Harnack-type inequality for linear fractional stochastic equations

Using the coupling method and Girsanov theorem, we prove a Harnack-type inequality for a stochastic differential equation with non-Lipschitz drift and driven by a fractional Brownian motion with Hurst parameter {H<\frac{1}{2}}. We also investigate this inequality for a stochastic differential equation driven by an additive fractional Brownian sheet.

Rate of convergence of uniform transport processes to a Brownian sheet

We give the rate of convergence to a Brownian sheet from a family of processes constructed starting from a set of independent standard Poisson processes. These processes have realizations that converge almost surely to the Brownian sheet, uniformly in the unit square.

Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube I 2 in R 2 . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total I 2 (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.