For real numbers $C,T_{1},T_{2},...$ we find all solutions $\mu$ to the stochastic fixed point equation $W \sim\sum_{j\ge 1}T_{j}W_{j}+C$, where $W,W_{1},W_{2},...$ are independent real-valued random variables with distribution $\mu$ and $\sim$ means equality in distribution. All solutions are infinitely divisible. The…

On complete, non-compact manifolds and infinite graphs, Faber-Krahn inequalities have been used to estimate the rate of decay of the heat kernel. We develop this technique in the setting of finite Markov chains, proving upper and lower $L^{\infty}$ mixing time…

Since the pioneering works of C.C. Heyde, A.V. Nagaev, and S.V. Nagaev in 1960's and 1970's, the precise asymptotic behavior of large-deviation probabilities of sums of heavy-tailed random variables has been extensively investigated by many people, but mostly it is…

In this paper we establish two-sided estimates for the density of the Feynman-Kac semigroups of stable-like processes with potentials given by signed measures belonging to the Kato class. We also provide similar estimates for the densities of two other kinds…

A stochastic calculus similar to Malliavin's calculus is worked out for Brownian excursions. The analogue of the Malliavin derivative in this calculus is not a differential operator, but its adjoint is (like the Skorohod integral) an extension of the Itô…

Consider reinforced random walk on a graph that looks like a doubly infinite ladder. All edges have initial weight 1, and the reinforcement convention is to add $\delta > 0$ to the weight of an edge upon first crossing, with…

We use the Malliavin calculus for Poisson processes in order to compute sensitivities for European and Asian options with underlying following a jump type diffusion. The main point is to settle an integration by parts formula (similar to the one…

We consider a branching system of $N$ Brownian particles evolving independently in a domain $D$ during any time interval between boundary hits. As soon as one particle reaches the boundary it is killed and one of the other particles splits…

This paper extends the notion of the $\Lambda$-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $\Lambda$-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the…

In the two-type Richardson model on a graph $G=(V,E)$, each vertex is at a given time in state $0$, $1$ or $2$. A $0$ flips to a $1$ (resp.\ $2$) at rate $\lambda_1$ ($\lambda_2$) times the number of neighboring $1$'s…