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26. CMB 1997 (vol 40 pp. 67)

Knight, Frank B.
 On a Brownian motion problem of T. Salisbury Let $B$ be a Brownian motion on $R$, $B(0)=0$, and let $f(t,x)$ be continuous. T.~Salisbury conjectured that if the total variation of $f(t,B(t))$, $0\leq t\leq 1$, is finite $P$-a.s., then $f$ does not depend on $x$. Here we prove that this is true if the expected total variation is finite. Category:60J65

27. CMB 1997 (vol 40 pp. 19)

 Lattice trees and super-Brownian motion This article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion called {\it integrated super-Brownian excursion\/} ($\ISE$), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is $\ISE$, in all dimensions. A connection is drawn between $\ISE$ and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point. Categories:82B41, 60K35, 60J65