Regularity of Gaussian processes on Dirichlet spaces
The authors bring an interesting new contribution to the very classical question of the regularity of Gaussian processes. For that, they consider a compact connected space endowed with a Radon measure \(\mu\) having full support, and with a conservative Markov semi-group \((P_t)\) with infinitesimal generator \(A\) and ``carré du champ'' \(\Gamma\), such that the topology of \(M\) is generated by the metric \(\,\varrho(x,y):= \sup\big\{f(x)-f(y)\,\big|\,x,y\in M,\, \Gamma(f,f)\leq 1\big\}\). They also assume that \((M,\varrho,\mu)\) satisfies the doubling property, and that a Poincaré inequality holds. They moreover consider a centred Gaussian process \(Z\equiv (Z_x,\,x\in M)\) with covariance function \(K\) commuting with \((P_t)\), \ and Besov spaces \(\,B^s_{\infty,q}\) (for \(s>0, 1\leq q\leq \infty\)). Then the authors establish the following nice and new characterization of the regularity of \(Z\,\): if \(\,\sup_{x\in M}\limits \|K(x,{\cdot}) \|_{B^s_{\infty,\infty}} <\infty\,\) and \(\,0<\alpha<s/2\,\), then almost every trajectory of \(Z\) belongs to \(\,B^\alpha_{\infty,1}\,\); \ and conversely, the almost sure \(\,B^\alpha_{\infty,\infty}\)-regularity of \(Z\) entails \(\,\sup_{x\in M}\limits \|K(x,{\cdot}) \|_{B^{2\alpha}_{\infty,\infty}} <\infty\,\). This statement is completed by an Itô-Nisio representation of \(Z\), yielding a Wiener-like probability measure on \(\,B^\alpha_{\infty,1}\,\), and by a characterization of the above regularity condition on \(K\) in terms of the associated reproducing kernel Hilbert space\(\,\): \(H_K\subset B^{s/2}_{\infty,\infty}\). The particular case of \(M\) being homogeneous, and more particularly a sphere, is emphasized.
scientific article