We give sharp estimates for the transition density of the isotropic stable Lévy process killed when leaving a right circular cone.

In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.

We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses.

Sensitivity analysis of irreducible Markov chains considers an original Markov chain with transition probability matrix $P$ and modified Markov chain with transition probability matrix $P$. For their respective stationary probability vectors $\pi ,\tilde{\pi }$, some of the following charactristics are usually studied: $\parallel \pi -\tilde{\pi }{\parallel }_p$ for asymptotical stability [3], $|{\pi }_i-{\tilde{\pi }}_i|,\frac{|{\pi }_i-{\tilde{\pi }}_i|}{{\pi }_i}$ for componentwise stability or sensitivity [1]. For functional transition probabilities, $P=P\left(t\right)$ and stationary probability vector $\pi \left(t\right)$, derivatives are also used for studying...

It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues,...

It is known that the time until a birth and death process reaches a certain level is distributed as a sum of independent exponential random variables. Diaconis, Miclo and Swart gave a probabilistic proof of this fact by coupling the birth and death process with a pure birth process such that the two processes reach the given level at the same time. Their coupling is of a special type called intertwining of Markov processes. We apply this technique to couple the Wright-Fisher diffusion with reflection...

Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms ${\left|\right|f\left|\right|}_{W^{\sigma ,2}}$ of a function f ∈ L²(E,μ) have the property $1/Cℰ(f,f)\le lim in{f}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le lim su{p}_{\sigma \nearrow 1}{\left(1-\sigma \right)\left|\right|f\left|\right|}_{W^{\sigma ,2}}\le Cℰ(f,f)$, where ℰ is the Dirichlet form relative to the fractional diffusion.

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: $(A_t^{-}:={\int }_0^t{1}_{X_s<0}\mathrm{d}s,t\ge 0)$. On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung.43 (2006) 171–246]).

We investigate the dissipativity properties of a class of scalar second order parabolic partial differential equations with time-dependent coefficients. We provide explicit condition on the drift term which ensure that the relative entropy of one particular orbit with respect to some other one decreases to zero. The decay rate is obtained explicitly by the use of a Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus. As a byproduct,...