##
**Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators.**
*(English)*
Zbl 0957.60002

Lecture Notes in Mathematics. 1718. Berlin: Springer. viii, 262 p. DM 74.00; öS 541.00; sFr 67.50; £25.50; $ 52.80 (1999).

Let \(E\) be an open subset of \(\mathbb{R}^d\). Consider a differential opertor on \(E\) of type
\[
Lf= \sum_{i,j=1}^d a_{ij} \frac{\partial^2f} {\partial x_i\partial x_j}+ \sum_{j=1}^d b_j \frac{\partial f}{\partial x_j}, \quad f\in{\mathcal A}, \tag{1}
\]
where \(a_{ij},b_j:E\to \mathbb{R}\) are measurable coefficient functions so that \((a_{ij})_{i,j}\) is symmetric and nonnegative definite, and \({\mathcal A}\) is a space of smooth functions (e.g. \({\mathcal A}= C_0^\infty(E)\)). Suppose that there exists a positive measure \(m\) on \(E\) and \(\alpha> 0\), such that
\[
\int Lf dm\leq \alpha\int f dm \quad \text{for all }f\in{\mathcal A},\;f\geq 0,\tag{2}
\]
where the integrals are assumed to exist. The monograph analyzes the question whether the operator \((L,{\mathcal A})\) has a unique extension \((\widehat{L}, D(\widehat{L}))\) which generates a \(C_0\)-semigroup \(T_t= e^{t\widehat{L}}\), \(t> 0\), on \(L^p (E,m)\) (provided, of course, \(p\in [1,\infty)\) is such that \((L,{\mathcal A})\) is an operator on \(L^p (E,m)\)).

One should note that analyzing the operator on \(L^p(E,m)\) rather than the classical \(L^p\)-space with respect to Lebesgue measure is in many cases much more convenient due to the fact that measures as in (2) (like e.g. infinitesimally invariant measures) are much more adapted to \(L\) than Lebesgue measure. There is a vast number of existence results for measures satisfying (2) for a given \(L\). The motivation for examining such kind of uniqueness questions for example comes from the interest in solving the Cauchy problem for \(L\) (whose solution is given by \(e^{t\widehat{L}}\), \(t> 0\)) or from probability theory, since in many cases \(e^{t\widehat{L}}\), \(t>0\), describes the transition probabilities of a Markov process. Non-uniqueness for such extensions \(\widehat{L}\) is, of course, well-known to come from different possible boundary conditions. But there are also two more causes for non uniqueness. One is the singularity of the coefficients and the other is nonfiniteness of dimension, i.e. the case \(d=\infty\). There are these two last aspects that this monograph focusses on, the first of which is geometrically motivated, while the second has important applications in the theory of stochastic (partial) differential equations and mathematical physics. Both aspects have been investigated in a large number of publications.

The monograph also surveys the state of the art concerning this fascinating problem, but mainly gives an abundance of new results, of which the following list gives a few instances:

– complete characterization (in terms of the coefficient) of uniqueness if \(d=1\) and \(m\) is a symmetrizing measure for \((L,{\mathcal A})\), i.e., \(L\) can be written in Sturm-Liouville form;

– development of a general probabilistic technique to produce examples of non-uniqueness for arbitrary \(d< \infty\), which gives deep insight as well as intuition;

– characterization of Markov uniqueness (i.e., uniqueness in the restricted sense that there exists at most one extension \((\widehat{L}, D(\widehat{L}))\) so that the \(C_0\)-semigroup \(T_t= e^{t\widehat{L}}\), \(t>0\), is sub-Markov, i.e. \(0\leq f\leq 1\Rightarrow 0\leq T_tf\leq 1\), \(t>0\)), in a number of finite-dimensional cases; the proof is based on the solution of the so-called maximality problem for Dirichlet forms, obtained in this monograph for the first time in such generality;

– first time ever counterexamples to Markov uniqueness in the infinite-dimensional case, disproving a number of conjectures formulated previously by other authors;

– a very clear presentation of two approaches to prove uniqueness in infinite dimensions with applications to models from mathematical physics;

– identification of a geometric structure associated with a general symmetric diffusion operator.

A number of background results are included in the text as three appendices to ease the reading.

One should note that analyzing the operator on \(L^p(E,m)\) rather than the classical \(L^p\)-space with respect to Lebesgue measure is in many cases much more convenient due to the fact that measures as in (2) (like e.g. infinitesimally invariant measures) are much more adapted to \(L\) than Lebesgue measure. There is a vast number of existence results for measures satisfying (2) for a given \(L\). The motivation for examining such kind of uniqueness questions for example comes from the interest in solving the Cauchy problem for \(L\) (whose solution is given by \(e^{t\widehat{L}}\), \(t> 0\)) or from probability theory, since in many cases \(e^{t\widehat{L}}\), \(t>0\), describes the transition probabilities of a Markov process. Non-uniqueness for such extensions \(\widehat{L}\) is, of course, well-known to come from different possible boundary conditions. But there are also two more causes for non uniqueness. One is the singularity of the coefficients and the other is nonfiniteness of dimension, i.e. the case \(d=\infty\). There are these two last aspects that this monograph focusses on, the first of which is geometrically motivated, while the second has important applications in the theory of stochastic (partial) differential equations and mathematical physics. Both aspects have been investigated in a large number of publications.

The monograph also surveys the state of the art concerning this fascinating problem, but mainly gives an abundance of new results, of which the following list gives a few instances:

– complete characterization (in terms of the coefficient) of uniqueness if \(d=1\) and \(m\) is a symmetrizing measure for \((L,{\mathcal A})\), i.e., \(L\) can be written in Sturm-Liouville form;

– development of a general probabilistic technique to produce examples of non-uniqueness for arbitrary \(d< \infty\), which gives deep insight as well as intuition;

– characterization of Markov uniqueness (i.e., uniqueness in the restricted sense that there exists at most one extension \((\widehat{L}, D(\widehat{L}))\) so that the \(C_0\)-semigroup \(T_t= e^{t\widehat{L}}\), \(t>0\), is sub-Markov, i.e. \(0\leq f\leq 1\Rightarrow 0\leq T_tf\leq 1\), \(t>0\)), in a number of finite-dimensional cases; the proof is based on the solution of the so-called maximality problem for Dirichlet forms, obtained in this monograph for the first time in such generality;

– first time ever counterexamples to Markov uniqueness in the infinite-dimensional case, disproving a number of conjectures formulated previously by other authors;

– a very clear presentation of two approaches to prove uniqueness in infinite dimensions with applications to models from mathematical physics;

– identification of a geometric structure associated with a general symmetric diffusion operator.

A number of background results are included in the text as three appendices to ease the reading.

Reviewer: M.Röckner (Bielefeld)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

31C25 | Dirichlet forms |

31-02 | Research exposition (monographs, survey articles) pertaining to potential theory |

60J60 | Diffusion processes |