1. CJM Online first
|Sharp Localized Inequalities for Fourier Multipliers|
In the paper we study sharp localized $L^q\colon L^p$ estimates for Fourier multipliers resulting from modulation of the jumps of LÃ©vy processes. The proofs of these estimates rest on probabilistic methods and exploit related sharp bounds for differentially subordinated martingales, which are of independent interest. The lower bounds for the constants involve the analysis of laminates, a family of certain special probability measures on $2\times 2$ matrices. As an application, we obtain new sharp bounds for the real and imaginary parts of the Beurling-Ahlfors operator .
Keywords:Fourier multiplier, martingale, laminate
Categories:42B15, 60G44, 42B20
2. CJM 2005 (vol 57 pp. 204)
|On the Duality between Coalescing Brownian Motions |
A duality formula is found for coalescing Brownian motions on the real line. It is shown that the joint distribution of a coalescing Brownian motion can be determined by another coalescing Brownian motion running backward. This duality is used to study a measure-valued process arising as the high density limit of the empirical measures of coalescing Brownian motions.
Keywords:coalescing Brownian motions, duality, martingale problem,, measure-valued processes
3. CJM 1999 (vol 51 pp. 372)
|Uniqueness for a Competing Species Model |
We show that a martingale problem associated with a competing species model has a unique solution. The proof of uniqueness of the solution for the martingale problem is based on duality technique. It requires the construction of dual probability measures.
Keywords:stochastic partial differential equation, Martingale problem, duality