Uniqueness and regularity for a system of interacting Bessel processes via the Muckenhoupt condition
The authors construct a strong Feller process with singular drift on the \(N\)-dimensional Weyl chamber NEWLINE\[NEWLINE\Sigma_N= \{x= (x^1,\dots, x^N)\in\mathbb{R}^N\mid x^0:= 0< x^1< x^2<\cdots< x^N< 1=: x^{N+1}\}NEWLINE\]NEWLINE equipped with the measure NEWLINE\[NEWLINEq_N(dx)= {1\over Z_\beta} \prod^N_{i=0} (x^{i+1}- x^i)^{{\beta\over N+1}} dx^1 dx^2\cdots dx^N,NEWLINE\]NEWLINE where \(\beta> 0\) is a free parameter and \(Z_\beta\) is a normalizing constant. The process is induced by the (pre-)Dirichlet form NEWLINE\[NEWLINE{\mathcal E}^N(f,f)= \int_{\Sigma_N}|\nabla f|^2(x) q_N\,(ds,\;f\in C^\infty(\overline\Sigma_N).NEWLINE\]NEWLINE It models a system of \(N\) particles with Coulomb interaction, performing an order preserving diffusion on the unit interval with reflecting boundaries and reflection between nearest neighbors. The process is a Euclidean semi-martingale if and only if the interaction is repulsive \((\beta\geq N+1)\), so that in this case regularity could be proved using the regularity criterion of \textit{M. Fukushima} [Electron. J. Probab. 4, Paper No. 18, 32 p. (1999; Zbl 0936.60067)].NEWLINENEWLINE In the case of attraction \((\beta< N+1)\), however, a method other than the construction via a system of Skorokhod SDEs had to be found. Noticing that in this case \(q_N\) can be extended to a measure on the full Euclidean space which lies in the Muckenhoupt class \({\mathcal A}_2\), the authors succeed by using the corresponding potential theory in Sobolev spaces with Muckenhoupt weights. The procedure resembles in parts the work of \textit{R. F. Bass} and \textit{P. Hsu} [Ann. Probab. 19, No. 2, 486--508 (1991; Zbl 0732.60090)], but the use of Muckenhoupt weights appears to be new.
scientific article; zbMATH DE number 6024132
scientific article