A random walk with exponentially varying step, modeling damped or amplified diffusion, is studied. Each step is equal to the previous one multiplied by a step factor s (01/s relating different processes. For s<1/2 and s>2, the process is retrodictive (i.e., every final position can be reached by a unique path) and the set of all possible final points after infinite steps is fractal. For step factors in the interval [1/2,2], some cases result in smooth density distributions, other cases present overlapping self-similarity and there are values of the step factor for which the distribution is singular without a density function.
Let M be a differentiable manifold endowed locally with two complementary distributions, say horizontal and vertical. We consider the two subgroups of (local) diffeomorphisms of M generated by vector fields in each of of these distributions. Given a stochastic flow ϕ t of diffeomorphisms of M , in a neighbourhood of initial condition, up to a stopping time we decompose ϕ t = ξ t • ψ t where the first component is a diffusion in the group of horizontal diffeomorphisms and the second component is a process in the group of vertical diffeomorphisms. Further decomposition will include more than two components; it leads to a maximal cascade decomposition in local coordinates where each component acts only in the corresponding coordinate.
Let M and N be manifolds equipped with connections Γ M and Γ N respectively and F : M → N be a smooth map. Let X be an M -valued semimartingale and Θ be an 1-form on N . We prove the following Itô formula in the context of Schwartz (second order) geometry,where the integrals are in the Itô sense, and β F is the fundamental form of F . Some applications are discussed.
We consider the stochastic transport linear equation and we prove existence and uniqueness of weak L p −solutions. Moreover, we obtain a representation of the general solution and a Wong-Zakai principle for this equation. We make only minimal assumptions, similar to the deterministic problem. The proof is supported on the generalized Itô-Ventzel-Kunita formula (see [15]) and the theory of Lions-DiPerna on transport linear equation (see [9]).Key words: Stochastic perturbation, Transport equation, Itô formula. MSC2000 subject classification: 60H10 , 60H15 .
IntroductionIn this article we establish global existence and uniqueness of solution of the transport linear equation with a stochastic perturbation. Namely, we consider the following equation:
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