The Annals of Probability

On the sample paths of Brownian motions on compact infinite dimensional groups

Alexander Bendikov and Laurent Saloff-Coste

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We study the regularity of the sample paths of certain Brownian motions on the infinite dimensional torus ${\mathbb T}^\infty$ and other compact connected groups in terms of the associated intrinsic distance. For each $\lambda\in (0,1)$, we give examples where the intrinsic distance $d$ is continuous and defines the topology of ${\mathbb T}^\infty$ and where the sample paths satisfy \[ 0<\liminf_{t\ra 0} \frac{d(X_0,X_t)}{t^{(1-\lambda)/2}}\le \limsup_{t\ra 0} \frac{d(X_0,X_t)}{t^{(1-\lambda)/2}}<\infty \] and \[ 0<\lim_{\varepsilon\to 0} \sup_{0<t<s<1 \atop t-s\le \varepsilon}\frac{d(X_s,X_t)}{(t-s )^{(1-\lambda)/2}} <\infty. \]

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Ann. Probab., Volume 31, Number 3 (2003), 1464-1493.

First available in Project Euclid: 12 June 2003

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Primary: 60J60: Diffusion processes [See also 58J65] 60B99: None of the above, but in this section 31C25: Dirichlet spaces 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

Invariant diffusions path regularity Gaussian convolution semigroups.


Bendikov, Alexander; Saloff-Coste, Laurent. On the sample paths of Brownian motions on compact infinite dimensional groups. Ann. Probab. 31 (2003), no. 3, 1464--1493. doi:10.1214/aop/1055425787.

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