Kakeya set conjecture

## Lecture 17. Some properties of the Itō integral

In these notes we lay out the basic theory of the Fourier transform, which is of course the most fundamental tool in harmonic analysis and also of major importance in related fields (functional analysis, complex analysis, PDE, number theory, additive combinatorics, representation theory, signal processing, etc.). The Fourier transform, in conjunction with the Fourier inversion formula, allows one to take essentially arbitrary (complex-valued) functions on a group $latex {G}&fg=000000$ (or more generally, a space $latex {X}&fg=000000$ that $latex {G}&fg=000000$ acts on, e.g. a homogeneous space $latex {G/H}&fg=000000$), and decompose them as a (discrete or continuous) superposition of much more symmetric functions on the domain, such as characters $latex {\chi: G \rightarrow S^1}&fg=000000$; the precise superposition is given by *Fourier coefficients* $latex {\hat f(\xi)}&fg=000000$, which take values in some dual object such as the Pontryagin dual $latex {\hat G}&fg=000000$ of $latex {G}&fg=000000$. Characters behave in a very simple manner…

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## Fractional Brownian motion

# Definition:

Given a* complete probability spac*e .

A *fractional Brownian motion* (fBm) of *Hurts parameter* is the *continuous centered Gaussian process* with *covariance* function

. (1)

The parameter characterizes all the important properties of the process.

Note that and then . As is Gaussian so he admits a *version* with continuous trajectories or *continuous modification*, according to the *Kolmogorov theorem*.

For , we set where is standard normal random variable.

The parameter controls the regularity of the trajectories, which are *Hölder continuous* of order , for any . more precisely,

*For all* *and* ,* there exists a nonnegative random variable* *such that* for all , and

This is simply the *modulus of continuity* for the trajectories of a fBm .

If , the covariance and the process is a *standard brownian motion.* in this case, the increments of this process in disjoint intervals are independents.

### Self-similarity

An -valued random process is –*self-similar* * if for any * * there exists * such that

and have the same distribution.

that means for, for every choise in we have,

for every in .

Since the covariance function of the fBm is homogenous of order , we deduce that the process is self-similar (Put ).

Note that

(2)

and it follow that the process has stationary increment, However it is not stationary itself.

Let and , it follow from (2) that

.

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