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Proof : The only problem is to prove formula (4), that is to prove that u is the sum of its Taylor expansion with respect to λ. In the case where A is a complex Banach algebra,
Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian
Keywords Brownian bridge, Brownian excursion, Brownian scaling, local time, Bessel process, zeros of Bessel functions, Riemann zeta function.. AMS subject classification 60J65,
To prove those results, we use the fact that the ISE has the same distribution (up to a constant scaling) as the total mass of an excursion of the Brownian snake conditioned to have
This process has values which are trajectories of standard Poisson process stopped at some random finite lifetime with Brownian evolution.. We use this Poisson snake to construct
This result has consequences for the Hausdorff dimension of the frontier of planar Brownian motion, and also for the Hausdorff dimension of the set of cut-points of planar
A consequence of the weak convergence of coalescing nonsimple random walks to the Brownian web is that, for the dual one-dimensional voter model with initial condition 1’s on
With this choice of ζ, the genealogical tree is the CRT (see [1]), and the Brownian snake can be seen as an embedding of the CRT in R d. We can now give the definition of the ISE
