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Zbl 1056.60029
Bendikov, A.; Saloff-Coste, L.
On the sample paths of diagonal Brownian motions on the infinite dimensional torus. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 40, No. 2, 227-254 (2004). ISSN 0246-0203

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 $\cal C$ 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 $\xi=(\xi_t)_{t\ge0}$ which is simply Brownian motion runned at twice the usual speed in classic probabilistic notation and wrapped around the circle. For any fixed sequence ${\bold a}=(a_1,\ldots)$ of positive numbers let us consider the product measures $\mu_t=\mu_t^{\bold a}=\bigotimes_1^\infty\nu_{a_it}$. 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 $X=X^{\bold a}=(X_t)_{t\ge0}$ which is simply the product of independent circle Brownian motions $X^i=(X^i_t)_{t\ge0}$ where $X^i_t=\xi_{a_it}$. The intrinsic distance is defined by $$d(x,y)=d^{\bold a}(x,y)= \sup\left\{f(x)-f(y):f\in{\cal C},\ \sum_1^\infty a_i \vert \partial_if \vert ^2\le1\right\} \text{ and } d(x)=d^{\bold a}(x)=d^{\bold a}(e,x),$$ 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 $\mu_t=\mu_t^{\bold a}$ is absolutely continuous with respect to the Haar measure $\mu$ and admits a continuous density $\mu_t^{\bold a}(x)$ given by $\mu_t^{\bold a}(x)=\prod_1^\infty \nu_{a_it}(x_i)$. Given a sequence ${\bold a}=(a_i)$ of positive numbers define $N(s)= N^{\bold a}(s)=\#\{i: a_i\le s\}$ 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 ${\bold a}=(a_i)$ be a sequence of positive numbers such that $N=N^{\bold a}$ 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 $X^{\bold a}$ 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 $$\limsup_{t\to 0} \frac {d(X_t)} {\sqrt{4t\log\log 1/t}}=1,\qquad \lim_{\varepsilon\to 0}\sup_{0<s<t \le1,\,t-s\le\varepsilon} \frac {d(X_s,X_t)} {\sqrt{4(t-2) \log(1/(t-s))}}=1$$ 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 $$0<\liminf_{t\to 0} \frac {d(X_t)} {\sqrt{tN(1/t)}}\le \limsup_{t\to 0} \frac {d(X_t)} {\sqrt {tN(1/t)}} <\infty,$$ and $$\lim_{\varepsilon\to 0}\sup_{0<s<t\le1, \,t-s \le\varepsilon} \frac{d(X_s,X_t)} {\sqrt{4(t-s)\log(1/(t-s))}}=1.$$ (4) If $\log\log s=O(N(s))$ at infinity, then $P_e$-almost surely $$0<\liminf_{t\to 0} \frac{d(X_t)} {\sqrt{tN(1/t)}} \le\limsup_{t\to 0} \frac{d(X_t)} {\sqrt{tN(1/t)}}<\infty.$$ (5) If $\log s=O(N(s))$ at infinity, then $P_e$-almost surely $$0< \lim_{\varepsilon\to 0}\sup_{0<s<t\le1,\,t-s\le \varepsilon} \frac{d(X_s,X_t)} {\sqrt{4(t-s)\log(1/(t-s))}}<\infty.$$ 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 $$\lim_{\varepsilon \to 0}\sup_{0<s<t\le1,\,t-s\le \varepsilon} \frac{d(X_s,X_t)} {\sqrt{4(t-s)\log(1/(t-s))}}=1.$$ (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 $\sqrt{tN(1/t)}$.
[Vakhtang V. Kvaratskhelia (Tbilisi)]

MSC 2000:: *60G17 Sample path properties
60B15 Probability measures on groups
60J60 Diffusion processes

Keywords: sample paths; modulus of continuity; diagonal Brownian motion; infinite-dimensional torus; intrinsic distance; law of iterated logarithm

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