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Mcconnell, Terry Robert 2014-12-16T06:18:01Z 2014-12-16T06:18:01Z 10000-01-01 1981 1981 This thesis is divided into two parts. The first part studies the control of the maximal function of N-dimensional Brownian motion, B(,t), by the maximal function of partially observed Brownian motion. Let R denote a fixed open subset of (//R)('N), G an arbitrary open subset, and T the first exit time of the Brownian motion from G. Define the maximal function, B(,T)('*), by (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) and the partially observed maximal function, B(,T)('*)(,(FDIAG)R), by Let x(,0) be a fixed point not belonging to the closure of R and p a positive number. Then there is a constant C(,1) so that the inequality (1) E('x(,0))(B(,T)('*))('p) (LESSTHEQ) C(,1)E('x(,0))(B(,T)('*)(,(FDIAG)R))('p) holds as G varies provided that there exists a function u, harmonic in R, and constants C(,2) > 0 and q > p such that (2) (VBAR)x(VBAR)('q) (LESSTHEQ) u(x) (LESSTHEQ) C(,2)(VBAR)x(VBAR)('q) + C(,2), x (ELEM) R. Conversely, if (1) holds then so does (2) with q replaced by p. This result has applications in complex analysis and probability. The second part considers the integrability of exit times of random walks in N-dimensions (N (GREATERTHEQ) 2). Let S(,n) = X(,1) + X(,2) + ... + X(,n) be the n('th) partial sum of independent, identically distributed random vectors, X(,1),X(,2),..., having mean zero, finite second moments, and covariance matrix equal to the identity. Let W be an open subset of (//R)('N) which is invariant under positive dilations, and (tau) the first exit time of S(,n) from W. If the boundary of W satisfies certain regularity conditions then the range of exponents p for which (tau)(' 1/2) has a finite p('th )moment is essentially the same as the corresponding range for standard Brownian motion. Made available in DSpace on 2014-12-16T06:18:01Z (GMT). No. of bitstreams: 1 8203525.pdf: 1711883 bytes, checksum: 7c9ad495d7943e58432997772f6d1e74 (MD5) Previous issue date: 1981 Embargo set by: Seth Robbins for item 71363 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 64 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981. http://hdl.handle.net/2142/71197 (UMI)AAI8203525 Mathematics Inequalities for Random Walk and Partially Observed Brownian Motion text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.