We establish a formula for the distribution of loop-erased random walks at certain random times. Several classical results on spanning trees, including Wilson's algorithm, follow easily, as well as a method to construct random Hamiltonian cycles.

We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The…

This investigation concerns the hyper-exponential jump-diffusion processes. Following the exposition of the two-sided exit problem by Kyprianou, A. E., and Asmussen, S. and Albrecher, H., this study investigates first passage functionals for these processes.The corresponding boundary value problems are solved…

A random variable is sampled from a discrete distribution. The missing mass is the probability of the set of points not observed in the sample. We sharpen and simplify McAllester and Ortiz's results (JMLR, 2003) bounding the probability of large…

In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and…

In the present paper we consider the fluctuations of the free energy in the random energy model (REM) on a moderate deviation scale. We find that for high temperatures the normal approximation holds only in a narrow range of scalings…

We present a direct derivation of the theorem of M. Maxwell and M. Woodroofe (Ann. Probab. vol. 28 (2000) 713-724), on martingale approximation of additive functionals of stationary Markov processes, from the non-reversible version of the Kipnis-Varadhan theorem.

We derive integral formulas, involving the Airy function, for moments of the time a two-sided Brownian motion with parabolic drift attains its maximum.

In this note we consider the point process of eigenvalues of the tensor product of two independent random unitary matrices of size m by m and n by n. When n becomes large, the process behaves like the superposition of…

In this note we provide a short and simple proof that every adapted measurable stochastic process admits a progressively measurable modification.