This note is concerned with the differentiation of heat semigroups on Riemannian manifolds. In particular, the relation $dP_tf=P_tdf$ is investigated for the semigroup generated by the Laplacian with Dirichlet boundary conditions. By means of elementary martingale arguments it is shown…

Let $\{P_t\}_{t\ge 0}$ be the transition semigroup of a diffusion process. It is known that $P_t$ sends continuous functions into differentiable functions so we can write $DP_tf$. But what happens with this derivative when $t\to 0$ and $P_0f=f$ is only…

P. Hitczenko, S.Kwapien, W.N.Li, G.Schechtman, T.Schlumprecht and J.Zinn stated the following conjecture. Let $\mu$ be a symmetric $\alpha$-stable measure on a separable Banach space and $B$ a centered ball such that $\mu(B)\le b$. Then there exists a constant $R(b)$, depending…

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…

We consider a continuous time random walk on the $d$-dimensional lattice $\mathbb{Z}^d$: the jump rates are time dependent, but symmetric and strongly elliptic with ellipticity constants independent of time. We investigate the implications  of heat kernel estimates on recurrence-transience  properties…

In 1971, Yamada and Watanabe showed that pathwise uniqueness holds for the SDE $dX= \sigma (X)dB$ when sigma takes values in the n-by-m matrices and satisfies $|\sigma (x)- \sigma (y)| < |x-y|\log(1/|x-y|)^{1/2}$. When $n=m=2$ and $\sigma$ is of the form…

Given a probability law $\pi$ on a set $S$ and a function $g : S \rightarrow R$, suppose one wants to estimate the mean $\bar{g} = \int g d\pi$. The Markov Chain Monte Carlo method consists of inventing and simulating…

We state and prove a simple quantitative bound on the total variation distance after k iterations between two Markov chains with different initial distributions but identical transition probabilities. The result is a simplified and improved version of the result in…