We produce a stochastic regularization of the Poisson-Sigma model of Cattaneo-Felder, which is an analogue regularization of Klauder’s stochastic regularization of the hamiltonian path integral [23] in field theory. We perform also semi-classical limits.

If the separability of the separable process definition is replaced by a set of optional (predictable) times, a new property is obtained, optional (predictable) separability. A well measurable (accessible) process is necessarily optionally (predictably) separable. Proofs of limit properties of a separable process become proofs of the same limit properties of a well measurable (predictable) process.

Existence of strong and weak solutions to stochastic inclusions $x_t-x_s\in {\int }_s^t{F}_{\tau }\left(x_{\tau }\right)d\tau +{\int }_s^t{G}_{\tau }\left(x_{\tau }\right)d{w}_{\tau }+{\int }_s^t{\int }_{{ℝ}^n}{H}_{\tau ,z}\left(x_{\tau }\right)q(d\tau ,dz)$ and $x_t-x_s\in {\int }_s^t{F}_{\tau }\left(x_{\tau }\right)d\tau +{\int }_s^t{G}_{\tau }\left(x_{\tau }\right)d{w}_{\tau }+{\int }_s^t{\int }_{\left|z\right|\le 1}{H}_{\tau ,z}\left(x_{\tau }\right)q(d\tau ,dz)+{\int }_s^t{\int }_{\left|z\right|>1}{H}_{\tau ,z}\left(x_{\tau }\right)p(d\tau ,dz)$, where p and q are certain random measures, is considered.

The martingale Hardy space $H_p\left({[0,1)}^2\right)$ and the classical Hardy space $H_p{(}^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a...

We consider a random walk in a random potential, which models a situation of a random polymer and we study the annealed and quenched costs to perform long crossings from a point to a hyperplane. These costs are measured by the so called Lyapounov norms. We identify situations where the point-to-hyperplane annealed and quenched Lyapounov norms are different. We also prove that in these cases the polymer path exhibits localization.

In the spirit of a classical result for Crump–Mode–Jagers processes, we prove a strong law of large numbers for fragmentation processes. Specifically, for self-similar fragmentation processes, including homogenous processes, we prove the almost sure convergence of an empirical measure associated with the stopping line corresponding to first fragments of size strictly smaller than η for 1≥η&gt;0.