Fourier spectra of measures associated with algorithmically random Brownian motion

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.Comment: 24 page

Some applications of the Menshov–Rademacher theorem

Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum _{n=1}^\infty a_n X_n$ is almost surely convergent. This paper extends the classical Menshov–Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate

Local time of Martin-Lof Brownian motion

In this paper we study the local times of Brownian motion from the point of view of algorithmic randomness. We introduce the notion of effective local time and show that any path which is Martin-L\"of random with respect to the Wiener measure has continuous effective local times at every computable point. Finally we obtain a new simple representation of classical Brownian local times, computationally expressed

Local times of Brownian motion

After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry and Fourier analysis and the properties of local times of Brownian motion, we study the Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure of one-dimensional Brownian motion X at the level a, that is the measure defined by the Brownian local time La at level a, and μ is its restriction to the random interval [0, L−1 a (1)], then the Fourier transform of μ is such that, with positive probability, for all 0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the best possible. Consequently, each Brownian level set, reduced to a compact interval, is with positive probability, a Salem set of dimension 1/2. We also show that the zero set of X reduced to the interval [0, L−1 0 (1)] is, almost surely, a Salem set. Finally, we show that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0. Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion, local times, level sets, Fourier transform, inverse local times.Decision SciencesPh. D. (Operations Research

Some applications of the Menshov–Rademacher theorem

Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper extends the classical Menshov--Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate

Latent relationships between Markov processes, semigroups and partial differential equations

This research investigates existing relationships between the three apparently unrelated subjects: Markov process, Semigroups and Partial difierential equations. Markov processes define semigroups through their transition functions. Conversely particular semigroups determine transition functions and can be regarded as Markov processes. We have exploited these relationships to study some Markov chains. The infnitesimal generator of a Feller semigroup on the closure of a bounded domain of Rn; (n ^ 2), is an integro-diferential operator in the interior of the domain and verifes a boundary condition. The existence of a Feller semigroup defined by a diferential operator and a boundary condition is due to the existence of solution of a bounded value problem. From this result other existence suficient conditions on the existence of Feller semigroups have been obtained and we have applied some of them to construct Feller semigroups on the unity disk of R2.Decision SciencesM. Sc. (Operations Research

Generalisation of Fractional-Cox-Ingersoll-Ross Process

Generalisation of Fractional-Cox-Ingersoll-Ross Proces