On the sample paths of diagonal Brownian motions on the infinite dimensional torus
Let \(T=R/2\pi Z\) be the circle group and \(T^\infty=\prod_1^\infty T_i\) be the countable product of circles \(T_i\). The group \(T^\infty\) is equipped with the product topology and its normalized Haar measure \(\mu\). Let \(\mathcal C\) be the set of all smooth functions on \(T^\infty\) which depend only on a finite number of coordinates. On the circle \(T\) denote by \(\nu_t\) the standard heat kernel measure associated to the infinitesimal generator \((\frac d{dx})^2\). The convolution semigroup \((\nu_t)_{t>0}\) is associated with a stochastic process \(\xi=(\xi_t)_{t\geq0}\) which is simply Brownian motion runned at twice the usual speed in classic probabilistic notation and wrapped around the circle. For any fixed sequence \({\mathbf a}=(a_1,\ldots)\) of positive numbers let us consider the product measures \(\mu_t=\mu_t^{\mathbf a}=\bigotimes_1^\infty\nu_{a_it}\). The family \((\mu_t)_{t>0}\) forms a convolution semigroup of measures on \(T^\infty\) and \(\mu_t\) is the marginal at time \(t\) of a diffusion process \(X=X^{\mathbf a}=(X_t)_{t\geq0}\) which is simply the product of independent circle Brownian motions \(X^i=(X^i_t)_{t\geq0}\) where \(X^i_t=\xi_{a_it}\). The intrinsic distance is defined by \[ d(x,y)=d^{\mathbf a}(x,y)= \sup\left\{f(x)-f(y):f\in{\mathcal C},\;\sum_1^\infty a_i | \partial_if | ^2\leq1\right\} \text{ and } d(x)=d^{\mathbf a}(x)=d^{\mathbf a}(e,x), \] where \(e=(0,0,\ldots)\) is the neutral element in \(T^\infty\). \(d\) is continuous and defines the topology of \(T^\infty\) if and only if \(\sum_1^\infty \frac 1{a_i}<\infty\). The last condition is assumed to hold throughout the paper under review. Under this condition for any \(t>0\) the measure \(\mu_t=\mu_t^{\mathbf a}\) is absolutely continuous with respect to the Haar measure \(\mu\) and admits a continuous density \(\mu_t^{\mathbf a}(x)\) given by \(\mu_t^{\mathbf a}(x)=\prod_1^\infty \nu_{a_it}(x_i)\). Given a sequence \({\mathbf a}=(a_i)\) of positive numbers define \(N(s)= N^{\mathbf a}(s)=\#\{i: a_i\leq s\}\) and for a given function \(f:(0, \infty)\to(0,\infty)\) define the transform \(f^{\#}\) by \(f^{\#}(z)= \int_0^z f(x)\,\frac{dx}{x}\). The main result of the paper is Theorem 4.4. Let \({\mathbf a}=(a_i)\) be a sequence of positive numbers such that \(N=N^{\mathbf a}\) is slowly varying. Then \(\log\mu_t(e)\sim(1/2)N^{\#}(1/t)\) as \(t\) tends to \(0\) and the sample paths of the process \(X^{\mathbf a}\) have the following properties: (1) We always have \(P_e\)-almost surely \(\liminf_{t\to 0} (d(X_t)/\sqrt{tN(1/t)})<\infty\). (2) If \(N(s)=o(\log s)\) at infinity, then \(P_e\)-almost surely \[ \limsup_{t\to 0} \frac {d(X_t)} {\sqrt{4t\log\log 1/t}}=1,\qquad \lim_{\varepsilon\to 0}\sup_{0<s<t \leq1,\,t-s\leq\varepsilon} \frac {d(X_s,X_t)} {\sqrt{4(t-2) \log(1/(t-s))}}=1 \] and \[ \liminf_{t\to 0} \frac{d(X_t)} {\sqrt{4t\log\log 1/t}}=0. \] (3) If \(N(s)=o(\log s)\) and \(\log\log s=O(N(s))\) at infinity, then \(P_e\)-almost surely \[ 0<\liminf_{t\to 0} \frac {d(X_t)} {\sqrt{tN(1/t)}}\leq \limsup_{t\to 0} \frac {d(X_t)} {\sqrt {tN(1/t)}} <\infty, \] and \[ \lim_{\varepsilon\to 0}\sup_{0<s<t\leq1, \,t-s \leq\varepsilon} \frac{d(X_s,X_t)} {\sqrt{4(t-s)\log(1/(t-s))}}=1. \] (4) If \(\log\log s=O(N(s))\) at infinity, then \(P_e\)-almost surely \[ 0<\liminf_{t\to 0} \frac{d(X_t)} {\sqrt{tN(1/t)}} \leq\limsup_{t\to 0} \frac{d(X_t)} {\sqrt{tN(1/t)}}<\infty. \] (5) If \(\log s=O(N(s))\) at infinity, then \(P_e\)-almost surely \[ 0< \lim_{\varepsilon\to 0}\sup_{0<s<t\leq1,\,t-s\leq \varepsilon} \frac{d(X_s,X_t)} {\sqrt{4(t-s)\log(1/(t-s))}}<\infty. \] The authors note that the different cases in this theorem are not exclusive. Broadly speaking for a slowly varying \(N\) there are three cases to consider: (a) If \(N\) is smaller than \(\log\log\), then we obtain a classical Lévy-Khinchin law of iterated logarithm \[ \limsup_{t\to 0} \frac {d(X_t)} {\sqrt{4t\log\log 1/t}}=1 \] and a classical Lévy modulus of continuity \[ \lim_{\varepsilon \to 0}\sup_{0<s<t\leq1,\,t-s\leq \varepsilon} \frac{d(X_s,X_t)} {\sqrt{4(t-s)\log(1/(t-s))}}=1. \] (b) If \(N\) is larger than \(\log \log\) but smaller than \(\log\), then we still have a classical Lévy modulus of continuity, but the Lévy-Khinchin-type result is not classical any more (\(d(X_t)\) is now controlled by the function \(\sqrt{tN(1/t)})\). (c) If \(N\) is larger than \(\log\), but still slowly varying, then all regularity behaviors are controlled by the function \(\sqrt{tN(1/t)}\).
scientific article; zbMATH DE number 2081278