On the structure of strong Markov continuous local Dirichlet processes (Q2769675)

A continuous strong Dirichlet process \(\mathbf Y = \{Y_t\}_{t\geq 0}\) is a continuous adapted process with decomposition \(Y_t = Y_0 + M_t + Q_t\) where \(\mathbf M = \{M_t\}_{t\geq 0}\) is a continuous \(L^2\)-martingale and \(\mathbf Q = \{Q_t\}_{t\geq 0}\) is a continuous adapted process of zero energy (i.e., all quadratic variational sums converge in \(L^1\) to \(0\) as the partitions refine). \(\mathbf Y\) is a local continuous strong Dirichlet process, if the stopped processes \(\{Y_t^{T_n}\}_{t\geq 0}\) are continuous strong Dirichlet processes for a suitable sequence of stopping times \(T_n \to \infty\). In this case the decomposition is still unique with a local martingale \(\mathbf M\) and a zero-quadratic variation process \(\mathbf Q\). The latter means that each \(\{Q_t^{T_n}\}_{t\geq 0}\) is of zero energy. Throughout the paper only state spaces \(I\subset \mathbb R\) are considered; \(R\) denotes the set of regular points of \(\mathbf Y\). A consequence of the above definition is the fact that every continuous strong local Dirichlet process \(\mathbf Y\) is invariant under a time change \(\mathbf \tau = \{\tau_t\}_{t\geq 0}\), provided that the process is \(\mathbf \tau\)-continuous, i.e., flat when \(\Delta\tau_t > 0\). NEWLINENEWLINENEWLINEThe aim of the paper is to find concrete formulae for the zero-energy part \(\mathbf Q\). Since \(\mathbf Y\) is, in general, not a semimartingale, the usual stochastic calculus techniques are not applicable. Under mild additional assumptions, \textit{S. Assing} and \textit{W. M. Schmidt} [``Continuous strong Markov processes in dimension one'' (1998; Zbl 0914.60008)] showed the existence of an increasing scale function \(p : I\to \mathbb R\) such that \(\{p(\mathbf Y)\}_{t\geq 0}\) is a semimartingale. If \(p^{-1}\) is sufficiently nice, the process \(\mathbf Y = p^{-1}\circ p (\mathbf Y)\) can be studied using stochastic analysis for the semimartingale \(p(\mathbf Y)\), in particular, the Bouleau-Yor generalization of Itô's formula. Following this route, the authors prove under some additional assumptions that the derivative of \(p^{-1}\) exists, \((p^{-1})' \in L^2_{\text{ loc}}(p(R))\), and \((p^{-1})'{\mathbf 1}_{p(R)} \in L^1_{\text{ loc}}(p(I))\). Moreover, \(p^{-1}\) has the following concrete representation \(dp^{-1} = (p^{-1})'{\mathbf 1}_{p(R)} dy + {\mathbf 1}_{p(I)\setminus p(R)}dy\). This allows to apply a generalized Bouleau-Yor formula [see \textit{J. Wolf}, ``Zur stochastischen Analysis stetiger lokaler Dirichletprozesse'' (1996; Zbl 0879.60061)] and to conclude that \(\mathbf Q\) is represented by a stochastic integral driven by the local time of \(\mathbf Y\) plus some deterministic functions of certain occupation times.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].