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\begin{document}
%====================END OF PREAMBULA=====================%
\centerline {\Large\bf Applications of Quantum Probability}
\medskip
\centerline{\Large\bf to Classical Stochastics}
\vspace{.25in}
\centerline{\sc Alexander M. Chebotarev}
\medskip
\centerline{Quantum Statistics Department, Moscow State University,}
\centerline{Moscow 119899, Russia}
\bigskip
\begin{itemize}
\item[]
New nonexplosion conditions for
Markov processes are derived from the general operator form of
the conservativity condition for a quantum dynamical semigroup.
\end{itemize}

\bigskip
     
\section{Introduction}

In this paper an operator-valued generalization of the Kolmogorov -- Feller
equation is discussed. The most remarkable feature of this generalization
is that it also covers diffusion and Heisenberg equations as particular
cases [L]. It is known that if intensity of jumps (or drift velocity, or
diffusion coefficient)as a function on the phase space $X$ of the
corresponding Markov process is unbounded, then the probability
measure of trajectories
hitting singularities of the coefficients may be positive. If there exist
a number of different
ways to define  the behaviour of the process after hitting a singular
point then the formal evolution equation for the
transition probability of the
Markov jump process has a linear manifold of solutions. In the class of
solutions $P(x,t|\Gamma)$ satisfying  the initial condition
$P(x,0|\Gamma)=I_{\Gamma}(x)$ there exists a minimal solution such that
$$
P^{min}(x,t|X) < 1
$$
(see [F]). A similar phenomenon exists for the
operator-valued generalization
of the Kolmogorov -- Feller equation. Examples of
Markov jump processes which explode or
escape to infinity with positive probability
are given in section 2 following [C1], [CF1].
In section 3 we describe an algebraic structure of operator-valued
extensions
of the basic evolution equations for Markov processes.
Section 4 contains domain assumptions for coefficients of
infinitesimal maps of semigroups in von Neumann algebras.
In section 5  we introduce
conditions which are necessary and sufficient for conservation of
the total probability by the minimal solution for evolution equation [C2].
We call them {\em conservativity conditions}. If the minimal solution
of the evolution equation is conservative (i.e. it preserves identity) then
the corresponding Markov process is called {\em regular}. Section 6
contains a constructive form of sufficient conservativity conditions and
give some applications to classical stochastic processes [C3], [CF2].
In Section 7 we present some new examples of regularity
conditions  derived for Markov processes from 
the general operator form of the conservativity condition for
a quantum dynamical semigroup.
\bigskip

\section{Conditions sufficient for the regularity of Markov jump processes}

Consider a Markov jump process in a Polish space $X$ with an abelian
group operation $\circ$. We denote by $\kappa(x)$ the intensity
of jumps from a point $x \in  X$  and by $m(B|x)$ we denote the distribution
of jumps from a point $x$:
$$
{\bf P}\{ value\, of\, the\, jump \in B, \,initial\, point\,
of\, the\, jump =x\}  = m(B|x).
$$
We consider $\kappa(x),\, m(B|x)$ as parameters of this Markov jump process.
If the intensity of jumps is unbounded, then the
probability measure of trajectories
producing an infinite number of jumps can be positive. Hence the
process escapes from any open set $D\in X$,
where the intensity is bounded by a positive probability
bounded below uniformly w.r.t. $D$.

Condition which is necessary and sufficient for regularity of such a process
was proved originally in [GS]. The Markov jump process with parameters
$\kappa(x),\, m(B|x)$ is regular if and only if
$$
{\bf P}\biggl\{\sum^{\infty}_{n=1} {1\over {\kappa(x_n)}}=\infty \biggr\} = 1,
\eqno (2.1)
$$
where $\{x_n\}^{\infty}_1$ is the Markovian chain with transition
probability  $m(B|x):$
$$
{\bf P}\{x_{n+1} \in B \circ x_n |x_n\}=m(B|x_n),\quad x_0=x.
$$
In nontrivial cases this condition is not constructive because it requires
bounds for the joint distribution for functions of dependent random variables
$x_n=q_n\circ \cdots \circ q_1 \circ x,$
where $q_k$ are random independent jumps. To derive constructive
conditions, one can note that (2.1) is equivalent to the following:
$$
\inf \limits_{\alpha \in (0,1]} \lim \limits_{m \to \infty}
{\m}{1\over 1+
\Bigl({\sum \limits^{m}_{n=1}
{1\over {\kappa(x_n)}}\Bigr)^{\alpha}}}  =0.
\eqno (2.2)
$$
The proof immediately follows from the Kolmogorov inequality 
(see [C1], Lemma 3.2 ).
The next observation has the form of nontrivial inequality:
$$
{\m}{1\over 1+
\Bigl({\sum \limits^{m}_{n=1}
{1\over {\kappa(x_n)}}\Bigr)^{\alpha}}} \le
{1\over 1+
\Bigl({\sum \limits^{m}_{n=1}
\bigl({1\over {{\m}\kappa(x_n)^\alpha}}\bigr)^{1\over \alpha}\Bigr)^{\alpha}}}
\eqno(2.3)
$$
([C1], Lemma 3.1). From this inequality we conclude that condition
(2.2) holds if
$$
\sup \limits_{\alpha \in (0,1]}
\lim \limits_{m \to \infty}
\Biggl({\sum \limits^{m}_{n=1}
{1\over\bigl( {{\m}\kappa(x_n)^\alpha}
\bigr)^{1\over \alpha}}\Biggr)^{\alpha}} = \infty
\eqno(2.4)
$$
([C1], theorem 3.1). So, condition (2.4) is sufficient
for the regular behaviour of the
Markov jump process with parameters $\kappa, m$. Case $\alpha <1$ is essential
when the function $\kappa(q)$ of a random jump $q: {\bf P}_x\{q
\in B\} =m(B|x)$
does not have a first moment.

The necessary condition follows from the Jenssen inequality
$$
{\m}{1\over 1+
\Bigl({\sum \limits^{m}_{n=1}
{1\over {\kappa(x_n)}}\Bigr)^{\alpha}}} \ge
{1\over \Bigl(1+
{\sum \limits^{m}_{n=1}
{\m}{1\over {\kappa(x_n)}}\Bigr)^{\alpha}}}.
\eqno(2.5)
$$
It follows from (2.5) that (2.2) holds only if the series diverges:
$$
\sum \limits^{\infty}_{n=1}
{\m}{1\over {\kappa(x_n)}}= \infty.
\eqno(2.6).
$$
This series is equal to  the  mean value of the time required to produce
an infinite sequence of jumps.

Criteria (2.4), (2.6) can be easily applied if jumps have a
space-homogeneous
stable distribution. Being applied to a space-homogeneous Markov jump
process with the Cauchy distributed jumps in $\r^n$
$$
m(B|x)={1\over \pi^n} \int_B \prod_i {dq_i\over (1+|q_i|^2)},
$$
criteria (2.4) shows that all processes with parameters $\kappa, m$
are regular whenever
$$
\kappa(x) \le C\,(1+|x|) \log (2+|x|).
$$
If
$$
\kappa(x) \ge C\,(1+|x|) \log^{1+\epsilon} (2+|x|), \quad
\forall \epsilon>0,
$$
then process escapes to infinity with a positive probability. The proof
for $n \ge 2$ was given in [C1].

For normally distributed jumps in $\r^n$
$$
m(B|x)= {1\over (2\pi)^{n/2}} \int_B  e^{-{|q|^2\over 2}} dq
$$
the Markov jump process is regular if
$$
\kappa(x) \le C\,(1+|x|)^2 \log (2+|x|).
$$
and unregular if
$$
\kappa(x) \ge C\,(1+|x|)^2 \log^{1+\epsilon} (2+|x|), \quad
\forall \epsilon>0,
$$
for dimensions $n \ge 3$ (see also [C1]).

      Another nontrivial example gives  the so-called Az\'ema-Emery
martingale (see [E1], [E2] for the basic properties) related to the
noncommutative generalization of the Brownian motion (see [P]
for details) with the infinitesimal map
$$
Lf(x)=(\beta x)^{-2}(f(cx)-f(x)- \beta x {d\over dx}f(x)),
\quad c=1+ \beta, \quad \beta \ne 0, \quad x\in \r.
\eqno(2.7)
$$
It describes scaled jumps with a singular drift. As we see later,
this process is regular if the following series
$$
\sum \limits_{n=1}^{\infty} c^{2n} \xi_{n+1} \prod \limits_{j=1}^n
(1+\xi_j)=\sum \limits_{n=1}^{\infty} {1\over \kappa_n},
\eqno(2.8)
$$
diverges with probability one, where
$\xi_j$ are independent random variables with distribution
$$
{\bf P}\{\xi \in B\}=sign(\beta)\,\int_Bd(1+\xi)^{1\over2\beta},
$$
where $\xi \in \r_+$ if $\beta<0$, and $\xi \in [0,1]$ if $\beta>0.$
In this particular case
$\kappa_n=\bigl(c^{2n} \xi_{n+1} \prod \limits_{j=1}^n (1+\xi_j)\bigr)^{-1}$
are positive random numbers. Since
$\xi_j$ are independent and identically distributed,
then the regularity condition for the Az\'ema-Emery
martingale is of the form
$$
\sum_n \biggl({\m}\kappa_n^{\alpha}\biggr)^{-{1\over \alpha}}=
\biggl({\m}\xi^{-\alpha}\biggr)^{-{1\over \alpha}} \sum_n \biggl((1+\beta)
(1-2\alpha \beta)^{1\over 2\alpha}\biggr)^{2n}= \infty,
\eqno(2.9)
$$
where $(1-2\alpha \beta)={\m}(1+\xi)^{-\alpha}$. Since
$\lim\limits_{\alpha \to 0}(1-2\alpha \beta)^{1\over 2\alpha}=exp(-\beta),$
the series (2.9) diverges for all $\beta<\beta^*$, where $\beta_*$ is
the critical point which is unique solution for equation
$(1+\beta)exp(-\beta)=-1.$
\vskip 0.3cm

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In [CF2] it was proved, that the Az\'ema-Emery martingale is not regular for
$\beta>\beta^*$. So,  the behaviour of the process at the critical point
$\beta^*$ is still unknown. Now we will introduce definitions which
explain the origin of conditions (2.4), (2.6) and
describe an operator-valued extension of these conditions.


\section{Operator-valued extension of the Markov evolution}

Consider a locally compact abelian algebra $X$ with an invariant Haar measure
$\mu$ and let ${\cal H}={\cal L}_2(X,\mu)$. Denote by
${\cal B}({\cal H})$ the algebra of all
linear bounded operators in ${\cal H}$.
The algebra $\ca$=${\cal L}_{\infty}(X,\mu)$ of
essentially bounded functions can be considered as an abelian subalgebra
in ${\cal B}({\cal H})$ consisting of multiplication by
functions from ${\cal L}_{\infty}$. The transition
probabilities $P(x,t|\Gamma)$
are natural elements of ${\cal L}_{\infty}$, the corresponding
operators from $\ca$ will be denoted by $P_t(I_{\Gamma}),$ where $I_{\Gamma}$
is another element from $\ca$ corresponding to multiplication by
the indicator function $I_{\Gamma}(x).$  In this way
$$
P_t(I_{\Gamma}) \psi (x)=P(x,t|\Gamma) \psi(x) \quad \psi(x)\in {\cal H}.
$$   From this point of view the main linear operations over transition
probabilities on $X=\r^n$ induce algebraic
operations in ${\cal B}({\cal H})$ as follows.
\begin{itemize}
\item  {\em Shifts}:  \quad $P(x+q,t|\cdot)
\Longleftrightarrow D_q^* P_t(\cdot) D_q,$
where $D_q \psi (x)=\psi(x-q)$
and $D_q^*$ is the adjoint unitar operator in
${\cal H}={\cal L}_{2}(\r^{n});$

\item  {\em Scalings}: \quad $P((1+c)x,t|\cdot)
\Longleftrightarrow S_c^* P_t(\cdot) S_c,$
where $S_c \psi (x)=   (1+c)^{-{n\over 2}} \psi((1+c)^{-1}x)$
and $S_c^* \psi (x)=
(1+c)^{n\over 2} \psi((1+c)^{n\over 2}x)$ is the
adjoint unitar operator in ${\cal H}={\cal L}_{2}(\r^{n});$

\item  {\em First derivatives}: \quad $(b(x),\nabla)P(x,t|\cdot)
\Longleftrightarrow i[H_b,P_t(\cdot)]$,
where $H_b=-{i\over 2}\bigl((b(x),\nabla)+(\nabla,(b(x))\bigr)$
is the symmetrical  first-order differential
operator, and $[\cdot,\cdot]$ is the commutator.
In the  particular case
$b(x)=(\beta^2_1(x),\cdots,\beta^2_n(x))$ the  operator $H$
has another equivalent form:
$$
H=H^*=-i \sum_k \beta_k(x) {\partial \over \partial x_k} \beta_k(x),
$$
that is $H\psi(x)=
-i \sum \beta_k(x)
\left({\partial \over \partial x_k} \beta_k(x) \psi(x)\right).$

\item {\em Second derivatives}: \quad ${1\over2}(\nabla,A(x)\nabla)
P(x,t|\cdot)\Longleftrightarrow
\Phi(P_t(\cdot))-{1\over 2}\{\Phi(I),P_t(\cdot)\}={\cal L}(P)$,
where $\{\cdot,\cdot\}$ is anticommutator and  $\Phi(P)= -
(\nabla,A^{1\over 2}(x)PA^{1\over 2}(x)\nabla).$

\end{itemize}

For example, in one-dimensional case we have:
$$
i[H,P]\psi= \beta (\beta P \psi)'-P\beta (\beta\psi)'=
\beta (\beta'P\psi +\beta P'\psi+\beta P \psi ')-
P\beta (\beta'\psi +\beta\psi') = (\beta)^2 P' \psi,
$$
where $H$ is the first-order differential operator described above, and
$$
{\cal L}(P)\psi=-(aPa\psi ')' +{1\over2}(a^2(P\psi)')'+{1\over2}P(a^2\psi')'=
-a^2(P\psi ''+P' \psi ')-2aa'P\psi'+
$$
$$
+{1\over 2}a^2(P'' \psi +2P' \psi ' + P\psi '')+aa'(P'\psi+P\psi')+
{1\over2}Pa^2\psi''+Paa'\psi'=
$$
$$
=aa'P'\psi+{1\over2}a^2P''\psi= {1\over2}(a^2P')'\psi.
$$
Multidimensional cases are similar to these examples.

The extension of the infinitesimal operator (2.7) to an operator map reduces
to  the cases considered above:
$$
LP(x,t|\cdot) \Longleftrightarrow{\cal L}(P_t(\cdot))=
\Phi(P_t)-{1\over2}\{\Phi(I),P_t\}+i[H,P_t],
$$
$$
\Phi(P)={1\over \beta x} S_c^* P S_c {1\over \beta x}, \quad
H= {i\over2}({1\over \beta x}\partial_x +
\partial_x{1\over \beta x}\partial_x).
\eqno (3.1)
$$

In what follows, we shall use closure of $\Phi(\cdot)$ and $H$ as
 the notations for operator-valued
coefficients of the infinitesimal map ${\cal L}$, and the
dissipative operator
${1\over2}\Phi(I)+iH$ will be denoted by $G$. Operator $H$ is supposed to be
symmetric. So formally $G^*={1\over2}\Phi(I)-iH$ and hence
$$
{\cal L}(P)=\Phi(P)-G^*P- PG, \quad P\in {\cal B}({\cal H}),
\quad G={1\over2}\Phi(I)+iH.
\eqno(3.2)
$$

Similar  but  more  complicated  computations  put   an
infinitesimal operator $L$ of the general Markov process
$$L={1\over 2}(\nabla ,A(x)\nabla )+(b(x),\nabla )+
\int_{R^n\backslash\{0\}}
\eta (dq)|C(q,x)|^2(D^*_q-1-(q,\nabla ))+$$
$$\int_{R\backslash\{0\}}\nu(dr)|E(r,x)|^2
((1+r)^{-{1\over2} }S^*_r-1-r(x,\nabla ))$$
into correspondence with an
infinitesimal  map ${\cal L}(\cdot )$ such  that ${\cal L}(P)(x)=LP(x)$ on
 the set  of bounded smooth functions:
$$
{\cal L} (B)=\Phi(B)-{1\over2}\{\Phi(I),B\}+i[H,B], \;\;\Phi(\cdot
)=\sum^4_0\Phi_i(\cdot ),  \;\;{\cal H}=\sum^2_0{\cal H}_i,
\eqno(3.3)
$$
where
\begin{eqnarray*}
H_0&=&\int_{0<|q|\leq 1}\mu(dq)C^*(q,\cdot )\left( \sin({1\over
i}(\nabla ,q)) -{1\over i}(\nabla ,q)\right)C(q,\cdot ),\\
H_1&=&\int_{0<|r|\leq 1}\nu(dr)E^*(r,\cdot
)\left({S^*_r-S_r\over 2i}-{r\over i}(x,\nabla ) \right)E(r,\cdot ),\\
H_2&=&{1\over 2i}\left((V(\cdot),\nabla )+(\nabla ,V(\cdot
)\right), \\
V(x)&=&b(x)- \int_{|q|> 1}\mu(dq)q|C(q,x)|^2-
x\int_{|r|> 1}\nu(dr)|E(r,x)|^2,\\
\Phi_0(B)&=&\int_{0<|q|\leq 1}\mu(dq)C(q,\cdot )^*
(D_q^*-I)B(D_q-I)C(q,\cdot ),\\
\Phi_1(B)&=&\int_{0<|r|\leq 1}\nu(dr)E(r,\cdot )^*
(S_r^*-I)B(S_r-I)E(r,\cdot ),\\
\Phi_2(B)&=&-(\nabla ,A^{1/2}(x)BA^{1/2}(x)\nabla ), \\
\Phi_3(B)&=&\int_{|q|> 1}\mu(dq)C(q,\cdot )^*D^*_qBD_qC(q,\cdot ),\\
\Phi_4(B)&=&\int_{|r|> 1}\nu(dr)E(r,\cdot )^*S^*_rBS_rE(r,\cdot ).
\end{eqnarray*}

All maps denoted above by $\Phi_k(\cdot)$ belong to the class of so-called
{\em completely positive} linear maps in operator algebra. A formal
definition of a
completely positive (CP) map has several equivalent forms.
In the class of contractive maps the characteristic property
of complete positivity is given by the inequality
$$
Q(X^*X)\ge Q(X^*)Q(X)
$$
(see [L]), where $X\in {\cal B}({\cal H})$.  The CP-property is equivalent to
$$
           \sum_{ij}(\psi_i,\Phi(X_i^*X_j)\psi_j) \ge 0
$$
in the class of bounded maps
for arbitrary finite sequences
$\{\psi_i\}\in {\cal H},\;\{X_i\}\in{\cal B}({\cal H})$
(see [S]).

      A CP-map $Q(\cdot):{\cal B}({\cal H}) \to {\cal B}({\cal
H})$ is called {\em normal} if
$l.u.b._n Q(X_n)=Q(l.u.b.X_n)$ for any bounded increasing sequence
$X_n$ from ${\cal B}({\cal H})$. Here $l.u.b.$ means 
{\it the least upper bound}.
The set of all normal bounded CP-maps on ${\cal B}({\cal H})$
for a separable Hilbert space ${\cal H }$ is described by the Kraus
representation theorem:
$$
\Phi(B)= \sum \limits_k A^*_k\,B\,A_k,\quad \sum \limits_k A^*_k A_k
\in {\cal B}_+({\cal H})
$$
(see [K]).

A remarkable fact discovered by G.Lindblad in 1976 is that infinitesimal
maps of a completely positive norm continuous semigroups have structure
described by (3.2), where $\Phi(\cdot)$ is a completely positive map
and $H$ is a symmetrical operator. So the operator-valued extension of the
Markov evolution equation has the form of the Lindblad's evolution equation:
$$
{d\over dt}P_t(B)={\cal L }(P_t(B)),\quad B \in {\cal B}({\cal H}),
$$
where ${\cal L }$ is  the infinitesimal map (3.2). A solution for the Lindblad
equation is called {\em a dynamical semigroup}. $P_t(\cdot)$ is strongly
continuous in $t \in \r_+$, normal, contractive and CP w.r.t. operator
argument. A dynamical semigroup is called
{\em conservative } if $P_t(I)=I.$
This property generalizes  total probability $P(x,t|X)=1$ conservation law
by solutions of the Markov evolution equations. Despite the
formal property of the map ${\cal L(\cdot)}$
$$
{\cal L }(I)=0,\quad P_t(I)=I+\sum_k {t^k\over {k!}}{\cal L}(\dots{\cal L}(I))=I,
$$
which gives  conservative solutions for equations with bounded coefficients,
in the general case
the conservativity of an evolution can be violated as well as
the conservation law for total probability
 in the case of the Kolmogorov-Feller or diffusion equations
with unbounded coefficients. It is known that the total probability is
conserved by the Markov evolution if the minimal solution for the
evolution equation enjoys this property [F]. We prove a similar result in
the theory of dynamical semigroups.


\section{Main assumptions}

If    $P_t(\cdot)$ is a solution of the Lindblad evolution equation
with bounded coefficients, then it  also satisfy the following integral
equation, which one can consider as a Duhamel equation:
$$
P_t(B) =W_t^*BW_t + \int_0^t ds W_{t-s}^*\Phi(P_s(B))W_{t-s},
\eqno(4.1)
$$
where $W_t=exp(-Gt)$ is a contractive semigroup in
${\cal H}$ with  generator $G={1\over2}\Phi(I)+iH$.
The sequence $P_t^{ (n)}(B)$ such that $P_t^{ (0)}(B)=0$,
$$
P_t^{ (n)}(B) =
W_t^*BW_t + \int_0^t ds W_{t-s}^*\Phi(P_s^{ (n-1)}(B))W_{t-s},
\eqno(4.2)
$$
increases monotonically  and is uniformly bounded w.r.t. arbitrary
bounds for $\Phi(\cdot)$:
$$
P_t^{ (n)}(B)\le ||B||I.
$$
We use this observation to construct the minimal solution of the Lindblad
equation with unbounded coefficients.

Suppose that there exists a strongly continuous contractive semigroup $W_t$
with the infinitesimal operator $-G$ such that for some $N\ge 2$
$$
G\phi =iH\phi + {1\over2}\Phi(I)\phi,\quad H\phi=H^*\phi
\quad \forall \phi \in dom\, G^N,
\eqno(4.3)
$$
where $H$ is a densely defined symmetric operator and $\Phi(I)$ is
a positive selfadjoint operator such that $\Phi(I)\ge I.$ Suppose also that
$$
ess\, dom\,\Phi(I) \subseteq dom\,G^N \subseteq dom \,H\cap  \Phi(I)
\eqno(4.4)
$$
$$
dom\, G \subseteq  \Phi(I)^{ {1\over2}}.
\eqno(4.5)
$$

Note that ${\cal L}(\cdot)$ is invariant under shifts of the coefficient
$\Phi(\cdot) \to \Phi(\cdot)+ \lambda I$, so our assumption  $\Phi(I)\ge I$
is not restrictive. If
$$
i[\Phi(I),H] \ge -c\Phi(I) \quad  on \; dom \,G^N
\eqno(4.6)
$$
then (4.5) holds also. (4.4) is The most restictive assumption  is (4.4).
Anyway it holds if $H$ is relatively bounded w.r.t. $\Phi(I).$

It is known that there exists one to one correspondence between densely
defined bounded from below closable quadratic forms and
selfadjoint operators bounded
below . We shall denote by $\Phi_*(B)[\cdot]$
the quadratic
form generated by the selfadjoint operator $\Phi(B)$ and by
$\lbrack\!\lbrack \Phi_*(B) \rbrack\!\rbrack$ - the selfadjoint operator
corresponding to the closed quadratic form $\Phi_*(B)[(\cdot)]$.

Let $\b \in  {\cal H} $ be a dense Banach subspace in ${\cal H}$ such that
$$
\b \subseteq dom \,\Phi_*(I), \quad ||\cdot||_{\b} \ge ||\cdot||_{\cal H}
\eqno (4.7)
$$
$$
W_t \;is\; uniformly\;bounded\; and\;strongly\;continuous\; in\;\b
\eqno (4.8)
$$
$$
\Phi_*(\cdot)\in CPn_*(\b),
\eqno(4.9)
$$
where $CPn_*(\b)$ is a completion of $CPn_*(\cal H)$ w.r.t. locally convex
topology induced by seminorms
$$
p_{A,B}(\Phi_*)= \sup \limits_{X\in A,\,\phi \in B}|\Phi_*(X)[\phi]|.
\eqno(4.10)
$$
Here ${\cal A}\in {\cal B}({\cal H}), B\in \b$ are absolutely convex compact
subsets and $CPn({\cal H})$ is a set of completely positive normal maps,
$CPn_*(\cal H)$ is the positive cone of the corresponding quadratic forms:
$$
Q_*(B)[\phi]=(\phi,Q(B)\phi),\quad B\in {\cal B}({\cal H}), 
\quad \phi \in {\cal H}.
$$

      The main examples of $A$ and $B$ give absolutely convex
linear spans of  compact sets
$$\{W_{t-s}\phi\}_{s\in [0,t],\;\phi \in {\cal H}}\,\in {\cal H},\qquad
\{P_s(X)\}_{s\in[0,t],\;X\in {\cal B}({\cal H})} \in {\cal B(\cal H)}.$$
The main examples of $\b$ are $dom \,G^N$ and
$dom \,\Phi(I)^{1\over2}$ endowed
with the graphic norms. If for example $\b =dom G^N$ then (4.7) follows from
(4.6), and (4.8) always holds . We need then to check only (4.9)
which becomes the only restrictive assumption.

If $\b=dom\, \Phi(I)^{1\over2}$, then $\Phi_*(\cdot)\in CPn_*(\b)$,
and we need to need to check (4.8) which holds if we require the
relative bound (4.6).

Here are some important examples of maps
$\Phi_*(\cdot)\in CPn_*(\b)$ for $\b=dom \,\Phi(I)^{1\over2}$.
\begin{itemize}
\item[(i)]
$$
\Phi_*(B)[\phi]=\sum\limits_1^{ \infty}(a_k\phi,Ba_k\phi),
$$
where $\{a_k\}$ is a sequence of closed operators with a dense joint domain
${\cal D} \subseteq \cap_k dom\, a_k$ such that $\sum_k||a_k\phi||^2 <\infty
\quad \forall \phi \in \cal D.$
\item[(ii)]
$$
\Phi_*(B)[\phi]=\int\limits_{ \Omega}\mu(d\omega)
(a_{\omega}\phi,Ba_{\omega}\phi),
$$
where $\{a_{\omega}\}$ is a sequence of closed operators with a dense joint
domain
${\cal D} \subseteq \cap_{\omega} dom\, a_{\omega}$ such that
$||Xa_{\omega}\phi|| \in {\cal L}_2(\Omega,\mu) \quad \forall X 
\in {\cal B(\cal H)}
\quad \forall \phi \in \cal D.$
\end{itemize}

These examples include the following maps:
\begin{itemize}
\item [(1)]
$$
\Phi_*(B)[\phi]=(A^{1\over2}(\cdot)\nabla\phi,BA^{1\over2}(\cdot)\nabla\phi),
$$
where $A(x) $ is a matrix with a.e. finite measurable
coefficients and $A(x)>0$ uniformly on each compact set in $\r^n$;
\item [(2)]
$$
\Phi_*(B)[\phi]=\int\limits_{\r^n}\mu(dq)
(D_q C(\cdot,q)\phi(\cdot),BD_q C(\cdot,q)\phi(\cdot)),
$$
where $\int\limits_{\r^n}\mu(dq) |C(\cdot,q)|^2 =\kappa(x)< \infty$
on each compact set in $\r^n$.
\item [(3)]
$\Phi_*(B)[\phi] ={1\over \beta x} S^*_cBS_c {1\over \beta x}$, where
$\beta =1+c,$ and $S_c$ is defined by (2.7),(3.1).
\end{itemize}

\noindent{\bf Theorem 4.1.} {\em If assumptions (4.3-4.5),(4.7-4.9) hold, then
\begin{itemize}
\item[$1^{ \circ}.$] There exist the minimal
dynamical semigroup $P_t^{ min}(\cdot)$ such that
$$P_t^{ min}(B)=l.u.b.P_t^{(n)}(B) \quad \forall B\in {\cal B_+(H)}$$.
\item[$2^{ \circ}.$] $P_t^{ min}(B)$ satisfies the integral Lindblad equation
$$
P_t^{ min}(B)=W_t^*BW_t +
\biggl[\!\!\biggl[ \int_0^t ds W_{t-s}^*\Phi_*(P_s^{ min}(B))W_{t-s}
\biggr]\!\!\biggr].
\eqno(4.10)
$$
\item[$3^{ \circ}.$] $P_t(B)-P_t^{ min}(B)\ge 0,\quad \forall B\ge 0$
for any other dynamical semigroup $P_t(B)$ which satisfies (4.10).
\item[$4^{ \circ}.$] If $P_t^{ min}(\cdot)$ is conservative then
$P_t^{ min}(\cdot)$
is
the only solution of equation (4.10) in the class of dynamical semigroups.
\end{itemize}     }

Brakets $[\![  \cdot ]\!]$ in (4.10)  are used to correspond a bounded
operator to  densely defined quadratic form
$$
\int _0^t ds \Phi_*(P_s^{ min}(B))[W_{t-s}\psi]
\leq||B||(||\phi||^2-||W_t\psi||^2) \qquad \forall \psi \in dom \Phi(I).
$$


\section {Necessary and sufficient conditions for conservativity
of a dynamical semigroup}

The integral equations (4.2) or (4.10)
for a dynamical semigroup $P_t(\cdot)$  imply
the corresponding integral form of the resolvent
equation
$$
X=A_{\lambda}(B) + Q_{\lambda}(X),\quad X=
\int\limits_0^{\infty}e^{ -\lambda t}P_t(B)\,dt,\quad \lambda \in \r_+,
\eqno(5.1)
$$
where
$$
A_{\lambda}(B)=\int\limits_0^{\infty}e^{ -\lambda t}W^*_tB W_t\,dt,
\qquad
Q_{\lambda}(B)=\biggl[\!\!\biggl[
\int\limits_0^{\infty}e^{ -\lambda t}W^*_t\Phi_*(B) W_t\,dt\biggr]\!\!\biggr].
$$

Values of the infinitesimal map ${\cal L}$ on the
range of the resolvent map $\cal R$
consists of bounded operators.
Since the range ${\cal R}$ of the resolvent map does not depend on
the parameter
$\lambda$ then the explicit equation for
the range
of the minimal resolvent map follows from (5.1) 
and from the definition of the minimal
dynamical semigroup $P_t^{ min}(\cdot):$
$$
{\cal R}^{ min}= \left\{X: X=
\int_0^{\infty}e^{ -t}P_t(B)\,dt=
\sum\limits_0^{ \infty}Q^n(A(B))\quad \forall B \in {\cal B(H)}\right\},
\eqno(5.2)
$$
where $A(\cdot)=A_1(\cdot),\,\,Q(\cdot)=Q_1(\cdot)$. Hence $I \in dom\,
{\cal L}^{ min}$
if and only if $I\in {\cal R}^{ min}$. The conservativity of the minimal
dynamical semigroup depends on this property.

      {\bf Theorem 5.1.}(see [C2], [C3]) {\em Under the
assumptions of Theorem 4.1 the
following are
equivalent:
\begin {itemize}
\item[1.] $P_t^{ min}(\cdot)$ is conservative.
\item[2.] $I\in {\cal R}^{ min}.$
\item[3.] $Q^n(I)\to 0$ strongly as $n\to \infty$.
\item[4.] Equation $Q(B)=B$ does not have solutions in ${\cal B}_+({\cal H})$.
\item[5.] Equation ${\cal L}(B)_*=B_*$ does not have positive bounded
solutions on dom $G^N$.
\end{itemize}}
Condition 3 is the most efficient.
Consider an application of this condition to the regularity problem for
the Az\'ema-Emery martingale. Now we are able to explain the origin of
the series (2.8).
As we observed it in Section 3 the infinitesimal
operator generated by this process can be extended to the infinitesimal
map ${\cal L}(\cdot )$ of the dynamical semigroup with coefficients
$$
\Phi(B)={1\over \beta x} S_c^* B S_c {1\over \beta x}, \quad
H= {i\over 2}({1\over \beta x}\partial_x +
\partial_x{1\over \beta x}\partial_x).
$$

Hence $G={1\over 2}\Phi(I)+iH={1\over 2x^2}{1+\beta\over \beta^2}
-{1\over\beta x}{\frac{\partial}{\partial x}}$, and
semigroups $W_t=exp{(-Gt)},
\,\,\,W^*_t=exp{(-G^*t)}$
can be constructed explicitly by method of characteristics
as solutions for the first order linear PDEs:
$$(W_tf)(x) =\left(1+{2t\over\beta x^2}\right)^{-{1+\beta\over 4\beta}}f
\left(x\sqrt{1+{2t\over\beta x^2}}\right)I_{\Gamma_{t}}(x)$$
$$(W^*_tf)(x) =\left(1-{2t\over\beta x^2}\right)^{{1-\beta\over 4\beta}}f
\left(x\sqrt{1-{2t\over\beta x^2}}\right)I_{\Gamma_{-t}}(x),$$
where $\beta\in\r\backslash\{0\},\,\Gamma_t=\left\{x\in\r :x^2\geq -
{2t\over\beta}\right\}$
and $I_{\Gamma_t}(\cdot)$ is the indicator function of $\Gamma_t$.

In this case, the map $Q$ leaves invariant the space
$L_\infty(\r ,dx)$ and, since
$1\in L_\infty(\r ,dx)$
it is sufficient for us to know how does $Q$ act on
bounded functions.

A direct computation shows that, for $f\in L_\infty (\r ,dx)$
\quad $(Qf)(x)=$
$$\int\limits^\infty_0\!\!dte^{-t}\left(\!1-{2t\over\beta x^2}
\right)^{{1-\beta\over 4\beta}}
{f(cx(1-{2t\over\beta x^2})^{1\over2})\over\beta^2x^2(1-{2t\over\beta x^2})}
\left(\!1+{2t\over\beta x^2(1-{2t\over\beta x^2})}
\right)^{-{1+\beta\over 4\beta}}\!\!\!\!I_{\Gamma_{-t}}(x)
\dps I_{\Gamma_t}\left(x(1-{2t\over\beta x^2})^{1\over2}\right)
$$

Note that $I_{\Gamma_t}(x\sqrt{1-{2t\over\beta x^2}})
\equiv 1$ and,
when $\beta <0$, we have $I_{\Gamma_{-t}}(x)\equiv 1$.
In the case $\beta <0$, with the change of variables $s=-{2t\over\beta x^2}$
we obtain
$$
(Qf)(x)=\int^\infty_0e^{{\beta s x^2\over 2}}f\left( cx\sqrt{1+s}
\right)d(1+s)^{{1\over 2\beta}}=
E e^{{\beta{\xi}x^2\over 2}}f(cx\sqrt{1+{\xi}}),
$$
where ${\xi}$ is a positive random variable with the following
distribution
$$
P({\xi}\leq s)=1-(1+s)^{{1\over 2\beta}} \qquad (s\geq 0).
$$

In the general case we have
$$
Q^n(1)=\m\exp{{\beta x^2\over 2}}\left\{{\xi}_n+{\xi}_{n-1}
(1+{\xi}_n)c^2+\cdots +{\xi}_1(1+{\xi}_2)
\cdots (1+{\xi}_n)c^{2(n-1)}\right\}
$$
It is convenient to replace the numbering of random variables
to obtain the explicit expression for $Q^n(I)$:
$$
Q^n(1)=\m\exp{{\beta x^2\over 2}}\left\{\sum^n_{k=1}c^{2k}
{\xi}_{k+1}\prod^n_{\ell =1}
(1+{\xi}_\ell)\right\}
\eqno(5.3)
$$

Now it follows from (5.3)
that $Q^n(1)\mathop{\longrightarrow} 0$ as ${n\to\infty}$ if and only if
$$
P\left\{\sum^\infty_{k=1}c^{2k}{\xi}_{k+1}
\prod^k_{\ell =1}(1+{\xi}_\ell )
=\infty\right\}=1.
$$
This condition coincides with our previous requirement of
divergence of (2.8). Now let us consider a constructive form of conditions
sufficient for the conservativity.

\section{Review of conditions sufficient for the
conservativity of a minimal dynamical
semigroup}

As  was noticed in [C2] for a given CP-map $Q$, there exists a
positive selfadjoint operator $S\in {\cal C(H)}, \,S\ge I$ such that
$$
Q(I)=(I+S^{-1})^{-1},
\eqno(6.1)
$$
which plays essential role in what follows. In particular
$S=\kappa(\cdot)$=\{{\em multiplication by intensity of jumps\}} for the map
$Q{\cal }$ corresponding to an extension of the Kolmogorov-Feller evolution
equation, and  $S =-(\nabla,A(\cdot)\nabla)$ for an
extension of the symmetrical diffusion equation with diffusion coefficients
$A(x) =\{A_{ij}(x)\}$.

For any contractive CP-map $Q$ and for any
sequence of positive bounded and invertable operators $X_1,\dots,X_n$
such that operators $Q(I)^{ -1},Q(X_k)^{ -1},\, k\in\{1,
\dots,n\}$ exist, we have proved the inequality
$$
Q((I+X_1^{ -1}+\dots +X_n^{ -1})^{ -1}) \le
(Q(I)^{ -1}+ Q(X_1)^{ -1}+ \dots + Q(X_n)^{ -1})^{ -1}
\eqno(6.2)
$$
(see [C2]).

Formal iterations of (6.1), (6.2) give: \qquad $Q^n(I)=$
$$
=Q^{ n-1}((I+S^{-1})^{-1})\le
Q^{ n-2}((Q(I)^{ -1}+ Q(S)^{ -1})^{-1})=
Q^{ n-2}((I+S^{-1}+ Q(S)^{ -1})^{ -1})\le
$$
$$
\le Q^{ n-3}((Q(I)^{ -1}+ Q(S)^{ -1}+ Q^2(S)^{ -1})^{ -1})\le
\cdots\le (I+S^{-1}+ Q(S)^{ -1}+\cdots+ Q^n(S)^{ -1})^{ -1}.
$$
Hence $Q^n(I)\to 0,\,n \to \infty$  in the strong sense if
$$
(I+S^{-1}+ Q(S)^{ -1}+\cdots+ Q^n(S)^{ -1})^{ -1} \to 0
\eqno(6.3)
$$
 also converges strongly.

Regularized version of (6.3) is given by the following lemma.

{\bf Lemma 6.1.}[C2,C3] {\em A quantum dynamical semigroup with
the contractive CP-map $Q(\cdot)$ corresponding to the
infinitesimal map ${\cal L}(\cdot)$ is conservative if
$$
\sup\limits_{n} \inf_{\lambda \in (0,1]}\sum\limits_{1}^{ n}
(\phi,Q^n_{\lambda}(S_\lambda)\phi)^{ -1}=\infty\quad
\eqno(6.4)
$$
$\forall \phi$ from an arbitrary dense subset $D\in {\cal H},$ where
$S_{\lambda}=S(1+\lambda S)^{ -1}\in {\cal B}_+({\cal H})$.}

The operator $S$ in inequality (6.4) can be replaced by the operator
$\Phi(I)$ under the following restriction.

{\bf Lemma 6.2.}[C2,C3] {\em Let
$$
i(H\phi,\Phi(I)^{ -1}\phi)-i(\Phi(I)^{ -1}\phi,H\phi) \ge
-c||\phi||^{ 2}\quad \forall \phi \in dom \,{G}^N
\eqno(6.5)
$$
or  $i[H,\Phi(I)^{ -1}]\ge -cI$ on $dom\,{G}^N.$
Then $S\le(c+2)\Phi(I)$, and $ S_{\lambda}\le (c+2)\Phi_{\lambda}(I).$}
\vskip3mm
The last simplification follows from the assumption described in the next
theorem.

{\bf Theorem 6.3.} {\em Suppose that $\sup\limits_{\lambda \in (0,1]}
\Phi_*(\Phi_{\lambda}(I))[\phi]<\infty,\quad \forall \phi \in dom \,{G}^N.$
If there exists a constant $c\in \r$ such that for any $\lambda \in (0,1]$
$$
{\cal L}(\Phi(I))[\phi]=\sup\limits_{\lambda \in (0,1]}
\Phi_*(\Phi_{\lambda}(I))[\phi]-(G\phi,\Phi(I)\phi)-
(\Phi(I)\phi),G\phi)\le c\Phi_*(I)[\phi],
\eqno(6.6)
$$
then $\Phi(I)-Q(\Phi(I)) \ge -(c-1)Q(I)$ and the series (6.4) diverges.}

Under the assumptions of this theorem we have:
$$
Q^n(\Phi_{\lambda}(I))\le
Q(Q\dots(Q(\Phi_{\lambda_1}(I)))_{\lambda_2}
\dots)_{\lambda_n}\mid_{\sum \lambda_k=\lambda}\quad\le
$$
$$
\le Q(Q\dots(Q(\Phi_{\lambda_2}(I)+(c-1)I))\dots)_{\lambda_n}\le\dots\le
\Phi_{\lambda_n}(I)+(c-1)nI.
$$
Hence $(\phi,Q^n(S_\lambda)\phi)\le(\phi,(\Phi(I)+(c-1)nI)\phi $ and
$Q^n(I)=O(\log n)^{ -1}.$
Thus, we obtain the following result.

{\bf Theorem 6.4.} {\em Let the assumptions (4.3)--(4.5), (4.7)--(4.9), (6.4),
(6.6) be fulfilled. Then the minimal dynamical semigroup is conservative.

Let an infinitesimal map ${\cal L}(\cdot)$ be the extension (3.3)
of an infinitesimal operator $L$ of a Markov semigroup. Then the
Markov evolution
is regular if the corresponding minimal dynamical semigroup is conservative.}

\section{Examples}

(1)  Let
${\cal H}={\cal L}_2(\r^n),\quad H=-{i\over 2}\bigl((b(x),\nabla)
+(\nabla,(b(x))\bigr)$ and
$$
\Phi(B)=\int_{\r^n}m(dq)C^*(\cdot,q) D^*_qBD_qC(\cdot,q),
$$
where $m$ is a $\sigma$-finite measure on $\r^n$ and
$C(\cdot,\cdot):\r^{ 2n}:\to\c$ is a measurable function such that
$C(\cdot,q)\in {\cal L}_2(\r^n,m)$ and
$$
\kappa(x)=\int m(dq)|C(\cdot,q)|^2
$$
is a smooth locally bounded function. Suppose that there exists
the family of trajectories $q_{\tau}:\,\dot q_{\tau}=b(q_{\tau}),\,\, q_0=y,$
such that for each point $x\in\r^n$ there exist a finite or infinite
return time from infinity $T_{\infty}(x)$ such that
$$
\tilde q_0=x, \,\,\dot{\tilde q_{\tau}}
=-b(\tilde q_{\tau}),\,\lim\limits_{t\to T_{\infty}(x)}|\tilde q_t|=\infty
$$
for reversed trajectories $\tilde q_{\tau}$. Denote by $q_0(x,t)$ a point such
that $q_t=x$ if $q_0=q_0(x,t)$.

Suppose that for any compact set $K\subseteq \r^n$ there exists
$t_0=t_0(K)>0$
sufficiently small such that the set
$K_{t_0}=\cup_{t\le t_0}\{x:q_0(x,t)\in K\}$
is  also compact, where  $q_0(x,t)$ denotes a point such
that $q_t=x$ if $q_0=q_0(x,t).$ We assume, that the Jacobian
$$
D(x,t)=det\,\left( \frac{\partial q_0(x,t)}{\partial x}\right)
$$
is a smooth function on $K\times t_0(K)$ for each compact $K.$

{\bf Theorem 7.1.} {\em Let there exist a constant $c\in \r_+$ such that
\begin{itemize}
\item[(a)] $\kappa(x)\le c\kappa(q_0(x,t))\quad \forall t< T_{\infty}(x),$
\item[(b)] $\sup_x\kappa(x)^{-2}(b(x),\nabla)\kappa(x) <\infty$,
\item[(c)] $\sup_x\left(\int\mu(x,dq)(\kappa(x+q)-\kappa(x))+
        (b(x),\nabla)log(\kappa(x))\right) < \infty$,
\item[(d)] $|div \,b(x)|\le c(1+\kappa(x))|^m$ for some $m>0,$
\item[(e)] $\quad\int \mu(x,dq)\kappa(x+q)\le c(1+\kappa(x))^{2m-1},$
\end{itemize}
where
$$
\mu(x,B)=\frac{1}{\kappa(x)}\int\limits_{B}m(dq)|C(x,q)|^2
$$
is a measurable family of probability measures.
Then the minimal dynamical semigroup is conservative.}

Consider some particular cases.

$1^{\circ}.$ Let $b(x)\equiv 0$. Then $q_0(x,t)\equiv x$ and hence only
one restriction from (a-e) survives:
$$
\sup\limits_{x}\int\limits_{\r^n}
\mu(x,dq)(\kappa(x+q)-\kappa(x)) < \infty.
$$

$2^{\circ}.$ Let $n=1,\,b(x)=-\frac{1}{\alpha}|x|^{\alpha+1}.$ Then
$$
q_0(x,t)=\frac{x}{(1+|x|^{\alpha}t)^{\frac{1}{\alpha}}},\,\quad
T_{\infty} (x)=|x|^{-\alpha},\,
$$
$$
D(x,t)=(1-|x|^{\alpha}t)^{-\frac{1+\alpha}{\alpha}},\,
\quad T_0(K)=(diam\,K)^{-\alpha}.
$$
In this case the assumption (a) of Theorem 7.1 is fulfilled for $c=1$.
The assumption (b) is satisfied if the scalar product of the vector field
$b(x)$ and the gradient of the intensity $\kappa$ is not positive, i.e.
the drift is opposite to the
direction of larger  intensities of jumps . The assumption (d)
will be fulfilled if $\kappa(x)=\kappa_0(1+|x|^{\beta}),\, \beta \ge 0,$
for all $m>\frac{\alpha}{\beta}$. The same time, assumption (c)
is fulfilled if $1^{\circ}$ takes place and drift is opposite
to the direction of larger  intensities $\kappa(x)$.

Let $\beta >1$ and $\mu(x,B)$ is such that
$$
\int \mu(x,dq)f(q)=f(x+a \,sign(x)),\, a>0
$$
i.e. the jumps push trajectories to infinity and the drift moves it back
to the origin. Then assumption (e) is satisfied  if
$$
\sup\limits_{x\ge 0 } \left(\kappa_0(x+a)^{\beta}-x^{\beta})
-\frac{1}{\alpha}\frac{x^{\alpha+\beta}}{1+x^{\beta}}\right) <\infty  .
\eqno(7.1)
$$
Now from the expansion
$$
(x+a)^{\beta}=x^{\beta}(1+\frac{\beta\alpha}{x}+O(x^{-2}))\qquad |x|>1
$$
we conclude that (7.1) is fulfilled and {\bf for all}
$\alpha>\beta -1,$ {\bf and for any} $\kappa_0,\, \alpha$ {\bf if}
$\alpha=\beta-1,\,a\alpha\kappa_0\le 1$
{\bf the Markov process with the intensity of jumps}
$ \kappa(x)=\kappa_0(1+|x|^{\beta}),\, \beta \ge 0$ {\bf
and the drift velocity} $b(x)=-\frac{1}{\alpha}|x|^{\alpha+1}$
{\bf is regular, i.e. the drift resists successfully
the critical behaviour of one sided
jumps of  by $a$}.

(2) Let ${\cal H}$ be ${\cal L}_2(\r^n, dm)$ with an absolutely continuous
measure $m$:
$$
(\phi,\psi)=\int_{\r^n} \phi(x)^*\psi(x)b(x)dx,\quad m(b)=\int_{B}b(x)dx,
$$
where $b(x)$ is a positive measurable function from
${\cal L}_1^{loc}(\r^n).$ So the adjoint operator $\partial^*=\partial_x^*$
has components
$$
\partial^*_k=-\frac{1}{b(x)}\frac{\partial}{\partial x_k}b(x),
\qquad k=1,\dots, n
$$
and each positive operator $B\in {\cal L}_2(\r^n)$ corresponds to
the positive operator $b(x)^{-1}B\in {\cal L}_2(\r^n, dm).$

Consider the CP-map  $\Phi_*(\cdot)$ related to a Markov symmetrical
diffusion process in $\r^n$ with diffusion matrix $A(x)=a^2(x):$
$$
\Phi_*(\cdot)=\sum\limits_{1}^{n}(a_k(\cdot)\partial_k \psi(\cdot),B
a_k(\cdot)\partial_k \psi(\cdot)),
\eqno(7.2)
$$
where $\{a_k(x)\}^{n}_{k=1}$  are smooth positive functions and
$a_k(\cdot)$ operators of multiplication by the function $a_k(x)$.
We suppose that the functions $a_k, b$ are locally bounded and nondegenerate:
$$
{\lambda}^{-1}\le a_k(x)\le \lambda, \quad {\lambda}^{-1}\le b(x)\le \lambda,
\quad\lambda=\lambda(D),\quad\forall x\in D,
$$
for any compact set $D\subset \r^n.$

It is known (see [Fu], [I]) that in
this case
\begin{itemize}
{\em
\item[--] The quadratic form $\Phi_*(I)$
is closable in ${\cal H}={\cal L}_2(\r^n, dm)$;
\item[--] $C_0^{\infty}(\r^n)=ess\,dom \,\Phi_*(I)$;
\item[--] There exists a Markov jump process $\{\xi_t,T_{\infty},P_x\}$ with
a random escape to infinity time  $T_{\infty}(\omega)$ such that
$$
\lim\limits_{t\to T_{\infty}(\omega)} |\xi_t(\omega)|=\infty,
$$
$$
\lim\limits_{t\to 0}\frac{1}{t}\left(\int_{\r^n} 
{\bf P}_x\{\xi_t(\omega)\in dy,
t<T_{\infty}(\omega)\}f(y)-f(x)\right) =Lf(x)\quad \forall f
\in C_0^{\infty}(\r^n),
$$
}
\end{itemize}
where $L$ is the second order negative partial derivative operator
$$
L=-{1\over2}[\![\Phi_*(I)]\!]= {1\over2b(x)}
\sum\limits_{1}^{n}\frac{\partial}{\partial x_k} b(x)a_k^2(x)
\frac{\partial}{\partial x_k}
$$
corresponding to the minimal extension (the Friedrichs extension)
of the quadratic form $\Phi_*(I)$.

As was proved in [Kh], [I], the necessary and sufficient condition for
the regular behaviour of the Markov diffusion process (i.e.,
${\bf P}\{T_{\infty}(\omega)=\infty\}=1$) is equivalent to nonexistence
of a bounded positive solutions for the spectral problem
$$
LP(x)=P(x).
$$
If the regularity is violated then the positive bounded function
$$
P(x)={\m}_xe^{-T_{\infty}(\omega)}
$$
is such solution.

Denote by $\widehat P$ the multiplication by $P(\cdot)$ operator. Note that
$LP(\cdot)={\cal L}(\widehat P)$. Hence ${\cal L}(\widehat P)=\widehat P$
if $LP(x)=P(x)$. This means a violation of the condition 5 (theorem 5.1)
which is necessary for the conservativity of the minimal dynamical semigroup.
If the equation ${\cal L}(\widehat P)=\widehat P$ does not have any positive
bounded solutions then the equation $LP(x)=P(x)$ also has no
such solutions, and the Markov
diffusion process is regular. Consider conditions of the theorem 6.4
for this particular case:
$$
{\cal L}(B)=\Phi(B)-{1\over2}\left(\Phi(I)B+B\Phi(I)\right).
$$

As was pointed out above,
$C_0^{\infty}(\r^n)=ess\,dom \,\Phi_*(I)$ and hence
$$C_0^{\infty}(\r^n)=ess\,dom \,\Phi(I)^2=ess\,dom \,\Phi(\Phi(I)).$$
The difference $\Phi(I)^2-\Phi(\Phi(I))=-{\cal L}(\Phi(I))$ is the
second-order symmetric differential operator:
$$
\Phi(I)^2-\Phi(\Phi(I))=-{\cal L}(\Phi(I))=
{1\over b(x)}
\sum\limits_{j,k=1}^{n}\frac{\partial}{\partial x_j} b(x)a_j(x)
A_{j,k}(x)a_k(x)\frac{\partial}{\partial x_k},
$$
where $A(x)$ is the matrix with coefficients
$$
A_{j,j}(x)= a_j(x)\frac{\partial^2}{\partial x_j^2}\left(a_j(x)\right)+
a_j(x)\frac{\partial}{\partial x_j}\left(a_j(x)\frac{\partial}{\partial x_j}
\ln b(x)\right)+
$$
$$
+a_j^2(x)\frac{\partial^2}{\partial x_j^2}\left(\ln b(x)\right)-
\sum\limits_{k\ne j}\frac{1}{a_j(x)b(x)}\frac{\partial}{\partial x_k}
\left(a_k^2(x)b(x) \frac{\partial}{\partial x_k}a_j(x)\right),
$$
$$
A_{k,j}(x)=\frac{a_k(x)}{a_j(x)}\frac{\partial}{\partial x_k}
\left(\frac{1}{b(x)}\frac{\partial}{\partial x_j}a_j^2(x)b(x)\right)
\quad (k\ne j).
\eqno(7.3)
$$
Since $H\equiv 0$, our Theorem 6.4 gives the following condition,
which is
sufficient for the conservativity of the minimal dynamical semigroup:
$$
{\cal L}(\Phi(I))=-{1\over b(x)}
\sum\limits_{j,k=1}^{n}\frac{\partial}{\partial x_j} b(x)a_j(x)
A_{j,k}(x)a_k(x)\frac{\partial}{\partial x_k}\le
$$
$$
\le c\Phi(I))=-
{c\over b(x)}
\sum\limits_{1}^{n}\frac{\partial}{\partial x_k} b(x)a_k^2(x)
\frac{\partial}{\partial x_k} .
\eqno(7.4)
$$
The operator $\sum \frac{\partial^2}{\partial x_k^2}$ is negative.
Hence, (7.4) implies
\vskip3mm

{\bf Theorem 7.2.} {\em
The symmetrical diffusion with the formal generator
$$
L={1\over2b(x)}
\sum\limits_{1}^{n}\frac{\partial}{\partial x_k} b(x)a_k^2(x)
\frac{\partial}{\partial x_k}
$$
is regular and the corresponding minimal dynamical semigroup
is conservative if
$$
\sup\limits_{x\in \r^n}A(x)\le cI
\eqno(7.5)
$$
for some $c\in \r$,  where $A(x)$
is matrix with coefficients $A_{k,j}(x)$
given by equations (7.3)}.

 This condition seems to be unknown. In the
spherically symmetrical case it can be compared with the
results of Khas'minskii [Kh] and McKean [MK]. They proved
that the symmetrical diffusion
in $\r^n$ with the diffusion coefficient $a^2(r)$ is regular if and only if
the following integrals diverge:
$$
\int\limits_{0}^{\infty}dr\int\limits_{r}^{\infty} \frac{dr}{a^2(r)}=\infty
\quad (n=1),\qquad \int\limits_{0}^{\infty} \frac{rdr}{a^2(r)}
=\infty\quad(n>2).
\eqno(7.6)
$$
It follows from this,that up to multipliers of order
$O(\ln(\ln r)^{1+\epsilon})$ the upper bound
on the rate of increase  of the coefficient
$a^2(x)$ gives function $a^2(x)=r^2\ln(2+r).$
It is known that the symmetrical diffusion process
with coefficient $a^2(x)=r^2\ln^{1+\epsilon}(2+r)$ explodes with a positive
probability.

In order to apply (7.5) to this problem we put
$$
a_1(r)=a(r), \quad \partial_1=\frac{\partial}{\partial r},\quad
a_j(r)=0, \,i>1,\quad b(r)=r^{n-1}.
$$
Then $A_{i,j}(r)=0$ for all $i,j>1$ and
$$
A_{11}=a(r)a(r)''+\frac{n-1}{r}a(r)a(r)'
-\frac{n-1}{r^2}a^2(r).
$$
Hence {\bf a symmetrical diffusion in} $\r^n$ {\bf with a
smooth locally bounded
diffusion coefficient} $a(r)^2$  {\bf is regular if}
\quad$\sup_{r>1}\left(a(r)a(r)''+\frac{n-1}{r}a(r)a(r)'-
\frac{n-1}{r^2}a^2(r)\right)<\infty$.

For example, if $a(r)=r\ln^{{1\over2}}r$ then $A_{11}(r)=\frac{n}{2}-
(4\ln r)^{-1}<\infty$  as $r\to \infty$. This gives the regularity
of the symmetrical diffusion with the smooth diffusion coefficient
$a(r)$ such that $a(r)\le c(1+r)\ln^{{1\over2}}(2+r).$
On the other hand, if
$a(r)=r\ln^{{1\over2}+\epsilon}r,$ then
$A_{11}(r)=O(\ln^{2\epsilon}r)\to \infty  $
as $r\to \infty$. Hence, up to multipliers of the order $O(\ln^{\epsilon})$
the condition (7.5) gives the same result as (7.6).

\bigskip

\section{Bibliography}
\begin{itemize}
\item[[C1]] Chebotarev A.M. Conditions sufficient for regularity
of the Markov jump processes. Probability theory and its
applications, 1988, V.33, N.1, P.21-35.
\item[[C2]]  Chebotarev  A.M. Necessary and sufficient conditions for
conservativeness of dynamical semigroups. Journal of Soviet Mathematics,
1991, V.56, P.2697-2719.
\item[[C3]]  Chebotarev A.M. Sufficient conditions of the conservatism
of a minimal dynamical semigroup. Mathematical Notes, 1993, V.52, N3-4,
P. 1067-1077.
\item[[CF1]] Chebotarev A.M., Fagnola F. Sufficient conditions
for conservativity of quantum dynamical semigroup. Journal of
Functional Analyse, vol. 113, N 1, 1993, p. 131 - 153.
\item[[CF2]] Chebotarev A.M., Fagnola F. On a remarkable quantum dynamical
semigroup (to appear in {\sl S\'em. Prob.}, 1994).
\item[[E1]] Emery M. On the Az\'ema martingales.
{\sl S\'em. Prob.} {\bf  XXIII}, LNM 1372, 1989, P.67--87.
\item[[E2]] Emery M. Sur les martingales d'Az\'ema.
{\sl S\'em. Prob.} {\bf  XXIV} (1990).
\item[[I]] Ichihara K. Explosion problems for symmetric
diffusion processes.
Lecture Notes in Math. Springer-Verlag, 1986, V. 1203, P.75-89.
\item[[GS]] Gikhman I.I., Skorokhod A.V. Introduction to the
theory of random processes. Scripta Technica, W.B.Saunders Co.,
Philadelphia, 1969.
\item[[F]] Feller W. An introduction to probability theory and its 
applications, v.1-2, John Wiley \& Sons, 1977.
\item[[Fu]] Fukushima M. Dirichlet forms and Markov processes.
Kodanasha, Tokyo, 1980.
\item[[K]] Kraus K. Annals of Physics, New-York, 1971, V.64, p.311-331
\item[[Kh]] Khas'minskii R.Z. Ergodic problems of recurrent
diffusion processes and stabilization of solutions of the Cauchy
problem for a parabolic equation. Probability theory and its
applications, 1960, V.5, N.1, P.196-214.
\item[[L]] Lindblad G. On the generators of quantum dynamical semigroups.
Commun. Math. Phys., 1976, V.48, N2, P. 119-130.
\item[[MK]] McKean H.P. (Jr.) Stochastic Integrals. Academic
Press, N.-Y., 1969.
\item[[P]]  Parthasarathy K.R. Az\'ema martingales and quantum 
stochastic calculus. In: Proc. R.C.Bose Symp. (ed. R.R. Bahadur)
Wiley Eastern, New Delhi, 1990, P.551-569.
\item[[S]] Stinespring W.F. Positive functions on $C^*$
--algebras. Proc.of Amer. Math. Soc., 1955, V.6, N2, P. 211-218.


\end{itemize}
\end{document}
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