Markovian uniqueness on Bessel space (Q2703556)

In a previous paper [Stochastic Processes Appl. 83, 187-209 (1999)], the authors constructed an \(\mathbb{R}_+\)-valued 2-parameter Bessel process \(X= (X_{s,t})_{(s,t)\in \mathbb{R}^2_+}\) such that the \(C(\mathbb{R}_+, \mathbb{R}_+)\)-valued processes \((X_{.,t})_t\) and \((X_{s,.})_s\) are (essentially) Bessel processes of dimension \(d\) \((\text{BES}_d)\). An open problem was whether \(\text{Law}(X_{s,t})_{(s,t)}\) equals \(\text{Law}(X_{t,s})_{(s,t)}\) with permuted parameters. This problem has an affirmative answer once it is known that the Dirichlet generator \({\mathcal L}\) of \((X_{.,t})_t\) restricted to smooth cylinder functions enjoys the Markov uniqueness property. A symmetric linear operator \(S\) is called Markov unique, if the set of all extensions of \(S\) which lead to Dirichlet forms is a one-point set.NEWLINENEWLINENEWLINEThe central results of the paper are (1) a general criterion for Markov uniqueness and (2) its application to the above concrete situation: if \(d\geq 4\), Markov uniqueness holds for the two-parameter symmetric Bessel process. The uniqueness criterion is derived using a well known maximality property for Dirichlet forms. This is done in a rather general setting: any differential operator \(\partial\) on an \(L^2\)-space defines a symmetric operator \(S= -(1/2)\partial^*\partial\) and a (properly modified!) double dual \(D^\partial\) of \(\partial\). Then, in form sense \(\widehat S\leq -(1/2)(D^\partial)^* D^\partial\), for any extension \(\widehat S\) of \(S\) which induces a Dirichlet form. Markov uniqueness of \(S\) is then implied by \(\text{Dom}(D^\partial)= \text{Dom}(\overline\partial)\). This criterion is due to \textit{A. Eberle} [``Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators'' (1999; Zbl 0957.60002)]; the alternative proof in the present paper is, however, tailor-made with the particular application to the Bessel process in mind. It is an important feature that the authors provide concrete examples of Markov uniqueness for non-flat Dirichlet structures in infinite dimensions.NEWLINENEWLINENEWLINEIn order to check this abstract criterion in the above concrete situation the authors extend first the criterion to product structures and consider then the following two Dirichlet structures \((\nabla, C(\mathbb{R}_+,\mathbb{R}_+), \mu_d)\) consisting of the derivative \(\nabla\) belonging to \(\text{BES}_d\), i.e., \({\mathcal L}= -(1/2)\nabla^*\nabla\), and the law \(\mu_d\) of \(\text{BES}_d\) and the structure \(((\partial_t, \partial^W), \mathbb{R}_+\times W,m_d\times \mu^W)\) where \(\partial^W\) is the Malliavin derivative on the Wiener space \((W, \mu^W)\), and \(\partial_t\) is the usual derivative on \(\mathbb{R}_+\) with the symmetric initial measure of \(\text{BES}_d\). These structures can be related to each other in a way that respects the derivatives, and all considerations can be traced back to the situation on Wiener space.