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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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