This paper uses linear programming to numerically evaluate the Laplace transform of the exit time distribution and the resolvent of the moments of various Markov processes in bounded regions. The linear programming formulation is developed from a martingale characterization of the processes and the use of occupation measures. The LP approach naturally provides both upper and lower bounds on the...

A Feynman-Kac type formula of relativistic Schrödinger operators with unbounded vector potential and spin 1/2 is given in terms of a three-component process consisting of Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic field with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound ...

We study a spatial branching model, where the underlying motion is d-dimensional (d ≥ 1) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. We show that the quenched local growth rate is given by the branching rate in the ‘free’ region. We obtain the asymptotics (in probability) of the quenched global growth rates fo...

We consider 2-dimensional reflected Brownian motions in sharp thorns pointed downward with horizontal vectors of reflection. We present a decomposition of the process into a Brownian motion and a process which has bounded variation away from the tip of the thorn. The construction is based on a new Skorohod-type lemma.

Abstract We give some adequate extension, in the framework of a general Lévy process, of our previous construction of processes with one-dimensional martingale marginals, done originally in the set-up of Brownian motion. The Lévy process framework allows us to streamline our previous arguments, as well as to reach a larger class of such processes, even in the Brownian case. We give some illustr...

The purpose of this paper is to relate the recurrence and transience properties of a branching di usion process on a Riemannian manifold M to some properties of a linear elliptic operator on M (including spectral properties). There is a trade-o between the tendency of the Brownian motion to escape (if it is transient) and the birth process of the new particles. If the latter has a high enough i...

Modify the scatterer configuration of a planar, finitehorizon Lorentz process in a bounded domain. Sinai asked in 1981 whether for the diffusively scaled variant of the modified process convergence to Brownian motion still holds. The main result of the work answers Sinai’s question in the affirmative. Other types of local perturbations are also investigated: finite horizon periodic Lorentz proc...

This paper uses linear programming to numerically evaluate the Laplace transform of the exit time distribution and the resolvent of the moments of various Markov processes in bounded regions. The linear programming formulation is developed from a martingale characterization of the processes and the use of occupation measures. The LP approach naturally provides both upper and lower bounds on the...

The purpose of this paper is to establish the convergence in law of the sequence of “midpoint” Riemann sums for a stochastic process of the form f ′(W ), where W is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-ofvariable formula in law with a second order correction term which is an Itô integral of f ′′(W ) with respect to...