Ben Arous, Gérard; Gradinaru, Mihai; Ledoux, Michel Hölder norms and the support theorem for diffusions. (English) Zbl 0814.60075 Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 3, 415-436 (1994). The law \(P_ x\) of the solution of the Stratonovich stochastic differential equation \[ dx_ t = \sum^ m_{k=1} \sigma_ k (t,x_ t) \circ dw^ k_ t + b(t,x_ t)dt, \quad x_ 0 = x, \] where \(\sigma_ k (t,x)\), \(k=1, \dots, m\), \(b(t,x)\) are smooth vector fields on \(R^{d+1}\) and \((w^ 1, \dots, w^ m)\) is an \(m\)-dimensional Brownian motion, is considered. It is proved that the support of \(P_ x\) for the \(\alpha\)-Hölder topology, \(\alpha \in [0,1/2)\), coincides with the closure of \(\Phi_ x (L^ 2)\), where \(\Phi_ x\) is the mapping which associates to \(h \in L^ 2 = L^ 2 ([0,1] \times R^ m)\) the solution of the differential equation \[ dy_ t = \sum^ m_{k=1} \sigma_ k (t,y_ t) h^ k_ tdt + b(t,y_ t)dt, \quad y_ 0 = x. \] Reviewer: B.Grigelionis (Vilnius) Cited in 19 Documents MSC: 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J65 Brownian motion 46E15 Banach spaces of continuous, differentiable or analytic functions 60G15 Gaussian processes Keywords:Hölder topology; Stratonovich stochastic differential equation; Brownian motion PDFBibTeX XMLCite \textit{G. Ben Arous} et al., Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 3, 415--436 (1994; Zbl 0814.60075) Full Text: Numdam EuDML