Besov Spaces, Martingales, and Brownian motion process
by
Zoran R. Pop-Stojanovic and Murali Rao
University of Florida, Gainesville, Fl 32611

Since their introduction [Integral Representations of Functions and Imbedding Theorems by O.V. Besov, V.P. Ilin, and S.M. Nikolski, Wiley 1978], Besov Spaces are playing an ever increasing role in the treatment of Gaussian processes. Here are some of the results obtained in this area:

- Various techniques have been developed enabling one to connect potentials of the Brownian motion process with Besov spaces as in [Brownian Potentials and Besov Spaces, Z.R. Pop-Stojanovi\'c, M. Rao and H. Siki\'c, J. Math. Soc. Japan, 30, N0. 2, 1998]. The main result in this paper shows that the Brownian potentials of finite measures, given over bounded domains, belong to Besov spaces.

- In [Approximation en Norme Besov de la Solution d'une Equation Differentielle Stochastique, B. Roynette, Stochastics and Stochastics Reports, 49, 1994], the author shows that a strong solution of an Ito stochastic differential equation in R¹ can be obtained as the limit in Besov norm 1/2, p, \infty of the Euler and Milshtein approximation schemes. The significance of the Besov norm 1/2, p, \infty in this context lies in the fact that it is larger than L² norm customarily used.

- If one considers a martingale sequence (f_k) in l^q(L^p), that is, in a Besov space, what could be concluded about the convergence of this martingale?

- In the treatment of wavelet bases and thresholding, [Wavelets and the Theory of Nonparametric Function Estimation, I.M. Johnstone, Phil. R. Soc. Lond. A (1999)], the author uses Besov norm with smoothness index \alpha (=number of derivatives), and the homogeneity index p, plotted as 1/p as is customary in harmonic analysis, where each point (\alpha, 1/p) corresponds to a class of Besov spaces. The so-called Bump Algebra, [Meyer, 1990, Section 6.6], is an example of a Besov space.

These and other related results will be presented in our lectures.

Date received: May 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-71.