion, Prevalence, rho-Irregularity, Mixing
Klassifikation (DDC)
510 MathematikGaleati, Lucio: Pathwise methods in regularisation by noise. - Bonn, 2022. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-66831
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5-66831
@phdthesis{handle:20.500.11811/10089,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-66831,
author = {{Lucio Galeati}},
title = {Pathwise methods in regularisation by noise},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2022,
month = jul,
note = {This thesis concerns the study of regularisation by noise phenomena for ODEs and PDEs. In particular, it focuses on the use of so called pathwise techniques: our aim is to identify analytical properties, satisfied by typical realizations of the noise in consideration, which are sufficient to guarantee a regularising effect. The advantage of this strategy is that, whenever successful, it provides strong probabilistic results (e.g. path-by-path uniqueness) and works for a large class of stochastic processes. Moreover, the theory developed in this way goes beyond the probabilistic framework by studying the regularising effect of generic functions (in the sense of prevalence).
We start by studying abstract nonlinear Young differential equations, needed in later applications; conditions for existence and uniqueness of solutions, as well as regularity of the flow, are provided. Solvability of evolutionary nonlinear Young PDEs, especially of transport and parabolic type, is studied as well.
We then apply the above abstract theory to show the regularising effect of continuous additive perturbations on ODEs. It is shown that there is a non trivial trade-off between the regularity of the parturbation and its effect on the ODE, formalising rigorously the well-known principle "the rougher the noise, the better the regularisation". The results apply for paths sampled as a fractional Brownian motion, as well as for generic continuous functions. We also present a variant of such results, by allowing the presence of an additional multiplicative noise term, again of fractional type.
Third, we analyse the notion of ρ-irregularity of a path, in relation to its regularising effect on ODEs and PDEs; several analytical and geometric properties of ρ-irregular paths are derived, like their Holder roughness and the dimension of their image sets. Useful criteria for stochastic processes to be ρ-irregular are also presented.
Finally, we study the mixing and diffusion enhancing properties of generic shear flows; while this topic does not technically belong to the regularisation by noise context, the techniques developed in previous chapters applies here as well. We obtain a variant of the aforementioned principle that can be summarised as "the rougher the shear flow, the faster the mixing".},
url = {https://hdl.handle.net/20.500.11811/10089}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-66831,
author = {{Lucio Galeati}},
title = {Pathwise methods in regularisation by noise},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2022,
month = jul,
note = {This thesis concerns the study of regularisation by noise phenomena for ODEs and PDEs. In particular, it focuses on the use of so called pathwise techniques: our aim is to identify analytical properties, satisfied by typical realizations of the noise in consideration, which are sufficient to guarantee a regularising effect. The advantage of this strategy is that, whenever successful, it provides strong probabilistic results (e.g. path-by-path uniqueness) and works for a large class of stochastic processes. Moreover, the theory developed in this way goes beyond the probabilistic framework by studying the regularising effect of generic functions (in the sense of prevalence).
We start by studying abstract nonlinear Young differential equations, needed in later applications; conditions for existence and uniqueness of solutions, as well as regularity of the flow, are provided. Solvability of evolutionary nonlinear Young PDEs, especially of transport and parabolic type, is studied as well.
We then apply the above abstract theory to show the regularising effect of continuous additive perturbations on ODEs. It is shown that there is a non trivial trade-off between the regularity of the parturbation and its effect on the ODE, formalising rigorously the well-known principle "the rougher the noise, the better the regularisation". The results apply for paths sampled as a fractional Brownian motion, as well as for generic continuous functions. We also present a variant of such results, by allowing the presence of an additional multiplicative noise term, again of fractional type.
Third, we analyse the notion of ρ-irregularity of a path, in relation to its regularising effect on ODEs and PDEs; several analytical and geometric properties of ρ-irregular paths are derived, like their Holder roughness and the dimension of their image sets. Useful criteria for stochastic processes to be ρ-irregular are also presented.
Finally, we study the mixing and diffusion enhancing properties of generic shear flows; while this topic does not technically belong to the regularisation by noise context, the techniques developed in previous chapters applies here as well. We obtain a variant of the aforementioned principle that can be summarised as "the rougher the shear flow, the faster the mixing".},
url = {https://hdl.handle.net/20.500.11811/10089}
}
Die folgenden Nutzungsbestimmungen sind mit dieser Ressource verbunden:
