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1. CMB Online first
Short Probabilistic Proof of the Brascamp-Lieb and Barthe Theorems We give a short proof of the Brascamp-Lieb theorem, which asserts that
a certain general form of Young's convolution inequality is saturated
by Gaussian functions. The argument is inspired by Borell's stochastic
proof of the PrÃ©kopa-Leindler inequality and applies also to the
reversed Brascamp-Lieb inequality, due to Barthe.
Keywords:functional inequalities, Brownian motion Categories:39B62, 60J65 |
2. CMB 2010 (vol 53 pp. 503)
The Time Change Method and SDEs with Nonnegative Drift Using the time change method we show how to construct a solution to the stochastic equation $dX_t=b(X_{t-})dZ_t+a(X_t)dt$ with a nonnegative drift $a$ provided there exists a solution to the auxililary equation $dL_t=[a^{-1/\alpha}b](L_{t-})d\bar Z_t+dt$ where $Z, \bar Z$ are two symmetric stable processes of the same index $\alpha\in(0,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0<\alpha<1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.
Keywords:One-dimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence Categories:60H10, 60J60, 60J65, 60G44 |
3. CMB 2008 (vol 51 pp. 146)
Stepping-Stone Model with Circular Brownian Migration In this paper we consider the stepping-stone model on a circle with
circular Brownian migration. We first point out a connection between
Arratia flow on the circle and the marginal distribution of this
model. We then give a new representation for the stepping-stone
model using Arratia flow and circular coalescing Brownian motion.
Such a representation enables us to carry out some explicit
computations. In particular, we find the distribution for the first
time when there is only one type
left across the circle.
Keywords:stepping-stone model, circular coalescing Brownian motion, Arratia flow, duality, entrance law Categories:60G57, 60J65 |
4. 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 |
5. CMB 1997 (vol 40 pp. 67)
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 |