Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form ∑ k Ykαk(z t (ω)) where Yk are vector fields, is a positive number, z t is a 1 L0 diffusion process taking values in possibly a different manifold, αk are annihilators of ker(L0). Under Hörmander type conditions on L0 we prove that, as approaches zero, the stochastic processes y t conve...

where (0 l ) is the ensemble-averaging operation. The higher order moments can be determined by forming the ensemble average of different combinations of u(x, t) with the help of (1.4). Unfortunately, the Green’s function, G(x, t ; x’, t’), of (l.l), (1.2) and (1.3) cannot be obtained in terms of elementary or special functions except for the simplest cases. In general, we will have to obtain G...

In this paper we establish the existence and uniqueness of a solution for different types of stochastic differential equation with random initial conditions and random coefficients. The stochastic integral is interpreted as a generalized Stratonovich integral, and the techniques used to derive these results are mainly based on the path properties of the Brownian motion, and the definition of th...

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.

Abstract To model subsurface flow in uncertain heterogeneous or fractured media an elliptic equation with a discontinuous stochastic diffusion coefficient—also called random field—may be used. In case of one-dimensional parameter space, Lévy processes allow for jumps and display great flexibility the distributions However, various situations (e.g. microstructure modeling), space is not sufficie...