On transformations of Wiener space

[摘要] We consider transformations of the form(Tax)t=xt+∫0ta(s,x)dson the spaceCof all continuous functionsx=xt:[0,1]→ℝ,x0=0, wherea(s,x)is a measurable function[0,1]×C→ℝwhich is?˜s-measurable for a fixedsand?˜sis theσ-algebra generated by{xu,u≤t}. It is supposed thatTamaps the Wiener measureμ0on(C,?˜s)into a measureμawhich is equivalent with respect toμ0. We study some conditions of invertibility of such transformations. We also consider stochastic differential equations of the formdy(t)=dw(t)+a(t,y(t))dt, y(0)=0wherew(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.

[学科分类] 应用数学