The Brownian plane

Values of Brownian intersection exponents I: Half-plane exponents

This paper proves conjectures originating in the physics literature regarding the intersection exponents of Brownian motion in a halfplane. For instance, suppose that B and B′ are two independent planar Brownian motions started from distinct points in a half-plane H. Then as t → ∞,

3 The Brownian loop soup

We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to “chronologically add Brownian loops” to simple curves in the plane.

Exact Solution of the Plane Flow with Unsteady Vortex , Brownian Motion , Diffusion and Osmosis ”

Based on the conception "pseudo-potential" of the incompressible plane flow, an exact solution is given to the Euler's equation with an arbitrarily given potential force. With the KAM theory and the second order Melnikov function, it is proved that this solution describes infinitely many unsteady vortices distributed periodically on the whole plane and the Brownian motion appeared along the bor...

On Brownian Motion in a Fluid with a Plane Boundary

Compact Brownian surfaces I. Brownian disks

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BDL, 0 < L < ∞) of random metric spaces homeomorphic to the closed unit disk of R2, the space BDL being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangu...