We consider a stochastic flow on $\mathds{R}$ generated by an SDE with its drift being a function of bounded variation. We show that the flow is differentiable with respect to the initial conditions. Asymptotic properties of the flow are studied.Comment: 19 page

We consider a d-dimensional SDE with an identity diffusion matrix and a drift vector being a vector function of bounded variation. We give a representation for the derivative of the solution with respect to the initial data.2000 Mathematics Subject Classification. 60J65, 60H10.

We consider a Brownian motion on the plane with semipermeable membranes on n rays that have a common endpoint in the origin. We obtain the necessary and sufficient conditions for the process to reach the origin and we show that the probability of hitting the origin is equal to zero or one. , we can assume that ξ k ≤ π, k = 1, . . . , n. Indeed, otherwise we can introduce an additional ray c ad = {(r, ϕ) : r ≥ 0, ϕ = π}, and put γ ad = 0. Brought to you by | University of Arizona Authenticated Download Date | 5/29/15 3:42 PM On Brownian motion on the plane 141 So, throughout this section, we assume ξ k ≤ π/2, k = 1, . . . , n. Lemma 1.2. Given x 0 ∈ R 2 , there exists a unique strong solution of equation (0.1) up to the first time of hitting the origin.Proof.(i) The process (x(t)) t≥0 is an ordinary Wiener process up to the first time of hitting one of the rays c 1 , . . . , c n . Define

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И н сти тут геоф и зи к и им. С. И. С уб б о ти н а Н А Н Украины , К иев, У кр аина 2 Берлин, Герм ания П о ступ и ла 24 и ю ля 2018 г.