Density of orbits of Brownian trajectories under the action of the Lévy transformation. (Densité des orbites des trajectoires browniennes sous l’action de la transformation de Lévy.)(English. French summary)Zbl 1248.37014

Let $$W$$ be the space of all continuous maps $$w:[0,\infty )\rightarrow {\mathbb R}$$, $$w(0)=0$$, with the Wiener measure $$P$$. The Lévy transform $$W\rightarrow W$$ is defined as taking $$w$$ into $$|w(t)|-l(t,\omega )$$, where $$l(t,w)=\liminf_{\varepsilon \rightarrow 0}(2\varepsilon )^{- 1}\int_{_{0}}^{^{t}}1_{(|w(s)|\leq \varepsilon )}ds$$ is the local time in $$0$$. It preserves the Wiener measure. It is an unsolved problem whether the Lévy transform is ergodic or not. Recently, M. Malric succeeded in making a step towards its solution by proving the density, in the uniform convergence on compacts, of the iterates of this transform for almost all $$w$$ [“Density of paths of iterated Lévy transforms of Brownian motion”, ESAIM, Probab. Stat. 16, 399–424 (2012; doi:10.1051/ps/2011020), arXiv: math/0511154].
The authors present in this paper a detailed alternative demonstration of this result (Theorem 1). The paper starts with the following general result: if $$T$$ is a map preserving a probability $$\pi$$ (on a space $$E$$), $$X$$ is a $$\pi$$-distributed random variable and $$K$$ is a conditional distribution of $$X$$ with respect to $$T(X)$$, then $\bigcup_{n\in {\mathbb N}}(K^{n}(\cdot ,B)>0)\subset \bigcup_{n\in {\mathbb N}}T^{-n}(B)=\limsup_{n\rightarrow +\infty }T^{-n}(B) \text{ a.e. }$ If $$E$$ is topological with a denumerable basis, then, in order that the orbit by $$T$$ of $$\pi$$-almost every point in $$E$$ is dense, it is sufficient that every open set $$V\subset E$$ is accessible from almost every point by the $$K$$-Markov chain. Consider the set $$Q$$ of positive rationals ordered lexicographically by $$(a+b,a)$$ in the irreducible writing $$a/b$$ and denote, for an interval $$A\subset(0,\infty)$$, by $$q(A)$$ the smallest $$q\in A\cap Q$$ in the order in $$Q$$. For $$r\in W_{+}=\{w;\in W,w\geq 0\}$$ and for $$\varepsilon \in E=\{-1,1\}^{Q}$$, $$t>0$$ let $$I=(a,b)$$ be the interval defined by by $$r(a)=r(b)=0$$, $$r>0$$ on $$I$$, $$t\in I$$. Let $$(\varepsilon \cdot r)(t)=\varepsilon (q(I))r(t)$$. Let $$W_{+}=\{w\in W: w\geq 0\}$$, let $$P_{R}$$ be the distribution of the Brownian motion reflected in $$0$$, $$w_{-}(t):=\min_{[0,t]}w$$ for $$w\in W_{+}$$, $$d_{t}^{CU}(f,g):= \sup_{[0,t]}|f-g|$$, let $$d_{t}^{CZ}(f,g)$$ be the infimum over all $$\delta >0$$ with $$Z(f)\cap [0,t-\delta ]\subset Z(g)+(-\delta ,\delta )$$, where $$Z(f)=\{t;f(t)=0\}$$, symmetrized in $$f$$, $$g$$ (the topology generated by them plays an important role), $$V_{t}(f,\varrho ,\delta )=\{g:d_{t}^{CU}(f,g)<\varrho$$, $$d^{CZ}(f,g)<\delta \}$$, $$V_{t}(f,\varrho )= \{g;d_{t}^{CU}(f,g)<\varrho \}$$. The authors reduce the proof of Theorem 1 to that of the accessibility (Proposition 62), from almost all $$r\in W_{+}$$, of $$V_{t}(f,\varrho ,\delta )$$, for every $$t$$, $$\varrho$$, $$\delta$$ and piecewise affine $$f\in W_{+}$$, by the chain $$R^{(n)}$$, $$R^{(0)}=r$$, $$R^{(n+1)}= \varepsilon_{n+1}\cdot R^{(n)}-(\varepsilon_{n+1}\cdot R^{(n)})_{-}$$, where $$\varepsilon_{1},\dots,\varepsilon_{n},\dots$$ are independent, “uniformly distributed” $$E$$-valued random variables. Let $$D_{a}(r)$$ denote the first visit in $$0$$ of $$r\in W_{+}$$ after $$a$$, $$\theta_{D_{a}}$$ the corresponding translation on $$W_{+}$$, $$O_{a,b}$$ be the set of $$r\in W_{+}$$ with $$r(t)=0$$ for at least one $$t\in (a,b)$$. Proposition 62 is deduced from the accessibility of $$U_{a,b}(f,\varrho ,\delta )= \{r;\in O_{a,a+\delta }\cap O_{b-\delta ,b},\| r-f\|_{[a,b]}<\varrho \}$$, with $$f\in W_{+}$$ piecewise affine, $$f(a)=f(b)=0$$, from almost all $$r\in O_{a,a+\delta }$$ with $$\| r\|_{[a,D_{a}(r)]}<\varrho$$, by a process constructed (as $$R^{(n)}$$) with $$\varepsilon_{n,a}$$, defined as $$\varepsilon_{n}$$ with all $$\varepsilon (t)$$ with $$t\leq D_{a}(r)$$ replaced by $$1$$, instead of $$\varepsilon_{n}$$. The first step to this result is the accessibility by this process of $$B\cap \theta_{D_{a}}^{-1}(V_{t}(0,\varrho ,\delta))$$ from almost all $$r\in B$$, for $$B\in {\mathcal F}_{D_{a}}$$. Several continuity properties are before established, arriving to the continuity of $$(\varepsilon,r)\rightarrow \varepsilon_{a}\cdot r-(\varepsilon_{a}\cdot r)_{-}$$. At the end, relations with the contents of the paper of Malric [loc. cit.] are discussed.

MSC:

 37A50 Dynamical systems and their relations with probability theory and stochastic processes 60J65 Brownian motion 28D05 Measure-preserving transformations
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References:

 [1] M. Malric. Densité des zéros des transformés de Lévy itérés d’un mouvement brownien. C. R. Math. Acad. Sci. Paris 336 (2003) 499-504. · Zbl 1024.60034 [2] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. Preprint, 2005. Available at . · Zbl 1274.60171 [3] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. Preprint, 2007. Available at . · Zbl 1274.60171 [4] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. Preprint, 2009. Available at . · Zbl 1274.60171 [5] M. Malric. Density of paths of iterated Levy transforms of Brownian motion. ESAIM Probab. Stat. (2012). To appear. , published on line (03 février 2011). · Zbl 1274.60171 [6] K. Petersen. Ergodic Theory . Cambridge Univ. Press, Cambridge, 1989. Corrected reprint of the 1983 original. · Zbl 0676.28008 [7] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion . Springer, Berlin, 1991. · Zbl 0731.60002
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