It is shown that the fundamental solution of a hyperbolic partial differential equation with time delay has a natural probabilistic structure, i.e. is approximately Gaussian, as t → ∞. The proof uses ideas from the DeMoivre proof of the Central Limit Theorem. It follows that solutions of the hyperbolic equation look approximately like solutions of a diffusion equation with constant convection a...

We derive a probabilistic representation for solutions of matrix-valued transport equations that account for polarization e ects. Such equations arise in radiative transport for the Stokes parameters that model the propagation of light through turbulent atmospheres. They also arise in radiative transport for seismic wave propagation in the earth's crust. The probabilistic representation involve...

Model checking can tell us whether a system is correct; probabilistic model checking can also tell us whether a system is timely and reliable. Moreover, probabilistic model checking allows one to verify properties that may not be true with probability one, but may still hold with an acceptable probability. The challenge in developing a probabilistic model checker able to handle realistic system...

The probabilistic equivalent formulation of Dupire's PDE is the Put-Call duality equality. In local volatility models including exponential Lévy jumps, we give a direct probabilistic proof for this result based on stochastic flows arguments. This approach also enables us to check the probabilistic equivalent formulation of various generalizations of Dupire's PDE recently obtained by Pironneau [...

The paper contains several results showing advantages of probabilistic automata over deterministic ones. The advantages always are of one kind, namely, the complexity of probabilistic automata turns out to be considerably smaller. The paper contains both new results and results already published in the USSR (though, hardly these results are known outside the USSR). The economy of computer resou...

This article is a short introduction to the surprising appearance of Burgers equation in some basic probabilistic models.

The object of this paper is the uniqueness for a d-dimensional Fokker-Planck type equation with inhomogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so-called Barenblatt’s solution of the fast diffusion equation which is the partial differential equation ∂tu = ∂ 2 xxu m with m ∈]0, 1[. Together...

Abstract. The object of this paper is the uniqueness for a d-dimensional Fokker-Planck type equation with non-homogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic representation of the so called Barenblatt solution of the fast diffusion equation which is the partial differential equation ∂tu = ∂ 2 xx u with m ∈ (0, 1)...

A surprisingly simple and explicit expression for the waiting time distribution of the M/D/c queueing system is derived by a full probabilistic analysis, requiring neither generating functions nor Laplace transforms. Unlike the solutions known so far, this expression presents no numerical complications, not even for high traffic intensities. Finally, the result is proved explicitly to satisfy E...