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Hölder norms and the support theorem for diffusions. (English) Zbl 0814.60075
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 \[ dy_ t = \sum^ m_{k=1} \sigma_ k (t,y_ t) h^ k_ tdt + b(t,y_ t)dt, \quad y_ 0 = x. \]

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
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