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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
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