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A simple proof of the support theorem for diffusion processes. (English) Zbl 0807.60073

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXVIII. Berlin: Springer-Verlag. Lect. Notes Math. 1583, 36-48 (1994).
Let \((\Omega,{\mathcal F},P)\) be a standard Wiener space, \(\mathcal H\) denotes the Cameron-Martin space. The well-known Stroock-Varadhan characterization of the support of the law of a diffusion \(X\), as the closure of the set of controlled equations \({\mathcal S} = \{S(h)_ ., h\in {\mathcal H}\}\) is proved in the set of \(\alpha\)-Hölder continuous functions \((\alpha < 1/2)\). The method consists in reducing both inclusions of the support to one single general convergence theorem for processes involving a stochastic integral, an integral with respect to adapted interpolations \(\omega^ n\) of \(\omega\) and a deterministic integral. It implies that both \(\| S(\omega^ n) - X(\omega)\|_ \alpha\) and \(\| X(\omega - \omega^ n + h) - S(h) \|_ \alpha\) converge to zero in \(L^ 2\). By Girsanov’s theorem, the law of the transformation \(T_ n\) of \(\Omega\) defined by \(T_ n(\omega) = \omega - \omega^ n + h\) is absolutely continuous with respect to \(P\), so that \({\mathcal S} \subset \text{support }P \circ X^{-1}\). The converse implication is proved in the usual way.
The method presented in this paper has also been used by the authors in the setting of hyperbolic SPDE’s [Probab. Theory Relat. Fields 98, No. 3, 361-387 (1994; Zbl 0794.60061)]and of parabolic SPDE’s [Ann. Probab. (with V. Bally), to appear].
For the entire collection see [Zbl 0797.00020].
Reviewer: A.Millet (Paris)

MSC:

60J60 Diffusion processes
60G17 Sample path properties

Citations:

Zbl 0794.60061
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