Density of orbits of Brownian trajectories under the action of the Lévy transformation
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, \textit{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; \url{doi:10.1051/ps/2011020}), \url{arXiv: math/0511154}].NEWLINENEWLINE 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 NEWLINE\[NEWLINE\bigcup_{n\in {\mathbb N}}(K^{n}(\cdot ,B)>0)\subset \bigcup_{n\in {\mathbb N}}T^{-n}(B)=\limsup_{n\rightarrow +\infty }T^{-n}(B) \text{ a.e. }NEWLINE\]NEWLINE 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.
scientific article; zbMATH DE number 6042936