Thus, the only subgroup $H\leq G$ that contain an element of index $n-1$ is
$H=G$ itself. Thus, from the expression

$D_{\mathbb{Q},C_{n}}(s)=\sum_{H\leq
C_{n}}\mu_{C_{n}}(H,C_{n})B_{H}(s)\prod_{g\in
H-\\{1\\}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(o)s)^{1/[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}$

only the term associated to $H=C_{n}$ has a pole at $s=\frac{1}{n-1}$. This
implies

$\displaystyle\lim_{s\to\frac{1}{n-1}}\left(s-\frac{1}{n-1}\right)^{\bar{b}_{n-1}(\mathbb{Q},C_{n})}D_{\mathbb{Q},C_{n}}(s)$
$\displaystyle=\lim_{s\to\frac{1}{n-1}}\left(s-\frac{1}{n-1}\right)^{\bar{b}_{n-1}(\mathbb{Q},C_{n})}B_{C_{n}}(s)\prod_{g\in
C_{n}-\\{1\\}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(o)s)^{1/[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}$ $\displaystyle\neq 0$

by construction. ∎

This is a nifty trick, which has been used to show the leading term does not
vanish when abelian extensions are ordered by the product of ramified primes.
Essentially, only one term from the surjective sieve has a pole at the
location of interest, so there is no chance of cancellation.

For the remaining parts, we leverage Lemma 10.1 to prove that the poles do not
cancel. In parts (i,ii), _every_ term of the surjective sieve contributes the
same zeta factors. By factoring these out, we are able to prove
noncancellation of the pole.

###### Proof of Theorem 1.7(i,ii).

We remark that part (i) actually follows from part (ii), as

$\\{g\in G-\\{1\\}:\textnormal{ind}(g)<a(G)\\}=\emptyset$

by definition, and $\emptyset\subseteq\Phi(G)$ is vacuously true.

The Möbius function $\mu_{G}(H,G)$ for subgroups of $G$ is supported on
subgroups containing the Frattini subgroup. Thus, we can factor several zeta
functions out of the sum over $H\leq G$. This proves that the product

$D_{\mathbb{Q},G}(s)\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(g)s)^{-\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}$

is equal to

$\displaystyle\sum_{H\leq
G}\mu_{G}(H,G)\prod_{p<\infty}\left(\prod_{\begin{subarray}{c}g\in
G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\left(1-p^{-\textnormal{ind}(g)sf_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}\right)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)g_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}\right)$
$\displaystyle\times\left(\frac{1}{|H|}\sum_{f_{p}\in\textnormal{Hom}(G_{\mathbb{Q}_{p}},H)}|\textnormal{disc}(f_{p})|^{-s}\right)$
$\displaystyle=$ $\displaystyle\sum_{H\leq
G}\mu_{G}(H,G)\prod_{p<\infty}\left(\prod_{\begin{subarray}{c}g\in
G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\left(1-p^{-\textnormal{ind}(g)sf_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}\right)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)g_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}\right)$
$\displaystyle\times\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p}\in\textnormal{Hom}(G_{\mathbb{Q}_{p}},G)\\\
f_{p}(I_{p})\subseteq H\end{subarray}}|\textnormal{disc}(f_{p})|^{-s}\right)$

Corollary 3.3 implies that these products are absolutely convergent for ${\rm
Re}(s)\geq 1/d$. (In fact, for ${\rm Re}(s)>1/m$ for
$m\in\textnormal{ind}(G-\\{1\\})$ the smallest index with $d<m$).

Define the function $f:{\rm Sub}(G)\times\mathbb{N}\to\mathbb{C}$ so that
$n\mapsto f(J,n)$ is multiplicative, $f(J,p^{j})=0$ for each $j>1$, $f(1,p)=0$
for the trivial subgroup, and

$f(J,p)=\left(\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\left(1-p^{-\frac{\textnormal{ind}(g)f_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{d}}\right)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)g_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}\right)\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p}\in\textnormal{Hom}(G_{\mathbb{Q}_{p}},G)\\\
f_{p}(I_{p})=J\end{subarray}}|\textnormal{disc}(f_{p})|^{-1/d}\right)$

otherwise. This is chosen so that

$\displaystyle\lim_{s\to 1/d}D_{\mathbb{Q},G}(s)\prod_{\begin{subarray}{c}g\in
G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(g)s)^{-\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}$ $\displaystyle=\sum_{H\leq
G}\mu_{G}(H,G)\prod_{p<\infty}\left(1+\sum_{J\leq H}f(J,p)\right),$

where we already know the limit exists by Theorem 1.2. The function $f$
satisfies the hypotheses of Lemma 10.1 by Corollary 3.3. Thus, Lemma 10.1
implies that

$\displaystyle\lim_{s\to 1/d}D_{\mathbb{Q},G}(s)\prod_{\begin{subarray}{c}g\in
G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(g)s)^{-\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}$
$\displaystyle=\sum_{n=2}^{\infty}\mu(n)^{2}\sum_{\begin{subarray}{c}(J_{p})_{p\mid
n}\\\ \prod J_{p}=G\end{subarray}}\prod_{p\mid n}f(J_{p},p).$

To prove part (ii), it suffices to prove that this limit is nonzero. As this
is a sum of nonegative terms, we can bound the limit below by a single nonzero
term to complete the proof. Let $(p_{g})_{g\in G-\\{1\\}}$ be a sequence of
distinct primes congruent to $1\pmod{|\langle g\rangle|}$. Thus

$\displaystyle\lim_{s\to 1/d}D_{\mathbb{Q},G}(s)\prod_{\begin{subarray}{c}g\in
G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(g)s)^{-\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}$ $\displaystyle\geq\prod_{g\in
G-\\{1\\}}f_{\sigma}(\langle g\rangle,p_{g})$ $\displaystyle=\prod_{g\in
G-\\{1\\}}\left(\prod_{\begin{subarray}{c}h\in G-\\{1\\}\\\
\textnormal{ind}(h)<d\end{subarray}}\left(1-{p_{g}}^{-\frac{\textnormal{ind}(h)}{d}}\right)^{b_{\textnormal{ind}(h)}(\mathbb{Q},G)}\right)\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p_{g}}\in\textnormal{Hom}(G_{\mathbb{Q}_{p_{g}}},G)\\\
f_{p}(I_{p})=\langle
g\rangle\end{subarray}}|\textnormal{disc}(f_{p_{g}})|^{-1/d}\right)$
$\displaystyle=\prod_{g\in G-\\{1\\}}\left(\prod_{\begin{subarray}{c}h\in
G-\\{1\\}\\\
\textnormal{ind}(h)<d\end{subarray}}\left(1-{p_{g}}^{-\frac{\textnormal{ind}(h)}{d}}\right)^{b_{\textnormal{ind}(h)}(\mathbb{Q},G)}\right)p^{-\frac{\textnormal{ind}(g)}{d}}$
$\displaystyle>0.$

This concludes the proof of part (ii), as we know that

$\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(g)s)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}$

has a pole of order $\bar{b}_{d}(\mathbb{Q},G)$ at $s=1/d$ by construction. ∎

For part (iii), we allow a slightly more complicated scenario - the is a
$\zeta(a(G)s)$ factor that will appear on some terms of the sieve, and not on
other terms. But otherwise, all Dedekind zeta functions that appear will
appear on all terms of the sieve and can be factored out. We then take
advantage of the fact that $\zeta(x)<0$ for all real $x\in[0,1)$.

###### Proof of Theorem 1.7(iii).

The start of the proof is similar to Theorem 1.7(ii), except now we know that
for $G=C_{2}\times A$ each element $g\in G-\\{1\\}$ with
$\textnormal{ind}(g)<d$ is _either_ (1) contained in the Frattini subgroup
_or_ (2) of order $2$, so that $\textnormal{ind}(g)=a(G)$.

We factor out the zeta functions corresponding to $g$ with
$a(G)<\textnormal{ind}(g)<d$ similarly to part (ii), so that

$D_{\mathbb{Q},G}(s)\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
a(G)<\textnormal{ind}(g)<d\end{subarray}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(g)s)^{-\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}$

is equal to

$\displaystyle\zeta(a(G)s)\sum_{\begin{subarray}{c}H\leq G\\\ C_{2}\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}$
$\displaystyle\left(\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\left(1-p^{-\textnormal{ind}(g)sf_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}\right)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)g_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}\right)$
$\displaystyle\times\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p}\in\textnormal{Hom}(G_{\mathbb{Q}_{p}},G)\\\
f_{p}(I_{p})\subseteq H\end{subarray}}|\textnormal{disc}(f_{p})|^{-s}\right)$
$\displaystyle+\sum_{\begin{subarray}{c}H\leq G\\\ C_{2}\not\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}$
$\displaystyle\left(\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
a(G)<\textnormal{ind}(g)<d\end{subarray}}\left(1-p^{-\textnormal{ind}(g)sf_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}\right)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)g_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}\right)$
$\displaystyle\times\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p}\in\textnormal{Hom}(G_{\mathbb{Q}_{p}},G)\\\
f_{p}(I_{p})\subseteq H\end{subarray}}|\textnormal{disc}(f_{p})|^{-s}\right)$

Corollary 3.3 implies that the infinite products given here are convergent for
${\rm Re}(s)\geq 1/d$.

We partition the poset of subgroups of $G$ into two separate posets: the poset
of subgroups containing $C_{2}$, and the poset of subgroups not containing
$C_{2}$. We address each of the two sums in the way we proved part (ii).
Define $f_{0}:\\{C_{2}\leq H\leq G\\}\times\mathbb{N}\to\mathbb{R}_{\geq 0}$
to be multiplicative with $f_{0}(J,p^{j})=0$ for $j>1$, $f_{0}(1,p)=0$, and

$f_{0}(J,p)=\left(\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
\textnormal{ind}(g)<d\end{subarray}}\left(1-p^{-\textnormal{ind}(g)sf_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}\right)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)g_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}\right)\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p}\in\textnormal{Hom}(G_{\mathbb{Q}_{p}},G)\\\
f_{p}(I_{p})C_{2}=J\end{subarray}}|\textnormal{disc}(f_{p})|^{-s}\right)$

otherwise. Define $f_{1}:\\{H\leq A\\}\times\mathbb{N}\to\mathbb{R}_{\geq 0}$
to be multiplicative with $f_{0}(J,p^{j})=0$ for $j>1$, $f_{1}(1,p)=0$, and

$f_{1}(J,p)=\left(\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
a(G)<\textnormal{ind}(g)<d\end{subarray}}\left(1-p^{-\textnormal{ind}(g)sf_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}\right)^{\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)g_{p}(\mathbb{Q}(\zeta_{|\langle
g\rangle|})/\mathbb{Q})}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}\right)\left(\frac{1}{|G|}\sum_{\begin{subarray}{c}f_{p}\in\textnormal{Hom}(G_{\mathbb{Q}_{p}},G)\\\
f_{p}(I_{p})=J\end{subarray}}|\textnormal{disc}(f_{p})|^{-s}\right)$

otherwise. Taking these together, the limit

$\lim_{s\to 1/d}D_{\mathbb{Q},G}(s)\prod_{\begin{subarray}{c}g\in G-\\{1\\}\\\
a(G)<\textnormal{ind}(g)<d\end{subarray}}\zeta_{\mathbb{Q}(\zeta_{|\langle
g\rangle|})}(\textnormal{ind}(g)s)^{-\frac{b_{\textnormal{ind}(g)}(\mathbb{Q},G)}{[\mathbb{Q}(\zeta_{|\langle
g\rangle|}):\mathbb{Q}]}}$

exists by Theorem 1.2, and is equal to

$\displaystyle\zeta\left(\frac{a(G)}{d}\right)\sum_{\begin{subarray}{c}H\leq
G\\\ C_{2}\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}\left(1+\sum_{J\leq
H}f_{0}(J,p)\right)+\sum_{\begin{subarray}{c}H\leq G\\\ C_{2}\not\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}\left(1+\sum_{J\leq
H}f_{0}(J,p)\right).$

Lemma 10.1 implies that

$\displaystyle\sum_{\begin{subarray}{c}H\leq G\\\ C_{2}\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}\left(1+\sum_{J\leq
H}f_{0}(J,p)\right)$
$\displaystyle=\sum_{n=2}^{\infty}\mu(n)^{2}\prod_{\begin{subarray}{c}(J_{p})_{p\mid
n}\\\ \prod_{p}J_{p}=G\end{subarray}}\prod_{p\mid n}f_{0}(J_{p},p)$
$\displaystyle>0$

by isolating a single nonzero term in the same way as the proof of part (ii).
Similarly, the structure $G=C_{2}\times A$ for $|A|$ odd implies

$\displaystyle\sum_{\begin{subarray}{c}H\leq G\\\ C_{2}\not\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}\left(1+\sum_{J\leq
H}f_{1}(J,p)\right)$ $\displaystyle=-\sum_{\begin{subarray}{c}H\leq
A\end{subarray}}\mu_{G}(H,A)\prod_{p<\infty}\left(1+\sum_{J\leq
A}f_{1}(J,p)\right)$
$\displaystyle=-\sum_{n=2}^{\infty}\mu(n)^{2}\prod_{\begin{subarray}{c}(J_{p})_{p\mid
n}\\\ \prod_{p}J_{p}=A\end{subarray}}\prod_{p\mid n}f_{1}(J_{p},p)$
$\displaystyle<0$

by isolating a single term in the same way as the proof of part (ii).

Finally, we know that $a(g)/d\in[0,1)$ implies that $\zeta(a(G)/d)<0$, so that
the limit is a sum of two negative values

$\displaystyle\zeta\left(\frac{a(G)}{d}\right)\sum_{\begin{subarray}{c}H\leq
G\\\ C_{2}\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}\left(1+\sum_{J\leq
H}f_{0}(J,p)\right)+\sum_{\begin{subarray}{c}H\leq G\\\ C_{2}\not\subseteq
H\end{subarray}}\mu_{G}(H,G)\prod_{p<\infty}\left(1+\sum_{J\leq
H}f_{0}(J,p)\right)<0.$

This concludes the proof. ∎

###  10.3. Proof of Corollary 1.13

We first consider $C_{4M}$. When $(3,M)=1$, the minimum index $a(C_{4M})=2M$
is given by an element of order $2$. The next smallest index is $3M$ and is
given by an element of order $4$. Theorem 1.7(ii) shows that
$D_{\mathbb{Q},C_{4M}}(s)$ has a pole of order $b_{3M}(\mathbb{Q},C_{4M})=1$
at $s=\frac{1}{3M}$. Thus, it suffices to prove that
$\theta(\mathbb{Q},C_{4M})<\frac{1}{3M}$.

We apply Corollary 1.5 with $D=\frac{\ell}{4M(\ell-1)}$ for $\ell$ the
smallest prime dividing $M$, which is the reciprocal of the index of an
element of order $\ell$. This gives

$\displaystyle\theta(\mathbb{Q},C_{4M})$
$\displaystyle\leq\frac{1}{2M}-\frac{\frac{1}{2M}-\frac{\ell}{4M(\ell-1)}}{1+\frac{1}{3}\left(1-\frac{2M\ell}{4M(\ell-1)}\right)+\frac{2}{3}\left(1-\frac{3M\ell}{4M(\ell-1)}\right)}=\frac{5}{16M}+\frac{3}{32M\ell-48M}.$

This is smaller than $\frac{1}{3M}$ whenever

$\frac{3}{32\ell-48}<\frac{11}{16}.$

This is true if and only if $\ell>18/11$, and so is certainly true for
$\ell\geq 5$.

We next consider $C_{6M}$. The minimum index $a(C_{6M})=3M$ is given by an
element of order $2$. The next smallest index is $4M$ and is given by an
element of order $3$. Theorem 1.7(iii) shows that $D_{\mathbb{Q},C_{6M}}(s)$
has a pole of order $b_{4M}(\mathbb{Q},C_{6M})=1$ at $s=\frac{1}{4M}$. Thus,
it suffices to prove that $\theta(\mathbb{Q},C_{6M})<\frac{1}{4M}$.

Suppose first that $5\nmid M$, so that $(30,M)=1$. We apply Corollary 1.5 with
$D=\frac{1}{5M}$ being the reciprocal of index of an element of order $6$, the
next smallest order if $(30,M)=1$. This gives

$\displaystyle\theta(\mathbb{Q},C_{6M})$
$\displaystyle\leq\frac{1}{3M}-\frac{\frac{1}{3M}-\frac{1}{5M}}{1+\frac{1}{3}\left(1-\frac{3M}{5M}\right)+\frac{2}{3}\left(1-\frac{4M}{5M}\right)}=\frac{13}{57M}<\frac{1}{4M}.$

If we have $5\mid M$ with $(6,M)=1$, let $D=\frac{5}{24M}$ be the reciprocal
of the index of an element of order $5$. Then

$\displaystyle\theta(\mathbb{Q},C_{6M})$
$\displaystyle\leq\frac{1}{3M}-\frac{\frac{1}{3M}-\frac{5}{24M}}{1+\frac{1}{3}\left(1-\frac{3M}{24M/5}\right)+\frac{2}{3}\left(1-\frac{4M}{24M/5}\right)}=\frac{62}{267M}<\frac{1}{4M}.$

##  Appendix A Restricted Local Conditions and Alternate Orderings

The methods of this paper are readily applicable to the more generalized
question of counting abelian extensions ordered by other admissible invariants
and with restricted local conditions, as was done for the main term in [AO21].
We restricted the main body of the paper to discriminants so as not to
overwhelm the reader with notation. In particular, the results for restricted
local conditions would be quite involved to state explicitly.

###  A.1. Ordering by Admissible Invariants

Suppose we fix a weight function ${\rm wt}:G\to\mathbb{R}_{\geq 0}$ for which
${\rm wt}(g)=0$ if and only if $g=1$. We say that an invariant
$\textnormal{inv}:\prod_{p}H^{1}(K_{p},G)\to I_{K}$ is an invariant of weight
${\rm wt}$ if, for all but finitely many places,
$\nu_{p}(\textnormal{inv}(f))={\rm wt}(f(\tau_{p}))$ for $\tau_{p}$ a
generator of the tame inertia group at $p$. Such an invariant is certainly
admissible as defined in [Alb21], so that [AO21] give the asymptotic growth
rate of the function

$\\#\textnormal{Surj}_{\textnormal{inv}}(G_{K},T,\pi;X)=\\#\\{\pi\in
Z^{1}(G_{K},T(\pi)):f*\pi\text{ surjective,}\ |\textnormal{inv}(f)|\leq X\\}$

for any abelian $T\trianglelefteq G$ and homomorphism $\pi:G_{K}\to G$ with
$\pi(G_{K})T=G$.

We may define the power saving invariant for an arbitrary admissible ordering
in analogy with Definition 1.3 (also with an analogous twisted version).

######  Definition A.1.

Let $K$ be a number field, $G\subseteq S_{n}$ a transitive group, ${\rm inv}$
an admissible invariant, and $D_{K,G,{\rm inv}}(s)$ the generating series of
$G$-extensions of $K$ ordered by ${\rm inv}$. Define $\theta_{\rm inv}(K,G)$
to be the infimum of all real numbers $\theta$ for which

  1. (a)

$D_{K,G,{\rm inv}}(s)$ has a meromorphic continuation to ${\rm Re}(s)>\theta$
with at most finitely many poles $s_{1},s_{2},...,s_{r}$, and

  2. (b)

The number of $G$-extensions of $K$ satisfies the bound

$\\#\textnormal{Surj}_{\rm
inv}(G_{K},G;X)-\sum_{j=1}^{r}\underset{s=s_{j}}{\rm Res}\left(D_{K,G,{\rm
inv}}(s)\frac{X^{s}}{s}\right)=O(X^{\theta}),$

where the implied constant may depend on $K$ and $G\subseteq S_{n}$.

The discriminant has weight given by the index function. One might predict
that the results of this paper are true for any invariant with a weight
function, but this is not quite the case. What is true is the following:

######  Theorem A.2.

Let ${\rm wt}:G\to\mathbb{R}_{\geq 0}$ be a weight function which is invariant
under conjugation and invertible powers, i.e. ${\rm wt}(g)={\rm wt}(hgh^{-1})$
for any $g,h\in G$ and ${\rm wt}(g)={\rm wt}(g^{k})$ for any $g\in G$ and
$k\in\mathbb{Z}$ such that $\langle g\rangle=\langle g^{k}\rangle$.

Then the main results of this paper are true for the discriminant replaced by
an invariant of weight ${\rm wt}$, where the index function is replaced by
${\rm wt}$ in each expression.

The only real consideration is in Theorem 3.1. In order for the proof to go
through that the Euler factors of $\widehat{w_{s}}(h)$ are sums of ${\rm
tr}(\rho_{h|_{\mathbb{C}o}}(\textnormal{Fr}_{p}))p^{-{\rm wt}(o)s}$, the
weight function needs to factor through the Galois orbits of
$\textnormal{Hom}((\mathbb{Q}/\mathbb{Z})^{*},T)$. The Galois action on $T$ is
via conjugation inside of $G$, while the Galois action on
$(\mathbb{Q}/\mathbb{Z})^{*}$ is by invertible powers through the cyclotomic
map. If ${\rm wt}$ is invariant under both of these actions, then the proof
goes through.

This condition is a natural one. The inertia subgroups are only well-defined
up to conjugation, so by asking ${\rm wt}$ to be invariant under conjugation
we are really asking that the invariant is independent of the choice of
embedding for the decomposition group $G_{K_{p}}\hookrightarrow G_{K}$. It is
also the case that the tame inertia group does not have a canonical generator.
Thus, asking that ${\rm wt}$ is invariant under invertible powers implies that
the invariant is independent of the choice of generator of tame inertia. While
it is still possible to fix some of these choices and ask a counting question,
the results in this paper suggest that the corresponding generating series may
not be as nice.

One example of particular interest is the product of ramified primes
invariant, ${\rm ram}$. This is an invariant of weight ${\rm wt}(g)=1$
whenever $g\neq 1$. Theorem A.2 applied to Theorem 1.2, Corollary 1.5, and
Corollary 1.6 implies the following:

######  Corollary A.3.

Let $K$ be a number field, $G$ an abelian group, and ${\rm ram}$ the product
of ramified primes ordering. Then

  1. (i)

the generating series $D_{K,G,{\rm ram}}(s)$ is meromorphic on ${\rm
Re}(s)>0$, and in the right halfplane ${\rm Re}(s)>1/2$ has only the pole at
$s=1$ of order exactly

$b_{\rm ram}(K,G)=\\#\\{\text{ orbits of }G-\\{1\\}\text{ under }\chi\\},$

for $\chi:G_{K}\to\hat{Z}^{\times}$ the cyclotomic character.

  2. (ii)

$\theta_{\rm ram}(K,G)\leq 1-\frac{3}{6+[K:\mathbb{Q}](|G|-1)}$, and

  3. (iii)

the generalized Lindelöf Hypothesis implies that $\theta_{\rm ram}(K,G)\leq
1/2$.

This gives an unconditional power savings for each abelian $G$ of the form

$\\#\textnormal{Surj}_{\rm ram}(G_{K},G;X)=XP_{K,G,{\rm ram}}(\log
X)+O\left(X^{1-\frac{3}{6+[K:\mathbb{Q}](|G|-1)}+\epsilon}\right).$

The generalized Linedlöf hypothesis improves this to $O(X^{1/2+\epsilon})$.

###  A.2. Restricted Local Conditions

The fact that Theorem 3.1 applies to $\widehat{w_{s}}(h)$ for _any_ $h\in
H^{1}(K,T(\pi)^{*})$ implies that we can use it to produce meromorphic
continuations for any finite sum of $\widehat{w_{s}}(h)$. It is proven in
[AO21] that for a family of local conditions $\mathcal{L}=(L_{p})$ for
$L_{p}\subseteq H^{1}(K_{p},T(\pi))$ satisfying

  * •

$p\mapsto L_{p}$ is Frobenian

  * •

For all but finitely many $p$, $L_{p}$ is a union of coests of
$H_{ur}^{1}(K_{p},T(\pi))$ containing the identity coset,

the generating Dirichlet series for

$H_{\mathcal{L}}^{1}(K,T(\pi))=H^{1}(K,T(\pi))\cap\prod_{p}L_{p}$

ordered by an admissible invariant is a sum of finitely many of the
$\widehat{w_{\mathcal{L},s}}(h)$.

It is possible to keep a close track of the Euler factors of
$\widehat{w_{\mathcal{L},s}}(h)$ to prove a version of Corollary 3.3 where the
orbits in the product is determined by which ramification types are allowed by
$\mathcal{L}$. However, it is possible that, if $p\mapsto L_{p}$ is determined
by the splitting type of $p$ in a nontrivial extension, the analogous
statement to Corollary 3.3 will include non-integer rational powers of Artin
$L$-functions. This gives a meromorphic continuation of the generating series
to a _branch cut_ of the complex plane.

In order for Theorem 6.1 to produce a power savings, we need an honest
meromorphic continuation which means we would need to know which $\mathcal{L}$
give only integer powers of Artin $L$-functions in the analogous decomposition
to Corollary 3.3. Giving a complete classification would be to cumbersome for
this appendix, but it is clear that at least the following is true:

######  Theorem A.4.

Let $\mathcal{L}=(L_{p})$ be a family of local conditions $L_{p}\subseteq
H^{1}(K_{p},T(\pi))$ such that there exists a subset $H\subseteq G$ closed
under conjugation and invertible powers with $1\subsetneq H$ for which

$L_{p}=\\{f\in H^{1}(K_{p},T(\pi)):f(I_{p})\subseteq H\\}$

for all but finitely many primes $p$.

Then the counting function

$\\#\textnormal{Surj}_{\mathcal{L}}(G_{K},T,\pi;X)=\\#\left\\{f\in
Z^{1}(G_{K},T(\pi)):f*\pi\text{ surjective},\ f\in\prod_{p}L_{p},\
|\textnormal{disc}(f*\pi)|\leq X\right\\}$

satisfies the main results of this paper, where the definitions of $a(T)$,
$b_{d}(K,T(\pi))$ are modified to give $a_{H}(T)$ and $b_{H,d}(K,T(\pi))$ by
only considering the indices of $g\in H-\\{1\\}$, and the analogous power
saving bounds $\theta_{H}(K,T)$ has unconditional bounds analogous to
Corollary 1.5 and 1.17 with the sum in the denominator over only those $g\in
H-\\{1\\}$ with $\textnormal{ind}(g)<D$.

Checking that the lower order terms, or even the main term, do not vanish
becomes significantly more involved when we restrict to a family of local
conditions. Not only does the work in Section 10 become more difficult to
perform, but we now have to worry about Grunwald–Wang counter examples. For
the main term when $T=G$ has the trivial action this is considered by Wood
[Woo09], whereas for $T\neq G$ this was originally overlooked in [AO21] and
was corrected in a follow-up Corrigendum [AO23]. Essentially, there are
certain local conditions that are never achievable which causes not only the
main term, but the entire generating Dirichlet series to vanish.

An interesting question is whether there exist weaker Grunwald–Wang counter
examples for which the main term does not vanish, but subsequent lower order
terms do. In other words, are there any cases where the analogous statement to
Conjecture 1.8 for restricted local conditions is true for the rightmost pole
but false for another pole?

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The Stern-Gerlach Experiment
Translation of: “Der experimentelle Nachweis der Richtungsquantelung im
Magnetfeld”

Martin Bauer

Institute for Particle Physics Phenomenology, Department of Physics,

Durham University, Durham, DH1 3LE, United Kingdom

The following is a translation of the paper by Walther Gerlach and Otto
Stern111W. Gerlach u. O. Stern, Zeitschrift für Physik 9, 349–352, 1922.) that
reported the first evidence for the quantisation of atoms in a magnetic field.
The atoms have quantum states corresponding to a limited number of possible
angles between the directions of the angular momenta of the atoms and the
magnetic field, also called space quantisation. Wording and layout have been
chosen to be as close to the original as possible. For context we recommend
the recent review222H. Schmidt-Böcking, L. Schmidt, H. Lüdde, W. Trageser, A.
Templeton and T. Sauer, Eur. Phys. J. H 41, 327–364, 2016.).
I thank Phillip Helbig for substantial help in preparing this translation and
Chanda Prescod-Weinstein for bringing to my attention that there is no
available english translation of the original paper by Gerlach and Stern.

Experimental Evidence for Space Quantisation in a Magnetic Field.

By Walther Gerlach in Frankfurt a.M. and Otto Stern in Rostock.

Including 7 figures. (Recieved March 1, 1922.)

Recently333O. Stern, ZS. f. Phys. 7, 249, 1921) this journal published a
possible method to experimentally answer the question whether space
quantisation in a magnetic field exists. It was shown in a second
communication444W. Gerlach u. O. Stern, ibid. 8, 110, 1921.) that the normal
silver atom has a magnetic moment. We allow ourselves to report in the
following that the continuation of these investigations has led to establish
space quantisation in a magnetic field as a fact.
Experimental setup. Method and experimental setup are generally the same as in
our earlier experiments, but substantial improvements have been made to some
parts of it555These were worked out and tested collaboratively. The final
experiments had to be performed by one of us alone (G.) due to the departure
from Frankfurt of one of us (St.). ), which we will describe here in addition
to the

information provided earlier. The beam of silver atoms emerges from an
electrically heated small chamotte oven with a steel insert and a cover with a
1 mm2 circular hole. The distance between the hole in the oven and the first
beam aperture was increased to 2.5 cm to prevent clogging by occasional small
silver droplets sprayed from the oven as well as precipation from the atomic
beam. This first aperture is almost circular and its surface measures $3\cdot
10^{-3}$ mm2. 3.3 cm behind this circular aperture, the silver beam passes
through a slit aperture with a length of 0.8 mm and a width between 0.03 mm
and 0.04 mm. Both apertures are made from platinum sheet. The slit aperture is
positioned where the magnetic field begins. The opening of the slit aperture
is right above the knife-edge $S$ (see Fig. 1) and is adjusted with respect to
the circular aperture and the hole in the oven in such a way that the silver
beam travels in parallel along the 3.5 cm long knife-edge. Precisely at the
end of this knife-edge the silver beam hits a glass plate where it condenses.

The two apertures, the poles of the magnet and the glass plate are in a brass
housing with wall of thickness 1 cm, which are rigidly connected so that
pressure from the poles of the electromagnet doesn’t result in a deformation
of the housing nor cause a shift in the relative position of the apertures,
the poles, and the glass plate.

Two Volmer diffusion pumps and a Gaede Hg-pump as pre-pump are used for
evacuation. Through continuous pumping and cooling with solid carbon dioxide a
vacuum of $10^{-5}$mm Hg was achieved and maintained.

The “exposure time” was increased to eight hours without interruption. But as
a result of the very narrow apertures and the long beam, the silver film on
the glass plate was so thin that it had to be developed —as reported
previously— even after eight hours of vaporisation.
Results. First, Fig. 2 shows a photograph after 4$\nicefrac{{1}}{{2}}$ hours
of exposure time without a magnetic field; it’s magnified by a factor of 20.
Measurements of the original under the microscope using an ocular micrometer
resulted in the following dimensions: Length 1.1 mm, width at the narrowest
point 0.06 mm, at the widest point 0.10 mm. We see that the slit isn’t exactly
parallel. It should be noted, however, that the figure shows the slit
magnified by a factor of 40, since the “silver image” of the slit is already
twice the size; it is difficult to make such a slit in a frame only a few
millimetres in size.

Fig. 3 shows a photograph after eight hours of exposure time magnified by a
factor 20 (20 scale divisions of the imaged scale = 1 mm). This is the best
photograph we took. Two other photographs show the same result in all relevant
aspects, but don’t show this complete symmetry. It has to be said that
adjustment of these small apertures by optical methods is very difficult, so
that it takes some luck to obtain a perfectly symmetric photograph as shown in
Fig.3; errors in adjusting an aperture by just a few hundredths of a
millimeter are enough to completely ruin a photograph.

The results of the two other experiments are shown schematically in Fig. 4a
and 4b. In Fig. 4a the silver beam ran intentionally at a slightly greater
distance past the knife-edge than in the experiment of Fig.3. The slit
aperture wasn’t completely “filled” here. Fig. 4b shows the deposit of an
experiment both with and without a field on the same plate; the beam ran very
close to the knife-edge, but was moved 0.3 mm perpendicular to the field (Fig.
4c). Regarding clarity of the pictures, the complete splitting, and all other
details, these pictures are in no way inferior to those shown in Fig. 3.

The photographs show that the silver-atom beam is split in an inhomogeneous
magnetic field in two directions in the direction of the inhomogeneity, of
which one is attracted by the knife-edge pole and the other is repelled by the
knife-edge pole. The deposits show the following details (see the schematic
Fig. 5)

  * a)

The dimensions of the original were measured in a microscope: Length 1.1 mm,
width $a$ 0.11 mm, width $b$ 0.20 mm.

  * b)

The atomic beam splits into two discrete beams in a magnetic field. We found
no evidence for non-deflected atoms.

  * c)

The attraction is slightly stronger than the repulsion. The attracted atoms
get closer to the pole and therefore to the region of largest inhomogeneity,
so that the deflection while flying past is stronger. Fig. 3 and 4b show the
significantly larger deflection directly at the knife-edge of this one
magnetic pole. In the immediate vicinity of the knife-edge the attraction
becomes very large so that a bulge arises, with a sharp edge pointing towards
the knife-edge.

  * d)

The width of the deflected bands is larger than the width of the undeflected
image. The latter is simply the image of aperture $B_{2}$ projected by
aperture $B_{1}$ onto the glass plate. The deflected band is broadened
following the velocity distribution of the silver atoms.

  * e)

This fact strengthens the case for there being few if any undeflected atoms
[see b)]. The detection of undeflected atoms in a small area is much more
sensitive than that of deflected atoms in a large area. There appears to be no
magnetic axis perpendicular to the field direction.

We view these results as direct, experimental evidence for space quantisation
in a magnetic field.

A detailed account of the experiment and results in our thus far short
communications will be published in the Annalen der Physik, as soon as we have
precise measurements of the inhomogeneity of the magnetic field and can
provide quantitative information about the size of the magneton.

The electromagnet necessary for these experiments was procured with funds from
the foundation of the Kaiser Wilhelm Institute; to the director, Mr. A.
Einstein, we would like to express our heartfelt thanks. We further thank the
Association of Friends and Sponsors of the University of Frankfurt sincerely
for the abundant resources they gladly made available to fund the continuation
of the experiments.

Frankfurt a. M. and Rostock i. M., February 1922.


# Stability of stochastic dynamic systems of a random structure with Markov
switchings in the presence of concentration points ††thanks: Citation:
Lukashiv, T.; Malyk, I.V.; Chepeleva, M.; Nazarov, P.V. Stability of
stochastic dynamic systems of a random structure with Markov switchings in the
presence of concentration points. Pages…. DOI:000000/11111.

Taras Lukashiv
Multiomics Data Science Research Group, NORLUX Neuro-Oncology Laboratory
Department of Cancer Research
Luxembourg Institute of Health, L-1210, Luxembourg, Luxembourg
Department of Mathematical Problems of Control and Cybernetics,
Yuriy Fedkovych Chernivtsi National University, 58000, Chernivtsi, Ukraine
<EMAIL_ADDRESS>
Igor V. Malyk
Department of Mathematical Problems of Control and Cybernetics,
Yuriy Fedkovych Chernivtsi National University, 58000, Chernivtsi, Ukraine
<EMAIL_ADDRESS>
Maryna Chepeleva
Multiomics Data Science Research Group, Department of Cancer Research
Luxembourg Institute of Health, L-1445, Strassen, Luxembourg
Faculty of Science, Technology and Medicine, University of Luxembourg,
2 av. de l’Université, L-4365, Esch-sur-Alzette, Luxembourg
<EMAIL_ADDRESS>
Petr. V. Nazarov
Multiomics Data Science Research Group, Department of Cancer Research
Luxembourg Institute of Health, L-1445, Strassen, Luxembourg
<EMAIL_ADDRESS>

###### Abstract

This article aims to investigate sufficient conditions for the stability of
stochastic differential equations with a random structure, particularly in
contexts involving the presence of concentration points. The proof of
asymptotic stability leverages the use of Lyapunov functions, supplemented by
additional constraints on the magnitudes of jumps and jump times, as well as
the Markov property of the system solutions. The findings are elucidated with
an example, demonstrating both stable and unstable conditions of the system.

_K_ eywords system of random structure $\cdot$ Markov switching $\cdot$
concentration point $\cdot$ Lyapunov function $\cdot$ asymptotic stability

##  1 Introduction

In the vast majority of works with jump changes of the trajectory, it is
assumed that the distance between jumps is not less than some $\delta$, i.e.
$\|t_{k}-t_{k-1}\|>\delta$. According to this assumption, only a finite number
of jumps occur on a finite interval, which is an important condition for
proving the stability and exponential boundedness of the solution. In this
case, the conditions of existence, unity, and stability of the systems of
stochastic differential equations with jumps are reduced to the corresponding
statements for systems without jumps.

This work considers the case in which jumps can be concentrated at some point,
which leads to the following relationship

$\lim\limits_{k\rightarrow\infty}t_{k}=t_{\infty}<\infty.$

In this case, the cumulative effect of jumps can lead to a lack of stability
of the system. This effect can be illustrated by a simple example of an
ordinary differential equation

$dx(t)=-x(t)dt$

with jumps

$x(t_{k})=x(t_{k}-)(1+k^{2})$

at the points

$t_{k}=\frac{\alpha}{k},\alpha>0.$

It can be easily concluded that

$\lim\limits_{t\rightarrow\alpha-}\|x(t)\|=\infty$

for $x(0)\neq 0$. This simple example indicates that the size of the jump
plays an important role in the presence of condensation points in the system.

The models proposed in this work can be applied in the field of catastrophe
theory. Random events with a frequency that exponentially increases over a
finite time interval are considered there. At the same time, solving applied
problems, one should consider the possibility of the collapse of the system in
case of large disturbances and/or small intervals between disturbances $g$.
The conditions of instability in such cases may have the following form

$\|t_{k}-t_{k-1}\|<\delta_{min}$

for some $k$.

##  2 Problem statement

On the probabilistic basis $(\Omega,\mathfrak{F},F,\mathbf{P})$ [1], [2], we
consider a stochastic dynamic system of random structure given by a stochastic
differential equation (SDE)

$dx(t)=a(t,\xi(t),x(t))dt+b(t,\xi(t),x(t))dw(t),~{}t\in\mathbb{R}_{+}\backslash
K,$ (1)

with Markov switching

$\Delta x(t)=g(t_{k}-,\xi(t_{k}-),\eta_{k},x(t_{k}-)),~{}~{}~{}t_{k}\in
K=\\{t_{n}\Uparrow\\},$ (2)

and initial conditions

$x(0)=x_{0}\in\mathbb{R}^{m},~{}\xi(0)=y\in\mathbf{Y},\eta_{0}=h\in\mathbf{H}.$
(3)

Here $\xi(t),t\geq 0,$ is a Markov chain with a finite number of states
$\mathbf{Y}=\\{1,2,...,N_{\xi}\\}$ and generator
$Q=\\{\tilde{q}_{ij}\\},i,j=\\{1,...,N_{\xi}\\}$; $\\{\eta_{k},k\geq 0\\}$ is
a Markov chain with values in space $\mathbf{H}$ and with a transition
probability matrix $\mathbb{P}_{H}$;
$x:\left[0,+\infty\right)\times\Omega\rightarrow\mathbb{R}^{m}$; $w(t),t\geq
0,$ is a $m$-dimensional standard Wiener process; the processes $w,\xi$ and
$\eta$ are independent random processes [1], [2].

We denote by

$\mathfrak{F}_{t_{k}}=\sigma(\xi(s),w(s),\eta_{e},s\leq t_{k},t_{e}\leq
t_{k})$

the minimal $\sigma$-algebra with respect to which $\xi(t),\forall
t\in[0,t_{k}]$ and $\eta_{n},n\leq k$ are measured.

Measured by a set of variables functions
$a:\mathbb{R}_{+}\times\mathbf{Y}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$,
$b:\mathbb{R}_{+}\times\mathbf{Y}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}\times\mathbb{R}^{m}$,
$g:\mathbb{R}_{+}\times\mathbf{Y}\times\mathbf{H}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$
satisfy the boundedness condition and the Lipschitz condition

$\|a(t,y,x)\|^{2}+\|b(t,y,x)\|^{2}+\|g(t,y,h,x)\|^{2}\leq C(1+\|x\|^{2});$ (4)

$\|a(t,y,x_{1})-a(t,y,x_{2})\|^{2}+\|b(t,y,x_{1})-b(t,y,x_{2})\|^{2}\leq
L\|x_{1}-x_{2}\|^{2},x_{1},x_{2}\in\mathbb{R}^{m};$ (5)

$\|g(t_{k},y,h,x_{1})-g(t_{k},y,h,x_{2})\|^{2}\leq
L_{k}\|x_{1}-x_{2}\|^{2},x_{1},x_{2}\in\mathbb{R}^{m},\sum\limits_{k=1}^{\infty}L_{k}<\infty.$
(6)

Consider the case of a point of concentration of jumps, i.e.

$\lim\limits_{n\rightarrow\infty}t_{n}=t^{*}\in[0,T].$

Let’s assume that the following relations are true:

$\sum\limits_{k=1}^{\infty}\gamma_{k}<\infty,\gamma_{k}=\sup\limits_{x\in\mathbb{R}^{m},~{}y\in\mathbf{Y},h\in\mathbf{H}}\|g(t_{k},y,h,x)\|$
(7)

and

$\lim\limits_{\varepsilon\downarrow
0}\left(\ln\varepsilon+N_{\varepsilon}\sum\limits_{k=1}^{N_{\varepsilon}}L_{k}\right)=-\infty,N_{\varepsilon}:=inf\left\\{k\geq
1:\sum\limits_{m=k}^{\infty}\gamma_{m}<\varepsilon\right\\}.$ (8)

The conditions (4)-(8) guarantee the existence of a strong solution to the
Cauchy problem (1)-(3) [3].

We denote by

${\it\mathbf{P}}_{k}((y,h,x),\Gamma\times G\times\mathbf{C}):=$
$:=\mathbf{P}((\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\in\Gamma\times
G\times\mathbf{C}|(\xi(t_{k}),\eta_{k},x(t_{k}))=(y,h,x))$

the transition probability of the Markov chain
$(\xi(t_{k}),\eta_{k},x(t_{k}))$, that determine the solution to the problem
(1)-(3) on the $k$-th step.

Definition 1. Discrete Lyapunov operator $(lv_{k})(y,h,x)$ on a sequence of
measurable scalar functions
$v_{k}(y,h,x){\it:\mathbf{Y}}\times{\it\mathbf{H}}\times{\it\mathbb{R}}^{m}\to{\it\mathbb{R}}^{1},k\in{\it\mathbb{N}}\cup\\{0\\},$
for the SDE (1) with Markov switchings (2) is defined by the equality

$(lv_{k})(y,h,x):=$
$:=\int\limits_{{\it\mathbf{Y}}\times{\it\mathbf{H}}\times{\it\mathbb{R}}^{m}}{\it\mathbf{P}}_{k}((y,h,x)(du\times
dz\times dl))v_{k+1}(u,z,l)-v_{k}(y,h,x).$ (9)

Here $v_{k}(y,h,x),k\in{\it\mathbb{N}}$, is a Lyapunov function defined by the
following definition.

Definition 2. The Lyapunov function for the system (1)-(3) is a sequence of
non-negative functions $\left\\{v_{k}(y,h,x),k\geq 0\right\\},$ for whom

  1. 1.

for all $k\geq
0,y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}},x\in{\it\mathbb{R}}^{m}$ the
discrete Lyapunov operator $(lv_{k})(y,h,x)$ (9) is defined;

  2. 2.

if $r\to\infty$

$\bar{v}(r)\equiv\mathop{\inf}\limits_{k\in{\it\mathbb{N},y\in{\it\mathbf{Y}},}{h\in{\it\mathbf{H}},\|x\|\geq
r}}v_{k}(y,h,x)\to+\infty;$

  3. 3.

if $r\to 0$

$\underline{v}(r)\equiv\mathop{\sup}\limits_{{k\in{\it\mathbb{N}},y\in{\it\mathbf{Y}},}{h\in{\it\mathbf{H}},\|x\|\leq
r}}v_{k}(y,h,x)\to 0;$

where $\bar{v}(r)$ and $\underline{v}(r)$ are continuous and monotonic for
$r>0$.

Definition 3. A system with a random structure (1)-(3) is called:

– stable in probability, if for $\forall\varepsilon_{1}>0,\varepsilon_{2}>0$
it can specify $\delta>0$ such that the inequality $\|x\|<\delta$ implies the
inequality

${\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{t\geq
0}\|x(t)\|>\varepsilon_{1}\right\\}<\varepsilon_{2}$ (10)

for all $y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}}$;

– asymptotically stochastically stable, if it is stable in probability and for
any $\varepsilon>0$ exists $\delta_{2}>0$ such that

$\mathop{\lim}\limits_{T\to\infty}{\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{t\geq
T}\|x(t)\|>\varepsilon\right\\}=0$ (11)

for all $\|x\|<\delta_{2}$ , $y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}}$ and
$T\geq 0$.

Definition 4. A system with a random structure (1)-(3) is called:

– mean square stable, if for $\forall\varepsilon>0$ it can specify the
following $\delta>0$, that the inequality $\|x\|<\delta$ implies the
inequality

${\it\mathbf{E}}\|x(t)\|^{2}<\varepsilon$ (12)

for all $t\in[0,T],y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}}$;

– mean square asymptotically stable, if it is mean square stable for any $T>0$
and

$\mathop{\lim}\limits_{t\to\infty}\mathop{\sup}\limits_{y\in\mathbf{Y},h\in\mathbf{H}}{\it\mathbf{E}}\|x(t)\|^{2}=0.$
(13)

If (10)-(13) hold true for all $x\in\mathbb{R}^{m}$, then the system is stable
in the corresponding probabilistic sense on the whole.

For solving the problem (1)-(3) on the intervals $[t_{k},t_{k+1})$, the
following estimate is obtained.

Theorem 1. Let the coefficients $a,b$ of the equation (1) satisfy the
condition of uniform boundedness (4), and the condition (6) holds for the
function $g$.

Then for all $k\geq 0$ for a strong solution of the Cauchy problem (1)-(3)
holds the next inequality

${\it\mathbf{E}}\left\\{\mathop{\sup}\limits_{t_{k}\leq t\leq
t_{k+1}}\|x(t)\|^{2}\right\\}\leq
9e^{5C}(1+2L_{k+1})\left[\mathbb{E}\|x(t_{k})\|^{2}+5C(t_{k+1}-t_{k})\right].$
(14)

Proof of Theorem 1.

We use the same methodology as in [4], [5]. A strong solution of the Cauchy
problem (1), (3) for all $t\in[t_{k},t_{k+1}),k\geq 0$, can be written in the
integral form

$x(t)=x(t_{k})+\int\limits_{t_{k}}^{t}a(\tau,\xi(\tau),x(\tau))d\tau+\int\limits_{t_{k}}^{t}b(\tau,\xi(\tau),x(\tau))dw(\tau).$
(15)

After squaring the left and right sides of (15), calculating $\sup$, and
applying the Cauchy–Schwarz inequality, we obtain:

$\sup\limits_{t_{k}\leq t<t_{k+1}}\|x(t)\|^{2}\leq$ $\leq
3\cdot\sup\limits_{t_{k}\leq
t<t_{k+1}}\left\\{\|x(t_{k})\|^{2}+\|\int_{t_{k}}^{t}a(\tau,\xi(\tau),x(\tau))d\tau\|^{2}+\right.$
$+\left.\|\int_{t_{k}}^{t}b(\tau,\xi(\tau),x(\tau))dw(\tau)\|^{2}\right\\}\leq$

$\leq 3\left[\sup\limits_{t_{k}\leq
t<t_{k+1}}\|x(t_{k})\|^{2}+\sup\limits_{t_{k}\leq
t<t_{k+1}}\int\limits_{t_{k}}^{t}\|a(\tau,\xi(\tau),x(\tau))\|^{2}d\tau+\right.$
$+\left.\sup\limits_{t_{k}\leq
t<t_{k+1}}\|\int\limits_{t_{k}}^{t}\|a(\tau,\xi(\tau),x(\tau))dw(\tau)\|^{2}\right].$

To the last inequality, we apply the conditional mathematical expectation
operation with respect to the $\sigma$-algebra $\mathfrak{F}_{t_{k}}$ and,
taking into account the properties of the Ito integral, we obtain

$\mathbb{E}\left\\{\sup\limits_{t_{k}\leq
t<t_{k+1}}\|x(t)\|^{2}/\mathfrak{F}_{t_{k}}\right\\}\leq
3\left[\mathbb{E}\|x(t_{k})\|^{2}+C(t_{k+1}-t_{k})+C\int\limits_{t_{k}}^{t_{k+1}}\mathbb{E}\|x(\tau)\|^{2}d\tau+\right.$
$+\left.4C\int\limits_{t_{k}}^{t_{k+1}}\mathbb{E}\|x(\tau)\|^{2}d\tau\right]=$
$=3\left[\mathbb{E}\|x(t_{k})\|^{2}+5C(t_{k+1}-t_{k})+5C\int\limits_{t_{k}}^{t_{k+1}}\mathbb{E}\|x(\tau)\|^{2}d\tau\right].$

Using the Gronwall inequality, we obtain an estimate of

$\mathbb{E}\left\\{\sup\limits_{t_{k}\leq
t<t_{k+1}}\|x(t)\|^{2}/\mathfrak{F}_{t_{k}}\right\\}\leq$ $\leq
3\left[\mathbb{E}\|x(t_{k})\|^{2}+5C(t_{k+1}-t_{k})\right]e^{5C}.$

For $t=t_{k+1}$ the strong solution of the system (1)-(3), obviously, must
satisfy the inequality

$\mathbb{E}\left\\{\|x(t_{k+1})\|^{2}/\mathfrak{F}_{t_{k}}\right\\}\leq
3\left[\mathbb{E}\left\\{\|x(t_{k+1}-)\|^{2}/\mathfrak{F}_{t_{k}}\right\\}+\right.$
$+2\mathbb{E}\left\\{\|g(t_{k+1}-,\xi(t_{k+1}-),\eta_{k+1},x(t_{k+1}-))-g(t_{k+1}-,\xi(t_{k+1}-),\eta_{k+1},0)\|^{2}/\mathfrak{F}_{t_{k}}\right\\}+$
$\left.+2\mathbb{E}\left\\{\|g(t_{k+1}-,\xi(t_{k+1}-),\eta_{k+1},0)\|^{2}/\mathfrak{F}_{t_{k}}\right\\}\right]\leq$
$\leq 3\left[(1+2L_{k+1})\mathbb{E}\left\\{\sup\limits_{t_{k}\leq t\leq
t_{k+1}}\|x(t)\|^{2}/\mathfrak{F}_{t_{k}}\right\\}+C\right].$

Combining the last two inequalities, we get the desired estimate (14).

End of proof of Theorem 1.

Remark 1. We will consider the stability of the trivial solution $x\equiv 0$,
i.e. the satisfying of (4), if $C=0$ [6], [7], [8].

Remark 2. Note that the Lipschitz condition (5) was not used in the proof of
the theorem, i.e. any (not necessarily unique) solution to the problem (1)-(3)
satisfies the condition of the theorem.

Theorem 2. Let:

1) the conditions (4)-(6) are met;

2) the Lyapunov functions $v_{k}(y,h,x)$ and $a_{k}(y,h,x),k\geq 0,$ exist,
such that, based on the system, the following inequality

$(lv_{k})(y,h,x(t))\leq-a_{k}(y,h,x(t)),k\geq 0,$ (16)

is correct.

Then the system of random structure (1)-(3) is asymptotically stochastically
stable on the whole.

Proof of Theorem 2. Define by
${\mathfrak{F}}_{t_{k}}=\sigma(\xi(s),\eta_{e},s\leq t_{k},t_{e}\leq t_{k})$ a
minimal $\sigma$-algebra, relative to which are measured $\xi(t)$ for all
$t\in[0,t_{k}]$ and $\eta_{n}$ for $n\leq k$. The conditional mathematical
expectation is calculated by the formula

${\it\mathbf{E}}\left\\{{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\mathord{\left/{\vphantom{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1})){\mathfrak{F}}_{t_{k}}}}\right.\kern-1.2pt}{\mathfrak{F}}_{t_{k}}}\right\\}=$
$=\int_{{\it\mathbf{Y}}\times{\it\mathbf{H}}\times{\it\mathbb{R}}^{m}}{\it\mathbf{P}}_{k}((\xi(t_{k}),\eta_{k},x)(du\times
dz\times dl)v_{k+1}(u,z,l)).$ (17)

Then, by the definition of the discrete Lyapunov operator $(lv_{k})(y,h,x)$
(see (9)) from equality (17), considering (16), we get the inequality

${\it\mathbf{E}}\left\\{{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\mathord{\left/{\vphantom{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\mathfrak{F}_{t_{k}}}}\right.\kern-1.2pt}\mathfrak{F}_{t_{k}}}\right\\}=v_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))+$
$+(lv_{k})(\xi(t_{k}),\eta_{k},x(t_{k}))\leq\bar{v}(\|x(t_{k})\|).$ (18)

From Theorem 1 (because the existence of the second moment implies the
existence of the first moment) and from properties of the function $\bar{v}$
follows the existence of a conditional mathematical expectation of the left-
hand side of the inequality (18).

Now, using(17), (18), we write the discrete Lyapunov’s operator
$(lv_{k})(y,h,x)$, which given on the solutions (1)-(3):

$lv_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))={\it\mathbf{E}}\left\\{{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\mathord{\left/{\vphantom{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\mathfrak{F}_{t_{k}}}}\right.\kern-1.2pt}\mathfrak{F}_{t_{k}}}\right\\}-$
$-v_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))\leq-
a_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))\leq 0.$ (19)

Then, at $k\geq 0$ the next inequality holds

${\it\mathbf{E}}\left\\{{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\mathord{\left/{\vphantom{v_{k+1}(\xi(t_{k+1}),\eta_{k+1},x(t_{k+1}))\mathfrak{F}_{t_{k}}}}\right.\kern-1.2pt}\mathfrak{F}_{t_{k}}}\right\\}\leq
v_{k}(\xi(t_{k}),\eta_{k},x(t_{k})).$

This means that a sequence of random variables

$v_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))$

forms a supermartingale in relation to $\mathfrak{F}_{t_{k}}$ [9].

Taking the mathematical expectation of both parts of inequality (19), we
summarize the obtained expressions for $k$ from $n\geq 0$ to $N$, and
obviously, we have the next inequality:

${\it\mathbf{E}}\left\\{v_{N+1}(\xi(t_{N+1}),\eta_{N+1},x(t_{N+1}))\right\\}-{\it\mathbf{E}}\left\\{v_{n}(\xi(t_{n}),\eta_{n},x(t_{n}))\right\\}=$
$=\sum_{k=n}^{N}{\it\mathbf{E}}\left\\{lv_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))\right\\}\leq$
$\leq-\sum_{k=n}^{N}{\it\mathbf{E}}\left\\{a_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))\right\\}\leq
0.$ (20)

Since a random variable $\mathop{\sup}\limits_{t_{k}\leq t\leq
t_{k+1}}\|x(t)\|^{2}$ does not depend on events of $\sigma$-algebra
$\mathfrak{F}_{t_{k}}$ [10], then

${\it\mathbf{E}}\left\\{{\mathop{\sup}\limits_{t_{k}\leq t\leq
t_{k+1}}\|x(t)\|^{2}\mathord{\left/{\vphantom{\mathop{\sup}\limits_{t_{k}\leq
t\leq
t_{k+1}}\|x(t)\|^{2}\mathfrak{F}_{t_{k}}}}\right.\kern-1.2pt}\mathfrak{F}_{t_{k}}}\right\\}={E}\left\\{\mathop{\sup}\limits_{t_{k}\leq
t\leq t_{k+1}}\|x(t)\|^{2}\right\\},$ (21)

that is, the inequality (14) also holds for the simple mathematical
expectation

${\it\mathbf{E}}\left\\{\mathop{\sup}\limits_{t_{k}\leq t\leq
t_{k+1}}\|x(t)\|^{2}\right\\}\leq 3\mathbb{E}\|x\|^{2}.$

Next, we have

${\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{t\geq
0}\|x(t)\|>\varepsilon_{1}\right\\}=$
$={\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{n\in\mathbb{N}}\mathop{\sup}\limits_{t_{n-1}\leq
t\leq t_{n}}\|x(t)\|>\varepsilon_{1}\right\\}\leq$
$\leq{\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{n\in\mathbb{N}}3\|x(t_{n-1})\|>\varepsilon_{1}\right\\}\leq$
$\leq{\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{n\in\mathbb{N}}\|x(t_{n-1})\|>\frac{\varepsilon_{1}}{3}\right\\}\leq$
$\leq{\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{n\in\mathbb{N}}v_{n-1}(\xi(t_{n-1}),\eta_{n-1},x(t_{n-1}))\geq\bar{v}(\frac{\varepsilon_{1}}{3})\right\\}.$
(22)

If $\sup\|x(t_{k})\|\geq r$, then, based on the definition of the Lyapunov
function, the next inequality holds:

$\mathop{\sup}\limits_{k\geq
0}v_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))\geq\inf\limits_{k\geq
0,y\in{\it\mathbf{Y}},{h\in{\it\mathbf{H}},\|x\|\geq
r}}v_{k}(y,h,x)=\bar{v}(r).$ (23)

Now let’s use the well-known inequality for nonnegative supermartingales [1],
[9] to evaluate the right-hand side of (22):

${\it\mathbf{P}}\left\\{\mathop{\sup}\limits_{n\in\mathbb{N}}v_{n-1}(\xi(t_{n-1}),\eta_{n-1},x(t_{n-1}))\geq\bar{v}(\frac{\varepsilon_{1}}{3})\right\\}\leq$
$\leq\frac{1}{\bar{v}(\frac{\varepsilon_{1}}{3})}v_{k}(y,h,x)\leq\frac{\bar{v}(\|x\|)}{\bar{v}(\frac{\varepsilon_{1}}{3})}.$
(24)

Given inequality (22), inequality (24) make it possible to guarantee the
fulfillment of inequality (10) of stability in probability on the whole of the
system (1)-(3).

From the inequality (20) follows the estimate

${\it\mathbf{E}}\\{v_{N+1}(\xi(t_{N+1}),\eta_{N+1},x(t_{N+1}))\\}\leq
v_{0}(y,h,x)-$
$-\sum_{k=0}^{N}{\it\mathbf{E}}\\{a_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))\\}\leq
v_{0}(y,h,x),$ (25)

for all $N\geq
0,y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}},x\in{\it\mathbb{R}}^{m}$.

Since the sequence $\\{a_{k}\\},k\geq 0$ forms Lyapunov functions, there must
exist continuous strictly monotone functions $\underline{a}(r)$ and
$\bar{a}(r)$, which are zero if $r=0$ [11] and such that

$\bar{a}(\|x\|)\leq a_{k}(y,h,x)\leq\underline{a}(\|x\|)$ (26)

for $\forall k\in\mathbb{N},y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}}$ and
$x\in{\it\mathbb{R}}^{m}$.

Thus, from the convergence of the series on the left side of the inequality
(25) (which will be convergent in the case of convergence of the series
$\sum\limits_{k=1}^{\infty}L_{k}$) follows the convergence of the series
$\sum\limits_{k=0}^{\infty}{\it\mathbf{E}}\\{\bar{a}(\|x(t)\|\\}$ for $\forall
t\geq t_{k},y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}},x\in{\it\mathbb{R}}^{m}$.

Then, taking into account the continuity of $\underline{a}(r)$ and the
equality $\underline{a}(0)=0$, we have:

$\mathop{\lim}\limits_{k\to\infty}\|x(t)\|=0,t\geq t_{k}.$ (27)

And from (27) it follows tends to zero in probability of the sequence
$\bar{v}\left(\|x(t)\|\right)$ for $k\to\infty$ for all $t\geq
t_{k},y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}},x\in{\it\mathbb{R}}^{m}.$

So, from the properties of the Lyapunov function, we conclude that the non-
negative supermartingale $v_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))$ for
$k\to+\infty$ tends to zero in probability for all realizations of the process
$\xi$ and sequence $\eta_{k}$.

Further, the nonnegative bounded supermartingale has a bound with probability
1 [1]. Based on Theorem 1 (inequality (14) for the usual mathematical
expectation), we obtain the asymptotic stochastically stability on the whole
of the system (1)-(3) by the definition 3 (see (11)). Theorem 2 is proven.

End of proof of Theorem 2.

Theorem 3. Suppose that the conditions of Theorem 2 are satisfied, and the
Lyapunov functions $\\{v_{k}\\},\\{a_{k}\\},k\geq 0$ satisfy the inequalities

$c_{1}\|x\|^{2}\leq v_{k}(y,h,x)\leq c_{2}\|x\|^{2},$ (28) $c_{3}\|x\|^{2}\leq
a_{k}(y,h,x)\leq c_{4}\|x\|^{2}$ (29)

for some $c_{i}>0,i=\overline{1,4},$ for all
$k\in\mathbb{N},y\in{\it\mathbf{Y}},h\in{\it\mathbf{H}},x\in{\it\mathbb{R}}^{m}.$

Then the system of random structure (1)-(3) is asymptotically stable in the
mean square

Proof of Theorem 3. Using the inequality (19) for $k=0$, based on (28) it is
easy to obtain an inequality

${\it\mathbf{E}}\left\\{\|x(t_{N+1})\|^{2}\right\\}\leq\frac{1}{c_{1}}{\it\mathbf{E}}\\{v_{N+1}(\xi(t_{N+1}),\eta_{N+1},x(t_{N+1}))\\}\leq$
$\leq\frac{1}{c_{1}}{\it\mathbf{E}}\\{v_{0}(\xi(0),\eta_{0},x)\\}\leq\frac{c_{2}}{c_{1}}\|x\|^{2}$
(30)

for all $N\geq 0,x\in{\it\mathbb{R}}^{m}$ and the initial distributions of the
random vector $\\{\xi(0),\eta_{0}\\}$.

Hence, by definition 4 (see (12)), it follows a stability in the mean square
of the system of random structure (1)-(3).

Using the inequalities (20), (28) and (29), it is possible to obtain an
inequality

$\sum_{k=0}^{N}{\it\mathbf{E}}\left\\{\|x(t_{N+1})\|^{2}\right\\}\leq\frac{1}{c_{3}}\sum_{k=0}^{N}{\it\mathbf{E}}\\{a_{k}(\xi(t_{k}),\eta_{k},x(t_{k}))\\}\leq$
$\leq\frac{1}{c_{3}}{\it\mathbf{E}}\\{v_{0}(\xi(0),\eta_{0},x)\\}\leq\frac{c_{2}}{c_{3}}\|x\|^{2}.$

This inequality guarantees the convergence of the series whose members are
${\it\mathbf{E}}\left\\{\|x(t_{k})\|^{2}\right\\}$ for any initial data
$x(0)=x$ and initial distributions of the random vector
$\\{\xi(0),\eta_{0}\\}$.

Therefore,

$\mathop{\lim}\limits_{k\to\infty}\mathop{\sup}\limits_{y\in\mathbf{Y},h\in\mathbf{H},t\in[t_{k-1},t_{k})}{\it\mathbf{E}}\left\\{\|x(t)\|^{2}\right\\}=0,$

which proves Theorem 3.

End of proof of Theorem 3.

Theorem 4. [Corollary] If the conditions of Theorem 3 are fulfilled and the
inequality (28) holds, then the system of random structure (1)-(3) is stable
in the mean square on the whole.

##  3 Computation of the weak infinitesimal operator

Based on the method [12], we will obtain an expression for calculating the
explicit form of the weak infinitesimal operator (WIO) based on the system
(1)-(3), which plays the role of the Lyapunov operator.

Let $U(t,y,h,x)$ be such a scalar integral function, that the sequence

$\\{v_{k}(y,h,x)\equiv U(t_{k},y,h,x),k\geq 0\\}$

is a Lyapunov function.

It is possible to prove [1] that the pair $(\xi(t),x(t),t\geq 0,)$ is a Markov
process and it is possible to introduce WIO

$(\mathcal{L}U)(t,y,h,x):=\lim\limits_{\Delta t\downarrow 0}\frac{1}{\Delta
t}[\mathbf{E}_{y,h,x}^{(t)}\\{U(t+\Delta t,\xi(t+\Delta t),\eta(t+\Delta
t),x(t+\Delta t))-$ $-U(t,y,h,x)\\}],$ (31)

where $\mathbf{E}_{y,h,x}^{(t)}U=\mathbf{E}\\{U|\xi(t)=y,\eta(t)=h,x(t)=x\\}$,
$\eta(t):=\eta_{k}$ and $t_{k}\leq t<t_{k+1},k\geq 0$. It is natural to assume
that the function $U$, defined above, belongs to the domain of definition of
the operator $\mathcal{L}$, if the limit (31) exists in the sense of uniform
convergence in some neighborhood of the point $(y,x)$ uniformly by
$h\in\mathbf{H}$.

Let’s introduce the operator $\mathcal{L}_{0}$ which is related to Markov
switchings (2) at the moment $t_{k},k\geq 0$:

$(\mathcal{L}_{0}U):=\mathbb{I}_{t\in
K}\left[\int\limits_{\mathbf{H}}U(t,y,h,x)\mathbf{P}_{k}(h,dz)-U(t,y,h,x)\right],$
(32)

where $\mathbf{P}_{k}(h,dz)$ is the transition probability of the Markov chain
at the $k$-th step, $\mathbb{I}$ is the indicator of the set $K$.

At the moment $\tau$ of changing of the structure of the parameter $\xi$ of
the system $y_{i}\rightarrow y_{j}$ there is a jump-like change in the phase
vector $x$ with transition probability

$p_{ij}(\tau,x,A):=\mathbf{P}\\{x(\tau)\in
A|x(\tau-)=x,\xi(\tau-)=y_{i},\xi(\tau)=y_{j}\\},A\subset\mathbb{R}^{m}.$ (33)

Theorem 5. The weak infinitesimal operator $\mathcal{L}$ on the solutions of
the system (1)-(3) of the function $U$ is calculated by the formula

$(\mathcal{L}U)(t,y,h,x)=(\mathcal{L}_{t}U)(t,y,h,x)+(\mathcal{L}_{x}U)(t,y,h,x)+(\mathcal{L}_{y}U)(t,y,h,x)+(\mathcal{L}_{0}U)(t,y,h,x),$
(34)

where

$(\mathcal{L}_{t}U)(t,y,h,x)=\frac{\partial U(t,y,h,x)}{\partial t},$ (35)
$(\mathcal{L}_{x}U)(t,y,h,x)=(\nabla_{x}U,a(t,y,x))+\frac{1}{2}Sp(\nabla^{2}_{xx}Ub(t,y,x),b^{T}(t,y,x)),$
(36) $(\mathcal{L}_{y}U)(t,y,h,x)=\sum\limits_{i\neq
j}\left[\int\limits_{\mathbb{R}^{m}}U(t,y_{j},h,\zeta)p_{ij}(t,x,d\zeta)-U(t,y_{i},h,x)\right]q_{ij}.$
(37)

Here $(\cdot,\cdot)$ is scalar product; $(\Delta U)=(\frac{\partial
U}{\partial x_{1}},...,\frac{\partial U}{\partial x_{m}})^{T}$,
$\frac{\partial U}{\partial x_{i}},i={1,...,m}$ is the derivative of the
$i$-th coordinate of the vector $x\in\mathbb{R}^{m}$;
$\nabla^{2}_{xx}U=\left[\frac{\partial^{2}U}{\partial x_{i}\partial
x_{j}}\right]_{i,j=1}^{m}$ is a matrix of second derivatives; $Sp$ is a trace
of the matrix; $q_{ij}=-\frac{\tilde{q}_{ij}}{\tilde{q}_{i}}$;
$(\mathcal{L}_{0}U)(t,y,h,x)$ calculated by formula (32); $U$ is a function
differentiable with respect to $t$, which has derivatives of the 1st and 2nd
order by the last argument.

Proof of Theorem 5. By definition (31)

$(\mathcal{L}U)(t,y,h,x):=\lim\limits_{\Delta t\downarrow 0}\frac{1}{\Delta
t}[\mathbf{E}_{y,h,x}^{(t)}\\{U(t+\Delta t,\xi(t+\Delta t),\eta(t+\Delta
t),x(t+\Delta t))-$ $-U(t,y,h,x)\\}].$

Next,

$(\mathcal{L}U)(t,y,h,x):=\lim\limits_{\Delta t\downarrow 0}\frac{1}{\Delta
t}[\mathbf{E}_{y,h,x}^{(t)}\\{U(t+\Delta t,\xi(t+\Delta t),\eta(t+\Delta
t),x(t+\Delta t))-$ $-U(t,y,h,x)\pm U(t,\xi(t+\Delta t),\eta(t+\Delta
t),x(t+\Delta t))\pm$ $\pm U(t,y,\eta(t+\Delta t),x(t+\Delta t))\pm
U(t,y,h,x(t+\Delta t))\\}].$

Therefore, $\mathcal{L}$ can be represented as

$(\mathcal{L}U)(t,y,h,x):=\lim\limits_{\Delta t\downarrow 0}\frac{1}{\Delta
t}[\mathbf{E}_{y,h,x}^{(t)}\\{U(t+\Delta t,\xi(t+\Delta t),\eta(t+\Delta
t),x(t+\Delta t))-$ $-U(t,\xi(t+\Delta t),\eta(t+\Delta t),x(t+\Delta
t))\\}]+$ $+\lim\limits_{\Delta t\downarrow 0}\frac{1}{\Delta
t}[\mathbf{E}_{y,h,x}^{(t)}\\{U(t,\xi(t+\Delta t),\eta(t+\Delta t),x(t+\Delta
t))-$ $-U(t,y,\eta(t+\Delta t),x(t+\Delta t))\\}]+$ $+\lim\limits_{\Delta
t\downarrow 0}\frac{1}{\Delta
t}[\mathbf{E}_{y,h,x}^{(t)}\\{U(t,y,\eta(t+\Delta t),x(t+\Delta t))-$
$-U(t,y,h,x(t+\Delta t))\\}]+$ $+\lim\limits_{\Delta t\downarrow
0}\frac{1}{\Delta t}[\mathbf{E}_{y,h,x}^{(t)}\\{U(t,y,h,x(t+\Delta
t))-U(t,y,h,x\\}].$

Let’s consider each term separately.

The form of the first term $\mathcal{L}_{t}U$ is obvious.

Let’s establish the explicit form of the term $\mathcal{L}_{x}U$. Consider a
complete group of disjoint events constructed as follows: denote by $H_{i}$
the event which means that the structure (1) does not change in the interval
$(t,t+\Delta t]$, i.e. $\xi(\tau)=y_{i}$ at $\tau\in(t,t+\Delta t]$. Then with
an accuracy of $o(\Delta t)$ we obtain [13]

$\mathbf{P}(H_{i})=-q_{i}\Delta t.$

Next, denote by $H_{ij}$ event, which means that in the interval $(t,t+\Delta
t]$ a change $y_{i}\rightarrow y_{j}\neq y_{i}$ occurs. Then with accuracy up
to $o(\Delta t)$ we have

$\mathbf{P}(H_{ij})=-q_{ij}\Delta t.$

Denote by $\Delta_{i}U:=U(t+\Delta t,\xi(t+\Delta t),h,x(t+\Delta
t)-U(t,y_{i},h,x)$ and by $\Delta_{ij}U$ the increment $\Delta U$ upon
occurrence of the event $H_{ij}$. Let’s calculate the increments $\Delta_{i}U$
and $\Delta_{ij}U$ of the function $U$ when events $H_{i},H_{ij},i\neq j$,
occur, neglecting terms of order $o(\Delta t)$:

$\Delta_{i}H=\left[\frac{\partial U}{\partial
t}+(\nabla_{x}U,a(t,y_{i},x))+\frac{1}{2}Sp(\nabla^{2}_{xx}Ub(t,y_{i},x),b^{T}(t,y_{i},x))\right]\Delta
t+o(\Delta t).$ (38)

Here the partial derivatives are calculated at a point $(t,y_{i},x)$, where
$x$ is the solution of equation (1) with initial condition
$\xi(t)=y_{i},x(t)=x,s>t\geq 0$. Next, for $\mathcal{L}_{y}U$ in the case of a
change in the structure $y_{i}\rightarrow y_{j}$ in the interval $(t.t+\Delta
t]$ we will get an increase

$\Delta_{ij}U=U(t+\Delta t,y_{j},h,x(t+\Delta t))-U(t,y_{i},h,x)$ (39)

with the probability $q_{ij}\Delta t$.

The terms that illustrate the possibility of changing the structure of $\xi$
are not included in the last equality and there are no Markov switchings,
since after averaging they have the order of $o(\Delta t)$ and we can ignore
them.

To calculate $\mathbf{E}\\{\Delta U|\xi(t)=y_{i},\eta_{0}=h,x(t)=x\\}$, we use
the full probability formula

$\mathbf{E}\\{\Delta U|\xi(t-)=y_{i},\eta_{0}=h,x(t)=x\\}=$
$=\mathbf{E}\\{\mathbf{E}\\{\Delta
U|\xi(t)=y_{j},\xi(t-)=y_{i},\eta_{0}=h,x(t)=x\\}\\},$

where the external mathematical expectation on the right-hand side is
calculated by the variable $\xi$ at the moment $t$.

Ignoring terms of order $o(\Delta t)$, from (38) and (39) we obtain

$\mathbf{E}\\{\Delta U|\xi(t)=y_{i},\eta_{0}=h,x(t)=x\\}=$
$=\left[\frac{\partial U}{\partial
t}+(\nabla_{x}U,a(t,y_{i},x))+\frac{1}{2}Sp(\nabla^{2}_{xx}Ub(t,y_{i},x),b^{T}(t,y_{i},x))\right](1-q_{i}\Delta
t)\Delta t+$ $+\sum\limits_{i\neq
j}\left[\int\limits_{\mathbb{R}^{m}}U(t,y_{j},h,\zeta)p_{ij}(t,x,d\zeta)-U(t,y_{i},h,x)\right]q_{ij}\Delta
t+o(\Delta t).$

When calculating the third term, we used the property $x^{T}Bx=Sp(Bxx^{T})$
and the property of the Wiener process with respect to the covariance of the
increment [1], [9].

Using division by $\Delta t$ and passing to the boundary at $\Delta\downarrow
0$, we obtain the first, second, and third terms in (34). The idea of
calculating the fourth term $\mathcal{L}_{0}U$ can be found in [7], pp.
163-164. Theorem 5 is proved.

End of proof of Theorem 5.

##  4 Stability in probability on the whole of a linear stochastic system of
random structure

One-dimensional linear stochastic system of random structure given by SDE

$dx(t)=a(\xi(t))x(t)dt+b(\xi(t)),x(t)dw(t),~{}t\in\mathbb{R}_{+}\backslash K,$
(40)

with Markov switchings

$\Delta x(t)=g(t_{k}-,\xi(t_{k}-),\eta_{k},x(t_{k}-)),$ $t_{k}\in
K=\\{t_{n}\Uparrow\\},\lim\limits_{n\rightarrow\infty}t_{n}=t^{*}\in[0,T<\infty],$
(41)

and initial conditions

$x(0)=x_{0}\in\mathbb{R}^{1},~{}\xi(0)=y\in\mathbf{Y},\eta_{0}=h\in\mathbf{H},$
(42)

where $x\in\mathbb{R}^{1}$ is a strong solution of SDE (40); $t^{*}$ is a
concentration point; $\xi$ is a Markov chain with a finite number of states
$\mathbf{Y}=\\{1,2,...,N_{\xi}\\}$ and generator
$Q=\\{{q}_{ij}\\},i,j=\\{1,...,N_{\xi}\\}$; $\\{\eta_{k},k\geq 0\\}$ is a
Markov chain with values in space $\mathbf{H}$ and the transition probability
at the $k$-th step $\mathbb{P}_{k}(h,dz)$; $w(t),t\geq 0$ is a one-dimensional
standard Wiener process; the processes $w,\xi$ and $\eta$ are independent [1],
[2].

We obtain sufficient conditions for the stability of the system (40)-(42) in
probability on the whole.

Let’s choose a Lyapunov function in the form [13]

$v(\xi(t),h,x)=\gamma\xi(t)\|x\|^{\beta},\gamma>0.$ (43)

Let the functions $a(i)=a_{i}$, $b(i)=b_{i}$ be such that for all
$i=\\{1,...,N_{\xi}\\}$

$a_{i}-\frac{b^{2}_{i}}{2}<-\varepsilon.$ (44)

Then in (43)

$\beta=\varepsilon b^{-2},b=\max\limits_{i=\\{1,...,N\\}}\\{b_{i}\\}.$ (45)

We can show which restrictions must satisfy the transitional probabilities
$q_{ij}$ of the Markov chain $\xi$ and $\mathbb{P}_{k}(h,dz)$ of the Markov
chain $\eta$, so that the system (40)-(42) is stable in probability on the
whole.

Calculating $\mathcal{L}v$ on the solutions of the system (40)-(42), we obtain

$(\mathcal{L}v)(t_{k},y,h,x)=\gamma\|x\|^{\beta}\left\\{b_{i}\left(a_{i}+\frac{\beta-1}{2}b_{i}\right)+\sum\limits_{j\neq
i}^{k}(j-i)q_{ij}\right\\}+$ $+\int\limits_{\mathbf{H}}\gamma
i\|x+g(t_{k},y,h,x)\|^{\beta}\mathbb{P}_{k}(h,dz)-\gamma i\|x\|^{\beta}.$

Considering (43)-(45), at the point $(\xi(t)=i,x)$ we have

$(\mathcal{L}v)(t_{k},y,h,x)=\gamma\|x\|^{\beta}\left[-\frac{\beta
i\varepsilon}{2}+a_{i}\right]+$
$+i\gamma\left[\int\limits_{\mathbf{H}}\|x+g(t_{k},y,h,x)\|^{\beta}\mathbb{P}_{k}(h,dz)-\|x\|^{\beta}\right],$
(46)

where $a_{i}=\sum\limits_{j>i}^{k}(j-i)q_{ij},a_{k}=0$.

Assuming that for $\forall h\in\mathbf{H}$ of the Markov chain $\eta$ the
transition probability at the $k$-th step $\mathbb{P}_{k}(h,dz)$ such that

$\int\limits_{\mathbf{H}}\|x+g(t_{k},y,h,x)\|^{\beta}\mathbb{P}_{k}(h,dz)\leq
2\|x\|^{\beta},$ (47)

then the right-hand side of (46) will take the form

$(\mathcal{L}v)(t_{k},y,h,x)=\gamma\|x\|^{\beta}\left[-\frac{\beta
i\varepsilon}{2}+a_{i}+i\right]=\gamma\|x\|^{\beta}\left[-\frac{i(\beta\varepsilon+2)}{2}+a_{i}\right].$

The function (43) satisfies the condition $\mathcal{L}v<0$ if the expression
in square brackets is negative. Thus, we can formulate the following
statement.

Theorem 6. If the conditions (44), (45) are met and

$a_{i}<\frac{i(\beta\varepsilon+2)}{2},i=\\{1,...,N_{\xi}\\},$ (48)

then the solution of the system (40)-(42) is stable in probability on the
whole for all fixed $y\in\mathbf{Y}$ and $h\in\mathbf{H}$.

##  5 Model example

Consider the linear stochastic differential equation

$dx(t)=a(\xi(t))x(t)dt+b(\xi(t))x(t)dw(t),t\geq 0,r>0,$ (49)

with impulse action

$\Delta x\left(2-\frac{1}{k}\right)=x\left(2-\frac{1}{k}-\right)+e^{-\alpha
k\eta_{k}}\left(x\left(2-\frac{1}{k}\right)\wedge
1\right),k\rightarrow\infty,$ (50)

and initial condition

$x(0)=10,\xi(0)=y_{0}\in\mathbf{Y},\eta_{0}=1.$ (51)

Here $a$ and $b$ are constants that depend on Markov process $\xi$ with values
in dimensional space $(\mathbf{Y},\mathcal{Y})$ with generator $Q$, and
$\eta_{k},k\geq 0,$ is Markov chain with two non-absorbing states $h_{1}=0$
and $h_{2}=1$.

According to [3] the solution of the system (49)-(51) exists, for example,
when $\alpha=1.673$.

Case 1. Let’s consider the same coefficients as in [3]:

\- if $\xi=1$: $a=1,b=0.3$;

\- if $\xi=2$: $a=-0.5,b=2.1$;

\- $\eta_{k}\in\\{1,2\\}$.

In this case condition (44) is not hold for $i=1$ because

$1-\frac{0.3^{2}}{2}=0.955>0.$

Therefore, the solution can be unstable. Indeed, if we consider an example of
the realization of the solution of the system (49)-(51) with indicated
parameters, then we observe a rapid growth (see Figure 1 (a)).

Figure 1:  Solution trajectories: (a) case 1 (unstable), (b) case 2 (stable),
(c) case 3 (unstable with an extreme growth at $t=2$). The red line
corresponds to the system’s solution $x(t)$ evolution. Blue marks - moments of
impulse actions.

Case 2. Next, we consider the values of the coefficients:

\- if $\xi=1$: $a=-1,b=0.3$;

\- if $\xi=2$: $a=0.5,b=2$;

\- $\eta_{k}\in\\{1,2\\}$.

Condition (44) for $i=1$ has the next form

$-1-\frac{0.3^{2}}{2}=-1.045<-\varepsilon,$

and for $i=2$ has the form

$0.5-\frac{2^{2}}{2}=-1.5<-\varepsilon,$

and holds for $\varepsilon=0.1$.

According to (45)

$\beta=\frac{0.1}{2^{2}}=0.025.$

And (48) hold:

\- if $i=1$: $-1<\frac{1\cdot(0.025\cdot 0.1+2)}{2}=1.00125$;

\- if $\xi=2$: $0.5<\frac{2\cdot(0.025\cdot 0.1+2)}{2}=2.0025$.

So, all conditions of Theorem 6 are held and the solution of the system
(49)-(51) with indicated parameters is stable in probability on the whole.
Indeed, in the realization (see Figure 1 (b)) we observe a direction to zero
after the point $t=3$.

Case 3. Here the values of the coefficients are the same as in case 2, but the
impulse action has the next form

$\Delta x\left(2-\frac{1}{k}\right)=x\left(2-\frac{1}{k}-\right)+e^{\alpha
k\eta_{k}}\left(x\left(2-\frac{1}{k}\right)\wedge
1\right),k\rightarrow\infty.$

In this case condition (47) does not hold and we cannot guarantee stability in
the probability of solution of the system (49)-(51): we observe a very rapid
growth (see Figure 1 (c)).

##  6 Discussion

The condition (44) means that stability in probability can be ensured due to
larger values of the coefficients and the fulfillment of the condition (47)
even when the system is unstable

$dx(t)=a_{i}x(t)dt.$

For example, if we consider the second case of the model example with the
coefficients $a=0.5,b=2$, we will see that the solution of the system
corresponding to the deterministic part is not Lyapunov stable, but the
solution of the stochastic system, as was demonstrated, is stable in
probability.

##  7 Conclusion

In this paper, we explore sufficient conditions for the asymptotic stability
of stochastic differential equations with a random structure, particularly in
the context of jump condensation points. Our main result is presented in
Theorem 2, which leverages the second Lyapunov method and involves the
construction of corresponding Lyapunov functions. An important consideration
in analyzing systems of random structure is the relationship between the
magnitudes of jumps, denoted as $L_{k}$, and the jump times, denoted as
$\tau_{k}$. The implications of Theorem 2 are demonstrated through an example
system whose stability can be modulated by varying parameters. We also
highlight a remarkable observation that the system can maintain asymptotic
stability even if, for some fixed value of the random process $\xi(t)$, the
system described by equation (1) becomes unstable when jumps (2) are absent.

In future studies, we plan to investigate the stability of stochastic
differential equations with a random structure, particularly when the jump
moments, denoted as $\tau_{k}$, are random variables satisfying the condition

$P\left(\lim\limits_{k\rightarrow\infty}t_{k}=t_{\infty}<\infty\right)>0,$

This implies a non-zero occurrence of condensation points. Furthermore, the
weak independence between the jumps and their corresponding moments will also
be considered as part of this analysis.

## Acknowledgments

We would like to acknowledge the administrations of the Luxembourg Institute
of Health (LIH) and Luxembourg National Research Fund (FNR) for their support
in organizing scientific contacts between research groups in Luxembourg and
Ukraine.

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