$p$-Adic Markov process and the problem of the first return over balls
UDC 511.225, 519.217, 511.225.1, 303.532 We consider the pseudodifferential operator defined as $H^{\alpha}\varphi = \mathcal{F}^{-1}[(\langle \xi\rangle^{\alpha} - p^{r\alpha})\mathcal{F}_{\varphi}],$ where $ \langle \xi \rangle= (\max\{|\xi|_{p}, p^r\})^{\alpha}$ and study the Markov process assoc...
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2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507029662072832 |
|---|---|
| author | Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. Casas-Sánchez, Oscar Galeano-Peñaloza, Jeanneth Rodríguez-Vega, J. J. Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. |
| author_facet | Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. Casas-Sánchez, Oscar Galeano-Peñaloza, Jeanneth Rodríguez-Vega, J. J. Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. |
| author_sort | Casas-Sánchez, O. F. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:47:53Z |
| description | UDC 511.225, 519.217, 511.225.1, 303.532
We consider the pseudodifferential operator defined as $H^{\alpha}\varphi = \mathcal{F}^{-1}[(\langle \xi\rangle^{\alpha} - p^{r\alpha})\mathcal{F}_{\varphi}],$ where $ \langle \xi \rangle= (\max\{|\xi|_{p}, p^r\})^{\alpha}$ and study the Markov process associated to this operator. We also study the first passage time problem associated to $H^{\alpha}$ for $r<0.$
  |
| doi_str_mv | 10.37863/umzh.v73i7.464 |
| first_indexed | 2026-03-24T02:02:49Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i7.464
UDC 511.225, 519.217, 511.225.1, 303.532
O. F. Casas-Sánchez (Univ. Pedagógica y Tecnológica Colombia, Tunja, Colombia),
J. Galeano-Peñaloza, J. J. Rodrı́guez-Vega (Dep. Mat., Univ. Nac. Colombia, Bogotá D.C., Colombia)
\bfitp -ADIC MARKOV PROCESS
AND THE PROBLEM OF THE FIRST RETURN OVER BALLS
\bfitp -АДИЧНI МАРКОВСЬКI ПРОЦЕСИ
ТА ЗАДАЧА ПЕРШОГО ПОВЕРНЕННЯ ДЛЯ КУЛЬ
We consider the pseudodifferential operator defined as H\alpha \varphi = \scrF - 1[(\langle \xi \rangle \alpha - pr\alpha )\scrF \varphi ], where \langle \xi \rangle \alpha = (\mathrm{m}\mathrm{a}\mathrm{x}\{ | \xi | p, pr\} )\alpha
and study the Markov process associated to this operator. We also study the first passage time problem associated to H\alpha
for r < 0.
Розглядається псевдодиференцiальний оператор H\alpha \varphi = \scrF - 1[(\langle \xi \rangle \alpha - pr\alpha )\scrF \varphi ], де \langle \xi \rangle \alpha = (\mathrm{m}\mathrm{a}\mathrm{x}\{ | \xi | p, pr\} )\alpha , та
вивчається пов’язаний iз цим оператором марковський процес. Також вивчається задача часу першого проходу для
H\alpha при r < 0.
1. Introduction. Avetisov et al. have constructed a wide variety of models of ultrametric diffusion
constrained by hierarchical energy landscapes (see [2, 3]). From a mathematical point of view, in
these models the time-evolution of a complex system is described by a p-adic master equation (a
parabolic-type pseudodifferential equation) which controls the time evolution of a transition function
of a random walk on an ultrametric space, and the random walk describes the dynamics of the system
in the space of configurational states which is approximated by an ultrametric space (\BbbQ p).
The problem of the first return in dimension 1 was studied by Avetisov, Bikulov and Zubarev
in [4, 5], and in arbitrary dimension by Chacón-Cortés, Torresblanca-Badillo and Zúñiga-Galindo in
[8, 15]. In these articles, pseudodifferential operators with radial symbols were considered. More
recently, Chacón-Cortés [7] considers pseudodifferential operators over \BbbQ 4
p with nonradial symbol;
he studies the problem of first return for a random walk X(t, w) whose density distribution satisfies
certain diffusion equation.
In this paper we define the operator
H\alpha \varphi = \scrF - 1[(\langle \xi \rangle \alpha - pr\alpha )\scrF \varphi ]
for \varphi \in \bfS (\BbbQ p), where \langle \xi \rangle = \mathrm{m}\mathrm{a}\mathrm{x}\{ | \xi | p, pr\} . We also define the heat-kernel Zr as
Zr(x, t) :=
\int
\BbbQ p
\chi ( - x\xi ) e - t
\bigl(
\langle \xi \rangle \alpha - pr\alpha
\bigr)
d\xi . (1.1)
Heat kernels of this type have been studied in [6], where it is shown that function
u(x, t) = Zr(x, t) \ast \Omega (| x| p) =
\int
\BbbQ p
\chi ( - x\xi )e - t(\langle \xi \rangle \alpha - pr\alpha )\Omega (| \xi | p) d\xi
is a solution of the Cauchy problem
c\bigcirc O. F. CASAS-SÁNCHEZ, J. GALEANO-PEÑALOZA, J. J. RODRIGUEZ-VEGA, 2021
902 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
p-ADIC MARKOV PROCESS AND THE PROBLEM OF THE FIRST RETURN OVER BALLS 903
u \in C
\bigl(
[0,\infty ],\bfS (\BbbQ p)
\bigr)
\cap C1
\bigl(
[0,\infty ], L2(\BbbQ p)
\bigr)
,
\partial u
\partial t
(x, t) +
\bigl(
H\alpha u
\bigr)
(x, t) = 0, x \in \BbbQ p, t \in (0, T ], \alpha > 0,
u(x, 0) = \Omega (| x| p).
We show that Zr(x, t) is the transition density of a time and space homogeneous Markov process,
which is bounded, right-continuous and has no discontinuities other than jumps, see Theorem 4.1.
In [12] Kochubei considers the Vladimirov operator restricted to a ball BN and studies a Cauchy
problem. Despite he uses a different approach to the one given by Casas-Sánchez and Rodrı́guez-
Vega in [6], the kernel Zr (1.1) is the same. On the other hand, Khrennikov and Kochubei (see [11])
show that the family of operators Zr \ast \cdot is a strongly continuous contraction semigroup on L1(Br).
Among other properties, the kernel Zr(x, t) vanishes outside the ball of radius p - r, which implies
that the process remains supported in the same ball. For that reason, we are interested in the case
r < 0, and thus \BbbZ p \subseteq B - r. In these conditions it is possible to study the problem of the first return
by a trajectory of the stochastic process entering the unit ball. In order to solve this problem we
demand that r < 0 and the natural answer is that the trajectory is always recurrent. Observe that
this problem is different to the one solved by Bikulov in [5], where the author define the stochastic
quantity as the first passage time entering some domain Br(a), since his solution is not bounded, the
answer depends on the range of \alpha , whereas we do not have conditions on \alpha . Our work can be seen
as a continuation of the problem of first return for a stochastic process, considered by Avetisov in
[4], since we use the same techniques, but a different symbol.
The article is organized as follows. In Section 2, we collect some facts about p-adic numbers.
In Section 3, we define the pseudodifferential operator, we show it has an integral representation and
solve the Cauchy problem based on the results of [6]. Section 4 is dedicated to the p-adic Markov
process over balls. In Section 5, we study the problem of the first passage time entering the domain
\BbbZ p, we conclude that the process is always recurrent with respect to the unit ball, see Theorem 5.1.
2. Preliminaries. In this section, we fix the notation and collect some p-adic facts that we
will use through the article. For a detailed exposition on p-adic analysis the reader may consult
[1, 14, 16].
2.1. The field of \bfitp -adic numbers. Along this article p will denote a prime number. The field of
p-adic numbers \BbbQ p is defined as the completion of the field of rational numbers \BbbQ with respect to
the p-adic norm | \cdot | p, which is defined as
| x| p =
\left\{ 0, if x = 0,
p - \gamma , if x = pr
a
b
,
where a and b are integers coprime with p. The integer \gamma := \mathrm{o}\mathrm{r}\mathrm{d}(x), with \mathrm{o}\mathrm{r}\mathrm{d}(0) := +\infty , is called
the p-adic order of x.
Any p-adic number x \not = 0 has a unique expansion x = pord(x)
\sum \infty
j=0
xjp
j , where xj \in
\in \{ 0, 1, 2, . . . , p - 1\} and x0 \not = 0. By using this expansion, we define the fractional part of x \in \BbbQ p,
denoted \{ x\} p, as the rational number
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
904 O. F. CASAS-SÁNCHEZ, J. GALEANO-PEÑALOZA, J. J. RODRIGUEZ-VEGA
\{ x\} p =
\left\{
0, if x = 0 or \mathrm{o}\mathrm{r}\mathrm{d}(x) \geq 0,
pord(x)
\sum - ord(x) - 1
j=0
xjp
j , if \mathrm{o}\mathrm{r}\mathrm{d}(x) < 0.
For r \in \BbbZ , denote by Br(a) = \{ x \in \BbbQ p : | x - a| p \leq pr\} the ball of radius pr with center at a \in \BbbQ p,
and take Br(0) := Br.
2.2. The Bruhat – Schwartz space. A complex-valued function \varphi defined on \BbbQ p is called locally
constant if for any x \in \BbbQ p there exists an integer l(x) \in \BbbZ such that
\varphi (x+ x\prime ) = \varphi (x) for x\prime \in Bl(x). (2.1)
The space of locally constant functions is denoted by \scrE (\BbbQ p). A function \varphi : \BbbQ p \rightarrow \BbbC is called a
Bruhat – Schwartz function (or a test function) if it is locally constant with compact support. The
\BbbC -vector space of Bruhat – Schwartz functions is denoted by \bfS (\BbbQ p). For \varphi \in \bfS (\BbbQ p), the largest of
such numbers l = l(\varphi ) satisfying (2.1) is called the exponent of local constancy of \varphi .
Let \bfS \prime (\BbbQ p) denote the set of all functionals (distributions) on \bfS (\BbbQ p). All functionals on \bfS (\BbbQ p)
are continuous.
Set \chi (y) = \mathrm{e}\mathrm{x}\mathrm{p}(2\pi i\{ y\} p) for y \in \BbbQ p. The map \chi (\cdot ) is an additive character on \BbbQ p, i.e., a
continuous map from \BbbQ p into S (the unit circle) satisfying \chi (y0 + y1) = \chi (y0)\chi (y1), y0, y1 \in \BbbQ p.
2.3. Fourier transform. Given \xi and x \in \BbbQ p, the Fourier transform of \varphi \in \bfS (\BbbQ p) is defined as
(\scrF \varphi )(\xi ) =
\int
\BbbQ p
\chi (\xi x)\varphi (x)dx for \xi \in \BbbQ p,
where dx is the Haar measure on \BbbQ p normalized by the condition \mathrm{v}\mathrm{o}\mathrm{l}(B0) = 1. The Fourier
transform is a linear isomorphism from \bfS (\BbbQ p) onto itself satisfying (\scrF (\scrF \varphi ))(\xi ) = \varphi ( - \xi ). We will
also use the notation \scrF x\rightarrow \xi \varphi and \widehat \varphi for the Fourier transform of \varphi .
The Fourier transform \scrF [f ] of a distribution f \in \bfS \prime (\BbbQ p) is defined by
(\scrF [f ] , \varphi ) = (f,\scrF [\varphi ]) for all \varphi \in \bfS (\BbbQ p) .
The Fourier transform f \rightarrow \scrF [f ] is a linear isomorphism from \bfS \prime (\BbbQ p) onto \bfS \prime (\BbbQ p) . Furthermore,
f = \scrF [\scrF [f ] ( - \xi )] .
3. Pseudodifferential operators.
Definition 3.1. For all \alpha \in \BbbC , we define the following pseudodifferential operator:
H\alpha \varphi = \scrF - 1[(\langle \xi \rangle \alpha - pr\alpha )\scrF \varphi ], \varphi \in \bfS (\BbbQ p),
where \langle \xi \rangle \alpha = \mathrm{m}\mathrm{a}\mathrm{x}\{ | \xi | p, pr\} .
It is clear that the map H\alpha : \bfS (\BbbQ p) \rightarrow \bfS (\BbbQ p) is continuous. Also it is possible to show that the
pseudodifferential operator H\alpha has the following integral representation:
(H\alpha \varphi )(x) =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
p-ADIC MARKOV PROCESS AND THE PROBLEM OF THE FIRST RETURN OVER BALLS 905
=
1 - p\alpha
1 - p\alpha +1
\left[ pr(\alpha +1)
\int
| y| p\leq p - r
(\varphi (x - y) - \varphi (x))dy -
- p(\alpha +1)
\int
| y| p\leq p - r
\varphi (x - y) - \varphi (x)
| y| \alpha +1
p
dy
\right] .
Definition 3.2. Set \alpha k :=
2k\pi i
ln p
, k \in \BbbZ ,
K\alpha (x) :=
\left\{
\biggl[
1 - p\alpha
1 - p - \alpha - 1
| x| - \alpha - 1
p + pr(\alpha +1) 1 - p\alpha
1 - p\alpha +1
\biggr]
\Omega (pr| x| p) for \alpha \not = - 1 + \alpha k,
(1 - p - 1)\Omega (pr| x| p)((1 - r) - \mathrm{l}\mathrm{o}\mathrm{g}p| x| p) for \alpha = - 1 + \alpha k,
and, for \alpha = 0, we define K0 = \delta .
After some calculations it is possible to show the following result.
Theorem 3.1. The Fourier transform (as a distribution) of K\alpha is given by \langle \xi \rangle \alpha for all \alpha \in \BbbC .
Definition 3.3. For x \in \BbbQ p, t \in \BbbR , the heat kernel is defined as
Zr(x, t) :=
\int
\BbbQ p
\chi ( - x\xi ) e - t
\bigl(
\langle \xi \rangle \alpha - pr\alpha
\bigr)
d\xi .
The following properties are proved in [6].
Lemma 3.1. For \alpha > 0, t > 0, the following assertions hold:
(1) Zr(x, t) \in C(\BbbQ p,\BbbR ) \cap L1(\BbbQ p) \cap L2(\BbbQ p) for t > 0,
(2) Zr(x, t) \geq 0 for all x \in \BbbQ p,
(3)
\int
\BbbQ p
Zr(x, t) dx =
\int
| x| p\leq p - r
Zr(x, t) dx = 1,
(4) \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0+ Zr(x, t) \ast \varphi (x) = \varphi (x) for \varphi \in \bfS (\BbbQ p),
(5) Zr(x, t) \ast Z(x, t\prime ) = Z(x, t+ t\prime ) for t, t\prime > 0,
(6) Zr(x, t) \leq Ct| x| - 1
p
\bigl( \bigl\langle
px - 1
\bigr\rangle \alpha - pr\alpha
\bigr)
.
Observe that thanks to (3) the heat kernel is concentrated in the ball B - r.
If we set, for \varphi \in \bfS (\BbbQ p),
u(x, t) :=
\left\{ Zr(x, t) \ast \varphi (x), if t > 0,
\varphi (x), if t = 0,
(3.1)
then it is easy to see that u(x, t) \in \bfS (\BbbQ p) for t \geq 0, and also it is possible to show that, for t \geq 0,
\alpha > 0,
H\alpha (u(x, t)) = \scrF - 1
\xi \rightarrow x
\Bigl[
(\langle \xi \rangle \alpha - pr\alpha )e - t(\langle \xi \rangle \alpha - pr\alpha ) \widehat \varphi (\xi )\Bigr] .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
906 O. F. CASAS-SÁNCHEZ, J. GALEANO-PEÑALOZA, J. J. RODRIGUEZ-VEGA
Theorem 3.2. Consider the following Cauchy problem:
u \in C
\bigl(
[0,\infty ],\bfS (\BbbQ p)
\bigr)
\cap C1
\bigl(
[0,\infty ], L2(\BbbQ p)
\bigr)
,
\partial u
\partial t
(x, t) +
\bigl(
H\alpha u
\bigr)
(x, t) = 0, x \in \BbbQ p, t \in (0, T ], \alpha > 0,
u(x, 0) = \varphi (x), \varphi \in \bfS (\BbbQ p).
Then the function u(x, t) defined in (3.1) is the solution.
Proof. See Theorem 3.14 in [6].
Another interpretation for the fundamental solution Zr was obtained in [12].
4. \bfitp -Adic Markov process over balls. The space (\BbbQ p, | \cdot | p) is a complete non-Archimedean
metric space. Let \scrB be the Borel \sigma -algebra of pr\BbbZ p; thus (pr\BbbZ p,\scrB , dx) is a measure space. By
using the terminology and results of [9] (Chapters 2, 3), we set
p(t, x, y) := Zr(x - y, t), t > 0, x, y \in pr\BbbZ p,
and
P (t, x,B) =
\left\{
\int
B
p(t, x, y) dy for t > 0, x \in \BbbQ p, B \in \scrB ,
1B(x) for t = 0.
In this case the Markov process remains in the ball pr\BbbZ p, r < 0.
Lemma 4.1. With the above notation the following assertions hold:
(i) p(t, x, y) is a normal transition density,
(ii) P (t, x,B) is a normal transition function.
Proof. The result follows from Lemma 3.1 (see [9] (Section 2.1) for further details).
Lemma 4.2. The transition function P (t, x,B) satisfies the following two conditions:
(i) for each u \geq 0 and compact B,
\mathrm{l}\mathrm{i}\mathrm{m}
| x| p\rightarrow \infty
\mathrm{s}\mathrm{u}\mathrm{p}
t\leq u
P (t, x,B) = 0;
(ii) for each \epsilon > 0 and compact B,
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
\mathrm{s}\mathrm{u}\mathrm{p}
x\in B
P (t, x,\BbbQ p \setminus B\epsilon (x)) = 0.
Proof. (i) By Lemma 3.1 (6), we have
P (t, x,B) =
\int
B
Zr(x - y, t)dy \leq Ct
\int
B
| x - y| - 1
p
\bigl( \bigl\langle
p(x - y) - 1
\bigr\rangle \alpha - pr\alpha
\bigr)
dy =
= Ct| x| - 1
p
\bigl( \bigl\langle
px - 1
\bigr\rangle \alpha - pr\alpha
\bigr) \int
B
dy,
since, for x \in \BbbQ p \setminus B, we have | x| p = | x - y| p. Therefore, \mathrm{l}\mathrm{i}\mathrm{m}| x| p\rightarrow \infty \mathrm{s}\mathrm{u}\mathrm{p}t\leq u P (t, x,B) = 0.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
p-ADIC MARKOV PROCESS AND THE PROBLEM OF THE FIRST RETURN OVER BALLS 907
(ii) By using Lemma 3.1 (6), \alpha > 0, we have
P (t, x,\BbbQ p \setminus B\epsilon (x)) \leq Ct
\int
| x - y| >\epsilon
| x - y| - 1
p
\bigl( \bigl\langle
p(x - y) - 1
\bigr\rangle \alpha - pr\alpha
\bigr)
dy =
= Ct
\int
| z| >\epsilon
| z| - 1
p
\bigl( \bigl\langle
p z - 1
\bigr\rangle \alpha - pr\alpha
\bigr)
dz.
If p - r - 1 \leq \epsilon < | z| p or \epsilon < p - r - 1 \leq | z| p, then
\bigl\langle
p z - 1
\bigr\rangle \alpha
= pr\alpha and\int
| z| >\epsilon
| z| - 1
p
\bigl( \bigl\langle
p z - 1
\bigr\rangle \alpha - pr\alpha
\bigr)
dz = 0.
Thus,
P (t, x,\BbbQ p \setminus B\epsilon (x)) \leq Ct
\int
| x - y| >\epsilon
| x - y| - 1
p
\bigl( \bigl\langle
p(x - y) - 1
\bigr\rangle \alpha - pr\alpha
\bigr)
dy =
= Ct
\int
p - r - 1>| z| >\epsilon
| z| - 1
p
\bigl(
| p z - 1| \alpha p - pr\alpha
\bigr)
dz \leq
\leq Ctp - 1
\int
p - r - 1>| z| >\epsilon
| z| - 1 - \alpha
p dz =
= Ctp - 1C1.
Therefore,
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
\mathrm{s}\mathrm{u}\mathrm{p}
x\in B
P (t, x,\BbbQ p \setminus B\epsilon (x)) = 0.
Lemma 4.2 is proved.
Theorem 4.1. Zr(x, t) is the transition density of a time and space homogeneous Markov pro-
cess in pr\BbbZ p, called T(t, \omega ), which is bounded, right-continuous and has no discontinuities other
than jumps.
Proof. The result follows from [9] (Theorem 3.6) by using that (\BbbQ p, | x| p) is a semicompact
space, i.e., a locally compact Hausdorff space with a countable base, and P (t, x,B) is a normal
transition function satisfying conditions (i) and (ii).
5. The first passage time. From now on, we assume r < 0 and we study the problem of the
first return to the domain \BbbZ p.
By Theorem 3.2, the function
u(x, t) = Zr(x, t) \ast \Omega (| x| p) =
\int
\BbbQ p
\chi ( - x\xi )e - t(\langle \xi \rangle \alpha - pr\alpha )\Omega (| \xi | p) d\xi (5.1)
is a solution of
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
908 O. F. CASAS-SÁNCHEZ, J. GALEANO-PEÑALOZA, J. J. RODRIGUEZ-VEGA
\partial u
\partial t
(x, t) + (H\alpha u)(x, t) = 0, x \in \BbbQ p, t > 0,
u(x, 0) = \Omega (| x| p).
Among other properties, the function u(x, t) = Zr(x, t) \ast \Omega (| x| p), t \geq 0, is pointwise differentiable
in t and, by using the dominated convergence theorem, we can show that its derivative is given by
the formula
\partial u
\partial t
(x, t) =
\int
\BbbQ p
\chi p( - x\xi ) (\langle \xi \rangle \alpha - pr\alpha ) e - t(\langle \xi \rangle \alpha - pr\alpha )\Omega (| \xi | p) d\xi . (5.2)
Lemma 5.1. If \alpha > 0 and r < 0, then
0 < -
\int
1<| y| p\leq p - r
K\alpha (y)dy < 1.
Proof. We have
-
\int
1<| y| p\leq p - r
K\alpha (y)dy =
=
1 - p\alpha
1 - p\alpha +1
\left[ p\alpha +1
\int
1<| y| p\leq p - r
1
| y| \alpha +1
p
dy - pr(\alpha +1)
\int
1<| y| p\leq p - r
dy
\right] <
<
1 - p\alpha
1 - p\alpha +1
\left[ p\alpha +1
\int
1<| y| p
1
| y| \alpha +1
p
dy - pr(\alpha +1)(p - r - 1)
\right] =
=
1 - p - 1
1 - p - \alpha - 1
- 1 - p\alpha
1 - p\alpha +1
pr\alpha (1 - pr) =
= 1 - 1 - p\alpha
1 - p\alpha +1
(1 + pr\alpha (1 - pr)) < 1.
Now
-
\int
1<| y| p\leq p - r
K\alpha (y)dy =
=
1 - p\alpha
1 - p\alpha +1
\left[ p\alpha +1
\int
1<| y| p\leq p - r
1
| y| \alpha +1
p
dy - pr(\alpha +1)
\int
1<| y| p\leq p - r
dy
\right] >
>
p\alpha (p - 1)(1 - pr\alpha )
p\alpha +1 - 1
+
pr\alpha (1 - p\alpha )
p\alpha +1 - 1
> 0.
Lemma 5.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
p-ADIC MARKOV PROCESS AND THE PROBLEM OF THE FIRST RETURN OVER BALLS 909
The rest of this section is dedicated to the study of the following random variable, by using the
same techniques given in [4].
Definition 5.1. The random variable \tau \BbbZ p(\omega ) : Y \rightarrow \BbbR + defined by
\mathrm{i}\mathrm{n}\mathrm{f}\{ t > 0 | T(t, \omega ) \in \BbbZ p and there exists t\prime such that 0 < t\prime < t and T(t\prime , \omega ) /\in \BbbZ p\}
is called the first passage time of a path of the random process T(t, \omega ) entering the domain \BbbZ p.
Lemma 5.2. The probability density function for a path of T(t, \omega ) to enter into \BbbZ p at the instant
of time t, with the condition that T(0, \omega ) \in \BbbZ p is given by
g(t) =
\int
1<| y| p\leq p - r
K\alpha (y)u(y, t)dy. (5.3)
Proof. We first note that, for x, y \in \Omega (| z| p), we have
u(x - y, t) =
\int
\BbbZ p
\chi p( - (x - y) \cdot \xi )e - t(\langle \xi \rangle \alpha - pr\alpha ) d\xi =
=
\int
\BbbZ p
e - t(\langle \xi \rangle \alpha - pr\alpha ) d\xi =
\int
\BbbZ p
\chi p( - x \cdot \xi )e - t(\langle \xi \rangle \alpha - pr\alpha ) d\xi =
= u(x, t),
i.e., u(x - y, t) - u(x, t) \equiv 0 for x, y \in \BbbZ p.
The survival probability, by definition
S(t) := S\Omega (| x| p)(t) =
\int
\BbbZ p
u(x, t)dnx,
is the probability that a path of T(t, \omega ) remains in \BbbZ p at the time t. Because there are no external or
internal sources,
S\prime (t) =
Probability that a path of T(t, \omega )
goes back to \BbbZ p at the time t
- Probability that a path of T(t, \omega )
exits \BbbZ p at the time t
=
= g(t) - C \cdot S(t) with 0 < C \leq 1.
By using the derivative (5.2), we have
S\prime (t) =
\int
\BbbZ p
\partial u(x, t)
\partial t
dx =
= - 1 - p\alpha
1 - p\alpha +1
\left[ pr(\alpha +1)
\int
| x| p\leq 1
\int
1<| y| p\leq p - r
(u(x - y, t) - u(x, t))dydx -
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
910 O. F. CASAS-SÁNCHEZ, J. GALEANO-PEÑALOZA, J. J. RODRIGUEZ-VEGA
- p(\alpha +1)
\int
| x| p\leq 1
\int
1<| y| p\leq p - r
u(x - y, t) - u(x, t)
| y| \alpha +1
p
dydx
\right] =
= - 1 - p\alpha
1 - p\alpha +1
\left[ pr(\alpha +1)
\int
| x| p\leq 1
\int
1<| y| p\leq p - r
u(x - y, t) dydx -
- p(\alpha +1)
\int
| x| p\leq 1
\int
1<| y| p\leq p - r
u(x - y, t)
| y| \alpha +1
p
dydx
\right] +
+
1 - p\alpha
1 - p\alpha +1
\left[ pr(\alpha +1)
\int
| x| p\leq 1
\int
1<| y| p\leq p - r
u(x, t) dydx -
- p(\alpha +1)
\int
| x| p\leq 1
\int
1<| y| p\leq p - r
u(x, t)
| y| \alpha +1
p
dydx
\right] .
Now if y \in \Omega (pr| y| p) \setminus \Omega (| y| p) and x \in \Omega (| x| p), then u(x - y, t) = u(y, t), consequently,
S\prime (t) =
\int
1<| y| p\leq p - r
K\alpha (y)u(y, t)dy+
+
\int
1<| y| p\leq p - r
K\alpha (y)dy
\int
| x| p\leq 1
u(x, t) dx =
= g(t) - CS(t),
where
C = -
\int
1<| y| p\leq p - r
K\alpha (y)dy.
Lemma 5.2 is proved.
Proposition 5.1. The probability density function f(t) of the random variable \tau \BbbZ p(\omega ) satisfies
the nonhomogeneous Volterra equation of the second kind
g(t) =
\infty \int
0
g(t - \tau )f(\tau )d\tau + f(t).
Proof. The result follows from Lemma 5.2 by using the argument given in the proof of
Theorem 1 in [4].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
p-ADIC MARKOV PROCESS AND THE PROBLEM OF THE FIRST RETURN OVER BALLS 911
Proposition 5.2. The Laplace transform Gr(s) of g(t) is given by
Gr(s) =
\int
1<| y| p\leq p - r
K\alpha (y)
\int
| \xi | p\leq 1
\chi p( - \xi \cdot y)
s+ (\langle \xi \rangle \alpha - pr\alpha )
d\xi dy.
Proof. We first note that
e - stK\alpha (y)e
- t(\langle \xi \rangle \alpha - pr\alpha )\Omega (| \xi | p) \in \scrL 1
\bigl(
(0,\infty )\times \Omega (pr| \xi | p) \setminus \Omega (| \xi | p)\times \BbbQ p, dtdyd\xi
\bigr)
for s \in \BbbC with \mathrm{R}\mathrm{e}(s) > 0. The announced formula follows now from (5.1) and (5.3) by using
Fubini’s theorem.
Definition 5.2. We say that T(t, \omega ) is recurrent with respect to \BbbZ p if
P (\{ \omega \in Y : \tau \BbbZ p(\omega ) < \infty \} ) = 1. (5.4)
Otherwise, we say that T(t, \omega ) is transient with respect to \BbbZ p.
The meaning of (5.4) is that every path of T(t, \omega ) is sure to return to \BbbZ p. If (5.4) does not hold,
then there exist paths of T(t, \omega ) that abandon \BbbZ p and never go back.
Theorem 5.1. For all \alpha > 0, the processes T(t, \omega ) is recurrent with respect to \BbbZ p.
Proof. By Proposition 5.1, the Laplace transform F (s) of f(t) equals
Gr(s)
1 +Gr(s)
, where Gr(s)
is the Laplace transform of g(t), and thus
F (0) =
\infty \int
0
f(t)dt = 1 - 1
1 +Gr(0)
.
Hence in order to prove that T(t, \omega ) is recurrent is sufficient to show that
Gr(0) = \mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0
Gr(s) = \infty ,
and to prove that it is transient that
Gr(0) = \mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0
Gr(s) < \infty ,
Gr(s) =
\int
1<| y| p\leq p - r
\int
| \xi | p\leq pr
K\alpha (y)\chi ( - \xi y)
s
d\xi dy+
+
\int
1<| y| p\leq p - r
\int
pr<| \xi | p\leq 1
K\alpha (y)\chi ( - \xi y)
s+ | \xi | \alpha p - pr\alpha
d\xi dy =
=
pr
s
\int
1<| y| p\leq p - r
K\alpha (y)dy +
\int
1<| y| p\leq p - r
\int
pr<| \xi | p\leq 1
K\alpha (y)\chi ( - \xi y)
s+ | \xi | \alpha p - pr\alpha
d\xi dy =
=
pr
s
\int
1<| y| p\leq p - r
K\alpha (y)dy +
- r\sum
k=1
k - 1\sum
m=0
pk - m
s+ p - m\alpha - pr\alpha
\int
| u| p=1
K\alpha (p
- ku)du+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
912 O. F. CASAS-SÁNCHEZ, J. GALEANO-PEÑALOZA, J. J. RODRIGUEZ-VEGA
+
- r\sum
k=1
- r - 1\sum
m=k
pk - m
s+ p - m\alpha - pr\alpha
\int
| u| p=1
\int
| v| p=1
K\alpha (p
- ku)\chi ( - pm - kuv)dvdu.
Therefore, \mathrm{l}\mathrm{i}\mathrm{m}s\rightarrow 0Gr(s) = \infty and the process T(t, \omega ) is recurrent.
Theorem 5.1 is proved.
The meaning of this result is that every path is sure to return to \BbbZ p, this always holds and the
process is never transient, this agrees with the fact that the process is concentrated in pr\BbbZ p, r < 0.
References
1. S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, Theory of p-adic distributions. Linear and nonlinear models,
London Math. Soc. Lect. Note Ser. 370, Cambridge Univ. Press (2010).
2. V. A. Avetisov, A. Kh. Bikulov, Protein ultrametricity and spectral diffusion in deeply frozen proteins, Biophys. Rev.
Lett., 3, № 3, 387 – 396 (2008).
3. V. A. Avetisov, A. Kh. Bikulov, S. V. Kozyrev, V. A. Osipov, p-Adic models of ultrametric diffusion constrained by
hierarchical energy landscapes, J. Phys. A: Math. and Gen., 35, 177 – 189 (2002).
4. V. A. Avetisov, A. Kh. Bikulov, A. P. Zubarev, First passage time distribution and the number of returns for
ultrametric random walks, J. Phys. A: Math. and Theor., 42, Article 085003 (2009).
5. A. Kh. Bikulov, Problem of the first passage time for p-adic diffusion, p-Adic numbers, Ultrametric Anal. and Appl.,
2, № 2, 89 – 99 (2010).
6. O. F. Casas-Sánchez, J. J. Rodrı́guez-Vega, Parabolic type equations on p-adic balls, Bol. Mat., 22, № 1, 97 – 106
(2015).
7. L. F. Chacón-Cortés, The problem of the first passage time for some elliptic pseudodifferential operators over the
p-adics, Rev. Colombiana Mat., 48, № 2, 191 – 209 (2014).
8. L. F. Chacón-Cortés, W. A. Zúñiga-Galindo, Nonlocal operators, parabolic-type equations, and ultrametric random
walks, J. Math. Phys., 54, № 11, Article 113503 (2013).
9. E. B. Dynkin, Markov processes, vol. I. Springer-Verlag (1965).
10. A. Y. Khrennikov, A. N. Kochubei, On the p-adic Navier – Stokes equation, Appl. Anal. (2018); DOI:
10.1080/00036811.2018.1533120.
11. A. Y. Khrennikov, A. N. Kochubei, p-Adic analogue of the porous medium equation, J. Fourier Anal. and Appl.
(2017); DOI 10.1007/s00041-017-9556-4.
12. A. N. Kochubei, Linear and nonlinear heat equations on a p-adic ball, Ukr. Math. J., 70, № 2, 217 – 231 (2018).
13. A. N. Kochubei, Pseudo-differential equations and stochastics over non-Archimedean fields, Marcel Dekker, New
York (2001).
14. M. H. Taibleson, Fourier analysis on local fields, Princeton Univ. Press (1975).
15. A. Torresblanca-Badillo, W. A. Zúñiga-Galindo, Ultrametric diffusion, exponential landscapes, and the first passage
time problem, Acta Appl. Math., 1 – 24 (2018).
16. V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, p-Adic analysis and mathematical physics, Ser. Soviet and East Europ.
Math., vol. 1, World Sci., River Edge, NJ (1994).
Received 01.11.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
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| id | umjimathkievua-article-464 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:49Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/91/e05d62b27eef7aaac3cadda8cfbb6591.pdf |
| spelling | umjimathkievua-article-4642025-03-31T08:47:53Z $p$-Adic Markov process and the problem of the first return over balls p-adic Markov process and the problem of the first return over balls $p$-Adic Markov process and the problem of the first return over balls Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. Casas-Sánchez, Oscar Galeano-Peñaloza, Jeanneth Rodríguez-Vega, J. J. Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. Random walks ultradiffusion p-adic numbers non-archimedean analysis Random walks ultradiffusion p-adic numbers non-archimedean analysis UDC 511.225, 519.217, 511.225.1, 303.532 We consider the pseudodifferential operator defined as $H^{\alpha}\varphi = \mathcal{F}^{-1}[(\langle \xi\rangle^{\alpha} - p^{r\alpha})\mathcal{F}_{\varphi}],$ where $ \langle \xi \rangle= (\max\{|\xi|_{p}, p^r\})^{\alpha}$ and study the Markov process associated to this operator. We also study the first passage time problem associated to $H^{\alpha}$ for $r&lt;0.$ &nbsp; UDC 511.225, 519.217, 511.225.1, 303.532 $p$ -адичнi марковськi процеси та задача першого повернення для куль Розглядається псевдодиференціальний оператор $H^{\alpha}\varphi = \mathcal{F}^{-1}[(\langle \xi\rangle^{\alpha} - p^{r\alpha})\mathcal{F}_{\varphi}],$ де $ \langle \xi \rangle= (\max\{|\xi|_{p}, p^r\})^{\alpha}$ та вивчається пов'язаний із цим оператором марковський процес. Також вивчається задача часу першого проходу для $H^{\alpha}$ при $r&lt;0.$ Institute of Mathematics, NAS of Ukraine 2021-07-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/464 10.37863/umzh.v73i7.464 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 7 (2021); 902 - 912 Український математичний журнал; Том 73 № 7 (2021); 902 - 912 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/464/9064 Copyright (c) 2021 Jeanneth Galeano-Peñaloza, Oscar Casas-Sánchez, John Rodríguez-Vega |
| spellingShingle | Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. Casas-Sánchez, Oscar Galeano-Peñaloza, Jeanneth Rodríguez-Vega, J. J. Casas-Sánchez, O. F. Galeano-Peñaloza, J. Rodríguez-Vega, J. J. $p$-Adic Markov process and the problem of the first return over balls |
| title | $p$-Adic Markov process and the problem of the first return over balls |
| title_alt | p-adic Markov process and the problem of the first return over balls $p$-Adic Markov process and the problem of the first return over balls |
| title_full | $p$-Adic Markov process and the problem of the first return over balls |
| title_fullStr | $p$-Adic Markov process and the problem of the first return over balls |
| title_full_unstemmed | $p$-Adic Markov process and the problem of the first return over balls |
| title_short | $p$-Adic Markov process and the problem of the first return over balls |
| title_sort | $p$-adic markov process and the problem of the first return over balls |
| topic_facet | Random walks ultradiffusion p-adic numbers non-archimedean analysis Random walks ultradiffusion p-adic numbers non-archimedean analysis |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/464 |
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