On transformations of Wiener space. (1994) - Skorokhod, Anatoli V.
Abstract : We consider transformations of the form ( T a x ) t = x t + ∫ 0 t a ( s, x ) d s on the space C of all continuous functions x = x t : [ 0, 1 ] → ℝ, x 0 = 0, where a ( s, x ) is a measurable function [ 0, 1 ] × C → ℝ which is 𝒞 ˜ s -measurable for a fixed s and 𝒞 ˜ s is the σ -algebra generated by x u, u ≤ t . It is supposed that T a maps the Wiener measure μ 0 on ( C, 𝒞 ˜ s ) into a measure μ a which is equivalent with respect to μ 0 . We study some conditions of invertibility of such transformations. We also consider stochastic differential equations of the form d y ( t ) = d w ( t ) + a ( t, y ( t ) ) d t, y ( 0 ) = 0 where w ( t ) is a Wiener process. We prove that this equation has a unique strong solution if and only if it has a unique weak solution.