### Removable singularities for H... and for analytic functions with bounded Dirichlet integral.

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We give several necessary and sufficient conditions that a function $\phi $ maps the paths of one diffusion into the paths of another. One of these conditions is that $\phi $ is a harmonic morphism between the associated harmonic spaces. Another condition constitutes an extension of a result of P. Lévy about conformal invariance of Brownian motion. The third condition implies that two diffusions with the same hitting distributions differ only by a chance of time scale. We also obtain a converse of the above...

We study optimal stopping problems for some functionals of Brownian motion in the case when the decision whether or not to stop before (or at) time t is allowed to be based on the δ-advanced information ${\mathcal{F}}_{t+\delta}$, where ${\mathcal{F}}_{s}$ is the σ-algebra generated by Brownian motion up to time s, s ≥ -δ, δ > 0 being a fixed constant. Our approach involves the forward integral and the Malliavin calculus for Brownian motion.

The exit distribution for open sets of a path-continuous, strong Markov process in ${\mathbf{R}}^{n}$ is characterized as a weak star limit of successive spherical sweepings of measures, starting with the unit point mass. Then this is used to prove that two path-continuous strong Markov processes with identical exit distributions from balls when starting form the center, have identical exit distributions from all opens sets, provided they both exit a.s. from bounded sets. This implies that the only path-continuous,...

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