## Thesis etd-07262011-002405 |

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Thesis type

Tesi di laurea magistrale

Author

MAURELLI, MARIO

URN

etd-07262011-002405

Thesis title

Stochastic differential equations with rough coefficients

Department

SCIENZE MATEMATICHE, FISICHE E NATURALI

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Flandoli, Franco

Keywords

- Fokker-Planck equation
- generalized stochastic flow
- isotropic Brownian flow
- regularization by noise
- rough coefficients
- stochastic continuity equation
- stochastic differential equation
- Wiener chaos

Graduation session start date

30/09/2011

Availability

Full

Summary

This thesis deals with the study of the stochastic continuity equation (SCE)

on R^d

under low regularity assumptions on the coefficients. This is the equations for the mass associated to an SDE on R^d.

In the first chapter Wiener pathwise uniqueness (i.e. uniqueness for solutions adapted to Brownian filtration) is proved for the SCE,. We use Wiener chaos to reduce Wiener uniqueness for the SCE to uniqueness for the Fokker-Planck

equation (FPE). The method consists of projecting the

equation on the Wiener chaos spaces and using the shift effect of the projectors in order to discard the Ito integral.

The second chapter deals with the SCE for flows, following Le Jan-

Raimond’s approach. Here we need that the FPE admits a particular semigroup as solution, which is guaranteed by the theory of Dirichlet forms under

mild assumptions on the coefficients. Wiener chaos gives Wiener uniqueness

(as before) and also existence. Another method of existence is based on

filtering a weak solution X of the associated SDE with respect to a certain

cylindrical Brownian motion W.

In the third chapter, we consider the case of a rough drift . Here a

phenomenon of regularization by noise can be observed: the results in the first

chapter give immediately Wiener uniqueness for the SCE, while uniqueness

does not hold in the deterministic case without additional

hypotheses on the drift. We cite an example of this phenomenon.

We prove also that, in many cases, strong uniqueness (i.e. Uniqueness

with respect to every filtration, not only Brownian filtration) holds for the

SCE. This is not surprising since a strong uniqueness result (due to Krylov-

Röckner) holds for the SDE. First, extending Ambrosio’s approach, we associate to every measure-valued solution of the SCE a superposition solution

N. Then, starting

from N, we build a weak solution of the SDE. This correspondence and

Krylov-Röckner’s result imply strong uniqueness for the SCE.

The last chapter is about a particular class of generalized flows, the

isotropic Brownian flows (IBFs). An IBF is a family of Brownian motions, indexed by their starting points in Rd , which are invariant in law

for translation and rotation; it can be found as (possibly generalized) solution S of an SCE with isotropic infinitesimal covariance function

K. Here we consider K’s driven by two parameters

α and η, related respectively to the correlation of the two-point motion and

to the compressibility of the flow. Studying the distance between the motions

of two points (which is a 1-dimensional diffusion), we find that coalescence

and/or splitting occur, depending of the values of α, η and d. This analysis makes rigorous the results for a simple model of turbulence

and shows that this situation cannot be described classically, thus motivating

the theory of generalized flows.

Finally two appendices recall preliminaries and technical results.

on R^d

under low regularity assumptions on the coefficients. This is the equations for the mass associated to an SDE on R^d.

In the first chapter Wiener pathwise uniqueness (i.e. uniqueness for solutions adapted to Brownian filtration) is proved for the SCE,. We use Wiener chaos to reduce Wiener uniqueness for the SCE to uniqueness for the Fokker-Planck

equation (FPE). The method consists of projecting the

equation on the Wiener chaos spaces and using the shift effect of the projectors in order to discard the Ito integral.

The second chapter deals with the SCE for flows, following Le Jan-

Raimond’s approach. Here we need that the FPE admits a particular semigroup as solution, which is guaranteed by the theory of Dirichlet forms under

mild assumptions on the coefficients. Wiener chaos gives Wiener uniqueness

(as before) and also existence. Another method of existence is based on

filtering a weak solution X of the associated SDE with respect to a certain

cylindrical Brownian motion W.

In the third chapter, we consider the case of a rough drift . Here a

phenomenon of regularization by noise can be observed: the results in the first

chapter give immediately Wiener uniqueness for the SCE, while uniqueness

does not hold in the deterministic case without additional

hypotheses on the drift. We cite an example of this phenomenon.

We prove also that, in many cases, strong uniqueness (i.e. Uniqueness

with respect to every filtration, not only Brownian filtration) holds for the

SCE. This is not surprising since a strong uniqueness result (due to Krylov-

Röckner) holds for the SDE. First, extending Ambrosio’s approach, we associate to every measure-valued solution of the SCE a superposition solution

N. Then, starting

from N, we build a weak solution of the SDE. This correspondence and

Krylov-Röckner’s result imply strong uniqueness for the SCE.

The last chapter is about a particular class of generalized flows, the

isotropic Brownian flows (IBFs). An IBF is a family of Brownian motions, indexed by their starting points in Rd , which are invariant in law

for translation and rotation; it can be found as (possibly generalized) solution S of an SCE with isotropic infinitesimal covariance function

K. Here we consider K’s driven by two parameters

α and η, related respectively to the correlation of the two-point motion and

to the compressibility of the flow. Studying the distance between the motions

of two points (which is a 1-dimensional diffusion), we find that coalescence

and/or splitting occur, depending of the values of α, η and d. This analysis makes rigorous the results for a simple model of turbulence

and shows that this situation cannot be described classically, thus motivating

the theory of generalized flows.

Finally two appendices recall preliminaries and technical results.

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