Stochastic processes for bounded noise

The scalar differential inclusion ẋ ∈ f(x) + g(x)u, u ∈ [-1, 1], x(0) = x0 (0.1) is considered as a model of the dynamical system ẋ = f(x) perturbed by the bounded noise g(x)u, u ∈ [-1, 1], and the problem of constructing a nontrivial probability measure on the set L of solutions to (0.1) is studied. In particular, it is shown that: (i) every Markov process whose probability measure is supported on L is degenerate, in a sense to be specified (see Theorem 3.1); (ii) given a flow of probability measures μt on the reachable sets Rt of (0.1), satisfying a certain compatibility condition, a Markov process Xt is constructed such that its marginals are exactly μt and (0.1) is satisfied "from one side" (see Theorem 4.1); its finite-dimensional distributions are computed and the regularity of its sample paths is investigated (see Section 5.2); (iii) given a process of a type previously considered, another process Yt is constructed through its finite-dimensional distributions, and its distribution is shown to be supported exactly on L. Finally, a model example is considered (see Section 7).