Brownian motion with partial information
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- by Terry R. McConnell PDF
- Trans. Amer. Math. Soc. 271 (1982), 719-731 Request permission
Abstract:
We study the following problem concerning stopped $N$-dimensional Brownian motion: Compute the maximal function of the process, ignoring those times when it is in some fixed region $R$. Suppose this modified maximal function belongs to ${L^q}$. For what regions $R$ can we conclude that the unrestricted maximal function belongs to ${L^q}$? A sufficient condition on $R$ is that there exist $p > q$ and a function $u$, harmonic in $R$, such that \[ |x{|^p} \leqslant u(x) \leqslant C|x{|^p} + C,\qquad x \in R,\] for some constant $C$. We give applications to analytic and harmonic functions, and to weak inequalities for exit times.References
- Albert Baernstein II and B. A. Taylor, Spherical rearrangements, subharmonic functions, and $^*$-functions in $n$-space, Duke Math. J. 43 (1976), no.Β 2, 245β268. MR 402083
- D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19β42. MR 365692, DOI 10.1214/aop/1176997023
- D. L. Burkholder, One-sided maximal functions and $H^{p}$, J. Functional Analysis 18 (1975), 429β454. MR 0365693, DOI 10.1016/0022-1236(75)90013-0
- D. L. Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26 (1977), no.Β 2, 182β205. MR 474525, DOI 10.1016/0001-8708(77)90029-9
- D. L. Burkholder, Weak inequalities for exit times and analytic functions, Probability theory (Papers, VIIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1976) Banach Center Publ., vol. 5, PWN, Warsaw, 1979, pp.Β 27β34. MR 561465
- D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249β304. MR 440695, DOI 10.1007/BF02394573
- D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class $H^{p}$, Trans. Amer. Math. Soc. 157 (1971), 137β153. MR 274767, DOI 10.1090/S0002-9947-1971-0274767-6
- Burgess Davis, Brownian motion and analytic functions, Ann. Probab. 7 (1979), no.Β 6, 913β932. MR 548889
- J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431β458. MR 109961 T. R. McConnell, Inequalities for random walk and partially observed Brownian motion, Ph. D. Dissertation, University of Illinois.
- H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
- Paul-AndrΓ© Meyer, Processus de Markov, Lecture Notes in Mathematics, No. 26, Springer-Verlag, Berlin-New York, 1967 (French). MR 0219136
- Karl Endel Petersen, Brownian motion, Hardy spaces and bounded mean oscillation, London Mathematical Society Lecture Note Series, No. 28, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 0651556
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 271 (1982), 719-731
- MSC: Primary 60J65; Secondary 60G46
- DOI: https://doi.org/10.1090/S0002-9947-1982-0654858-5
- MathSciNet review: 654858