New classes of p-adic evolution equations and their applications

Abstract

In this article, we study new classes of evolution equations in the p-adic context. We establish rigorously that the fundamental solutions of the homogeneous Cauchy problem, naturally associated to these equations, are transition density functions of some strong Markov processes ${\mathfrak {X}}$ with state space the n-dimensional p-adic unit ball (${\mathbb {Z}}_{p}^{n}$). We introduce a family of operators $\{T_{t}\}_{t\ge 0}$ (obtained explicitly) that determine a Feller semigroup on $C_{0}({\mathbb {Z}}_{p}^{n})$. Also, we study the asymptotic behavior of the survival probability of a strong Markov processes ${\mathfrak {X}}$ on a ball $B_{-m}^{n}\subset {\mathbb {Z}}_{p}^{n}$, $m\in {\mathbb {N}}$. Moreover, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the mentioned above Feller semigroup.

1 Introduction

The interest on pseudo-differential operators in the p-adic context has been considerably strengthened during the last years due to its usefulness in the modeling of different kind of physical phenomenon. For example, modeling of geological processes (such as formation of petroleum micro-scale reservoir and fluid flows in porous media such as rock); the dynamics of complex systems such as macromolecules, glasses, proteins; the study of Coulomb gases, etc., see e.g., [5, 6, 12, 19, 24, 34, 35, 38], and the references therein.

On the other hand, in the archimedean setting, nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems and coagulation models. Nonlocal evolution equations of the form

$$\begin{aligned} u_{t}(x,t)=(J*u-u)(x,t)=\int _{{\mathbb {R}}^{n}}J(x-y)u(y,t)dy-u(x,t), \end{aligned}$$

(1.1)

and variations of it, have been recently widely used to model diffusion processes. Here, $J:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ be a nonnegative, radial, continuous function with $\int _{{\mathbb {R}}^{n}}J(z)dz=1$. Following [14], the model (1.1) can be interpreted as follows: if u(x, t) is thought of as a density at a point x at time t and $J(x-y)$ is thought of as the probability distribution of jumping from location y to location x, then $\int _{{\mathbb {R}}^{n}}J(y-x)u(y,t)dy=(J*u)(x,t)$ is the rate at which individuals are arriving at position x from all other places. In the same way, $-u(x,t)=-\int _{{\mathbb {R}}^{n}}J(y-x)u(x,t)dy$ is the rate at which they are leaving location x to travel to all other sites. This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies equation (1.1). Equation (1.1), is called nonlocal diffusion equation since the diffusion of the density u at a point x and time t does not only depend on u(x, t), but on all the values of u in a neighborhood of x through the convolution term $J*u$. For further details the reader may consult [1, 8, 10, 11, 16,17,18], et al.

In [6, 7] Avetisov, et al, developed class of p-adic pseudo-differential equations in dimension one (the p-adic counterpart of equation (1.1)) with the aim of studying the dynamics of a large class of complex systems such as macromolecules, glasses and proteins. In these models, the time-evolution of the system is controlled by a master equation of the form

$$\begin{aligned} \frac{\partial u\left( x,t\right) }{\partial t}=\int _{{\mathbb {Q}} _{p}}j\left( \left| x-y\right| _{p}\right) \left\{ u\left( y,t\right) -u\left( x,t\right) \right\} dy, \ \ \text { } t\ge 0\text {,} \end{aligned}$$

(1.2)

where ${\mathbb {Q}}_{p}$ is the field of p-adic numbers, $j:{\mathbb {Q}}_{p}\times {\mathbb {Q}}_{p}\rightarrow {\mathbb {R}}_{+}$ is the probability of transition from state y to the state x per unit time, and the function $u(x,t):{\mathbb {Q}}_{p}\times {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}_{+}$ is a probability density distribution.

In the latest years, equation (1.2) and variations of it, have been recently widely used to model the dynamics of a broad class complex systems through a random walk on a complex energy landscape, see e.g. [15, 20, 21, 26, 27, 29, 36], et al. An energy landscape (or simply a landscape) is a mapping of all possible conformations of a molecular entity, or the spatial positions of interacting molecules in a system. Mathematically, an energy landscape is a continuous function ${\mathbb {U}}:X \rightarrow {\mathbb {R}}$ that assigns to each physical state of a system its energy, where X is a topological space. The term complex landscape means that function ${\mathbb {U}}$ has many local minima. In this case the method of interbasin kinetics is applied, in this approach, the study of a random walk on a complex landscape is based on a description of the kinetics generated by transitions between groups of states (basins), see e.g. [9, 30, 31, 36]. By using these methods, a complex landscape is approximated by a disconnectivity graph (a rooted tree) and by a function on the tree describing the distribution of the activation energies. From all the above, we have that the dynamics of several complex systems can be described as a random walk on a rooted tree. For more details, the reader may consult [6, 7, 23, 36], and the references therein.

Given the non-archimedean topology of ${\mathbb {Q}}_{p}^{n}$, we have that every point of the ball $B_{r}^{n}(a)$, $a\in {\mathbb {Q}}_{p}^{n}$, $r\in {\mathbb {Z}}$, is its center and two balls in ${\mathbb {Q}}_{p}^{n}$ have nonempty intersection if and only if one is contained in the other. Moreover, any ball can be represented as disjoint union of balls of smaller radius, each of the latter can be represented in the same way with even smaller radius and so on, see e.g. [3, 37]. Geometrically, note that the above implies that every ball in ${\mathbb {Q}}_{p}^{n}$ can be identified with a rooted tree. For this reason, the p-adic numbers are the natural and essential structure for studying new mathematical models. For example, the spread of an infectious or contagious disease that takes into account social clusters in a situation of extreme social isolation, see e.g. [2, 24].

Motivated by the above we will center our attention on the p-adic integers ${\mathbb {Z}}_{p}^{n}$. We fix a function $J:{\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}_{+}$ that codify the structure of the energy landscapes studied in this article. This function is continuous, radial, $supp(J)={\mathbb {Z}}_{p}^{n}$ and $\int \nolimits _{{\mathbb {Z}}_{p}^{n}}J(||x||_{p})d^{n}x=1$. Under these considerations, we establish rigorously that the non-archimedean ultradiffusion equations

$$\begin{aligned} \frac{\partial u\left( x,t\right) }{\partial t}=\int _{{\mathbb {Q}}_{p}^{n}}J(x-y)u(y,t)dy-u(x,t)=(J*u-u)(x,t), \end{aligned}$$

determine ultradiffusion equations, i.e. we show that the fundamental solutions of the Cauchy problems naturally associated to these equations are transition density functions of some strong Markov processes ${\mathfrak {X}}$ with state space ${\mathbb {Z}}_{p}^{n}$ whose paths are right continuous and have no discontinuities other than jumps, see Theorem 3.

When posing a heat equation associated with certain models on a system closed, it is interesting from the physical–chemical point of view to determine if the mass of the system always remains constant over time (law of conservation of mass or principle of mass conservation). In our case, effectively the classical solution of the Cauchy problem fulfills the property of conservation of mass for each fixed $t\in [0,\infty )$ and moreover to satisfy the Comparison principle, see Lemma 4.

It should also be noted that our heat kernel (denoted by $Z(x,t):=Z_{t}(x)$, $x\in {\mathbb {Q}}_{p}^{n}$, $t>0$) is a probability measure over the p-adic integers ${\mathbb {Z}}_{p}^{n}$, see Theorem 1. Moreover, we are interested in knowing if the heat kernel Z(x, t) may be associated with Feller semigroups and transition functions of some strong Markov processes ${\mathfrak {X}}$. Inspired by this fact, in this article we will obtain explicitly a Feller semigroups $\{T_{t}\}_{t\ge 0}$ on the space of Banach $C_{0}({\mathbb {Z}}_{p}^{n})$ (space of continuous functions on ${\mathbb {Z}}_{p}^{n}$ that vanish at infinity) generated from the heat kernel Z(x, t), see Theorem 2. Moreover, we also obtain in an explicit way a transition function $p_{t}(x,\cdot )$ of the previously mentioned strong Markov processes ${\mathfrak {X}}$, see Theorem 3.

It is important to mention that in [4, 22, 25], Antoniouk, Khrennikov and Kochubei, also introduce Markov processes in a p-adic ball and moreover, study some properties in common with our heat kernels. However, certain differences should be noted, for example, the symbols of our pseudo-differential operators are supported at ${\mathbb {Q}}_{p}^{n}{\setminus } {\mathbb {Z}}_{p}^{n}$; at the time of studying the Cauchy problem, we are not restricted to considering only as initial condition test functions supported in the state space of our Markov processes (${\mathbb {Z}}_{p}^{n}$); and moreover, the semigroups treated in this article are studied on the space $C_{0}({\mathbb {Z}}_{p}^{n})$.

On the other hand, since ${\mathbb {Z}}_{p}^{n}$ is a union at most of a countable set of disjoint balls, see e.g. [3, Sections 1.8 and 1.10], we study the survival probability of the strong Markov processes ${\mathfrak {X}}$ on a ball $B_{-m}^{n}\subseteq {\mathbb {Z}}_{p}^{n}$, $m\in {\mathbb {N}}$, i.e., the probability that a path of ${\mathfrak {X}}$ remains on the ball $B_{-m}^{n}$ at the time t, see Theorem 4.

This article is organized as follows. In Sect. 2, we will collect some basic results on the p-adic analysis and fix the notation that we will use through the article. In Sect. 3, we study a class of non-archimedean pseudo-differential operators (denoted by ${\mathcal {A}}$) associated to our non-archimedean evolution equations. We also study some properties of the heat Kernel associated with these operators. In Sect. 4, we show that there a family of operators $\{T_{t}\}_{t\ge 0}$ (obtained explicitly) that determine a Feller semigroup on $C_{0}({\mathbb {Z}}_{p}^{n})$. In Sect. 5, we initially studied the homogeneous Cauchy problem and we will show that its fundamental solutions determine a transition function of some strong Markov processes with state space ${\mathbb {Z}}_{p}^{n}$. Our next objective, in this section, is to study the asymptotic behavior of the survival probability of a strong Markov processes ${\mathfrak {X}}$ on a ball $B_{-m}^{n}\subset {\mathbb {Z}}_{p}^{n}$, $m\in {\mathbb {N}}$, at the time t. Finally, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the above Feller semigroup.

2 Fourier analysis on ${\mathbb {Q}}_{p}^{n}$: essential ideas

Along this article p will denote a prime number. The field of $p-$adic numbers ${\mathbb {Q}}_{p}$ is defined as the completion of the field of rational numbers ${\mathbb {Q}}$ with respect to the $p-$adic norm $|\cdot |_{p}$, which is defined as

$$\begin{aligned} \left| x\right| _{p}=\left\{ \begin{array}{lll} 0\text {,} &{} \text {if} &{} x=0 \\ &{} &{} \\ p^{-\gamma }\text {,} &{} \text {if} &{} x=p^{\gamma }\frac{a}{b}\text {,} \end{array} \right. \end{aligned}$$

where a and b are integers coprime with p. The integer $\gamma :=ord(x)$, with $ord(0):=+\infty$, is called the $p-$adic order of x.

Any $p-$adic number $x\ne 0$ has a unique expansion of the form $x=p^{ord(x)}\sum _{j=0}^{\infty }x_{j}p^{j}$, where $x_{j}\in \{0,1,2,\dots ,p-1\}$ and $x_{0}\ne 0$. We extend the $p-$adic norm to ${\mathbb {Q}}_{p}^{n}$ by taking $||x||_{p}:=\max _{1\le i\le n}|x_{i}|_{p},\text { for }x=(x_{1},\dots ,x_{n})\in {\mathbb {Q}}_{p}^{n}.$

For $r\in {\mathbb {Z}}$, denote by $B_{r}^{n}(a)=\{x\in {\mathbb {Q}}_p^n;||x-a||_{p}\le p^{r}\}$ the ball of radius $p^{r}$ with center at $a=(a_{1},\dots ,a_{n})\in {\mathbb {Q}}_p^n$, and take $B_{r}^{n}(0)=:B_{r}^{n}$. The ball $B_{0}^{n}$ equals the product of n copies of $B_{0}={\mathbb {Z}}_{p}$, the ring of $p-$adic integers of ${\mathbb {Q}}_{p}$. We also denote by $S_{r}^{n}(a)=\{x\in {\mathbb {Q}}_p^n;||x-a||_{p}=p^{r}\}$ the sphere of radius $p^{r}$ with center at $a=(a_{1},\dots ,a_{n})\in {\mathbb {Q}}_p^n$, and take $S_{r}^{n}(0)=:S_{r}^{n}$. We will use $\Omega \left( p^{-r}||x-a||_{p}\right)$ to denote the characteristic function of the ball $B_{r}^{n}(a)$, $a\in {\mathbb {Q}}_{p}^{n}$, $r\in {\mathbb {Z}}$. In particular, $\Omega \left( ||x||_{p}\right)$ is the characteristic function of the ball $B_{0}^{n}={\mathbb {Z}}_{p}^{n}$. We will use the notation $1_{A}$ for the characteristic function of a set $A\subset {\mathbb {Q}}_{p}^{n}$. Along the article $d^{n}x$ will denote a Haar measure on ${\mathbb {Q}}_{p}^{n}$ normalized such that $\int _{{\mathbb {Z}}_{p}^{n}}d^{n}x=1.$

A complex-valued function f defined on ${\mathbb {Q}}_{p}^{n}$ is called locally constant if for any $x\in {\mathbb {Q}}_{p}^{n}$ there exist an integer $l:=l(x)$ such that $f(x)=f(x^{\prime })\text { for all } x^{\prime }\in B_{l}^{n}(x)$. Denote by $({\mathbb {Q}}_{p}^{n})$ the linear space of locally constant ${\mathbb {C}}$-value functions on ${\mathbb {Q}}_{p}^{n}$.

A function $\varphi :{\mathbb {Q}}_p^n\rightarrow {\mathbb {C}}$ is called a Bruhat–Schwartz function (or a test function) if it is locally constant with compact support. The ${\mathbb {C}}$-vector space of Bruhat–Schwartz functions is denoted by ${\mathcal {D}}({\mathbb {Q}}_{p}^{n})=:{\mathcal {D}}$. Let ${\mathcal {D}}^{\prime }({\mathbb {Q}}_{p}^{n})=:{\mathcal {D}}^{\prime }$ denote the set of all continuous functional (distributions) on ${\mathcal {D}}$. The natural pairing ${\mathcal {D}}^{\prime }({\mathbb {Q}}_{p}^{n})\times {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\rightarrow {\mathbb {C}}$ is denoted as $\left<T,\varphi \right>$ for $T\in {\mathcal {D}}^{\prime }({\mathbb {Q}}_p^n)$ and $\varphi \in {\mathcal {D}}({\mathbb {Q}}_p^n)$, see e.g. [3, Section 4.4].

Denote by $L_{loc}^{1}({\mathbb {Q}}_p^{n}):=L_{loc}^{1}$ the set of functions $f:{\mathbb {Q}}_p^{n}\rightarrow {\mathbb {C}}$ such that $f\in L^{1}(K)$ for any compact $K\subset {\mathbb {Q}}_p^{n}$. Every $f\in$ $L_{loc}^{1}$ defines a distribution $f\in {\mathcal {D}}^{\prime }\left( {\mathbb {Q}}_p^n\right)$ by the formula

$$\begin{aligned} \left\langle f,\varphi \right\rangle =\int _{{\mathbb {Q}}_p^n}f\left( x\right) \varphi \left( x\right\rangle d^{n}x. \end{aligned}$$

Such distributions are called regular distributions, see e.g. [3, Section 4.4], [37, Chapter VI]. Set $\chi _{p}(y)=\exp (2\pi i\{y\}_{p})$ for $y\in {\mathbb {Q}}_{p}$. The map $\chi _{p}(\cdot )$ is an additive character on ${\mathbb {Q}}_{p}$, i.e. a continuous map from $\left( {\mathbb {Q}}_{p},+\right)$ into S (the unit circle considered as multiplicative group) satisfying $\chi _{p}(x_{0}+x_{1})=\chi _{p}(x_{0})\chi _{p}(x_{1})$, $x_{0},x_{1}\in {\mathbb {Q}}_{p}$.

Given $x=(x_{1},\dots ,x_{n})$, $\xi =(\xi _{1},\dots ,\xi _{n})\in {\mathbb {Q}}_{p}^{n}$, we set $x\cdot \xi :=\sum _{j=1}^{n}x_{j}\xi _{j}$. If $f\in L^{1}({\mathbb {Q}}_p^n)$, its Fourier transform is defined by

$$\begin{aligned} ({\mathcal {F}} f)(\xi )={\mathcal {F}}_{x\rightarrow \xi }(f)={\widehat{f}}(\xi ):=\int _{{\mathbb {Q}}_p^n}\chi _{p}(\xi \cdot x)f(x)d^{n}x,\quad \text {for }\xi \in {\mathbb {Q}}_p^n. \end{aligned}$$

The inverse Fourier transform of a function $f\in L^{1}({\mathbb {Q}}_p^n)$ is

$$\begin{aligned} ({\mathcal {F}}^{-1}f)(x)={\mathcal {F}}^{-1}_{\xi \rightarrow x }(f)=\int _{{\mathbb {Q}}_p^n}\chi _{p}(-x \cdot \xi )f(\xi )d^{n}\xi ,\quad \text {for }x \in {\mathbb {Q}}_p^n. \end{aligned}$$

The Fourier transform is a linear isomorphism from ${\mathcal {D}}({\mathbb {Q}}_{p}^{n})$ onto itself satisfying

$$\begin{aligned} ({\mathcal {F}}({\mathcal {F}}f))(\xi )=f(-\xi ), \end{aligned}$$

for every $f\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})$, see e.g. [3, Section 4.8].

The Fourier transform ${\mathcal {F}}(f)={\mathcal {F}}_{x\rightarrow \xi }(f)={\widehat{f}}$ of a distribution f is defined by the relation

$$\begin{aligned} \left<{\mathcal {F}}(f),\varphi \right>=\left<f,{\mathcal {F}}(\varphi )\right>, \text { for all } \varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

The Fourier transform $f\rightarrow {\mathcal {F}}(f)$ is a linear isomorphism from ${\mathcal {D}}'({\mathbb {Q}}_{p}^{n})$ onto ${\mathcal {D}}'({\mathbb {Q}}_{p}^{n})$, see e.g. [3, Section 4.9].

3 Heat kernels

In this section, we study a class of non-archimedean pseudo-differential operators (denote by ${\mathcal {A}}$) on ${\mathcal {D}}({\mathbb {Q}}_{p}^{n})$. We also study some properties of the heat Kernel attached to operator ${\mathcal {A}}$. From now on denote by ${\mathbb {N}}=\{1,2,\ldots \}$ the set of natural numbers and by ${\mathbb {R}}_{+}=\{x\in {\mathbb {R}}:x\ge 0\}$ the set of non-negative real numbers. Along this article we fix a continuous and radial function $J:{\mathbb {Q}}_{p}^{n}\rightarrow {\mathbb {R}}_{+}$ such that $supp(J)={\mathbb {Z}}_{p}^{n}$ and $\int \nolimits _{{\mathbb {Z}}_{p}^{n}}J(||x||_{p})d^{n}x=1$.

Remark 1

The function J above mentioned satisfies the following conditions:

• (i) ${\widehat{J}}$ is a real-valued, radial and continuous function, satisfying $|{\widehat{J}}(||\xi ||_{p})|\le 1$ for all $\xi \in {\mathbb {Q}}_{p}^{n}$ and moreover, ${\widehat{J}}(0)=1$;

• (ii) $supp(1-{\widehat{J}}(||\xi ||_{p}))={\mathbb {Q}}_{p}^{n}\backslash {\mathbb {Z}}_{p}^{n}$;

Example 1

• (i) (Bessel Potentials) For $\alpha >0$, we define $J_{\alpha }$ as follows:

  $$\begin{aligned} J_{\alpha }(x)=\left\{ \begin{array}{ll} \frac{1-p^{-\alpha }}{1-p^{\alpha -n}}\left( ||x||_{p}^{\alpha -n}-p^{\alpha -n}\right) \Omega (||x||_{p})&{}\quad \text {if }\alpha \ne n \\ \\ (1-p^{-n})\log _{p}(\frac{p}{||x||_{p}})\Omega (||x||_{p}) &{}\quad \text {if }\alpha =n \end{array} \right. \end{aligned}$$

  for $x\in {\mathbb {Q}}_{p}^{n}$. By [32, Chapter III-Section 5] we have $J_{\alpha }\in L^{1}({\mathbb {Q}}_{p}^{n})$ and ${\widehat{J}}_{\alpha }(\xi )=(\max \{1,||\xi ||_{p}\})^{-\alpha }$. Then, ${\widehat{J}}_{\alpha }(\xi )$ is a decreasing function with respect to $||\cdot ||_{p}$. On the other hand, note that $J_{\alpha }$ is a nonnegative radial function and by a direct calculation one verifies that $\int _{{\mathbb {Z}}_{p}^{n}}J_{\alpha }(x)d^{n}x=1$. Therefore, $1-J_{\alpha }$ is an increasing function with respect to $||\cdot ||_{p}$.

• (ii) Take $M\in {\mathbb {N}}\backslash \{0\}$ and let $C_{k}$, $k=0,1,\ldots ,M$, be positive real numbers, such that $0<(1-p^{-n})\sum _{k=0}^{M}C_{k}\sum _{j=0}^{\infty }p^{-j(k+n)}<\infty$ and $\sum _{k=0}^{M}C_{k}||x||_{p}^{k}\ge 0$ for all $x\in {\mathbb {Z}}_{p}^{n}.$ Let $s:=(1-p^{-n})\sum _{k=0}^{M}C_{k}\sum _{j=0}^{\infty }p^{-j(k+n)}$ and define for $x\in {\mathbb {Q}}_{p}^{n}$,

  $$\begin{aligned} J(x)=\frac{1}{s}\left( \sum _{k=0}^{M}C_{k}||x||_{p}^{k}\right) \Omega (||x||_{p}). \end{aligned}$$

  Note that J is a nonnegative radial function such that $\int \nolimits _{{\mathbb {Q}}_{p}^{n}}J(||x||_{p})d^{n}x=1$. By a direct calculation one verifies that ${\widehat{J}}$ is a decreasing function with respect to $||\cdot ||_{p}$, since

  $$\begin{aligned} {\widehat{J}}(||\xi ||_{p})=\left\{ \begin{array}{ll} (1-p^{-n})||\xi ||_{p}^{-n}\sum \nolimits _{j=ord_{p}(\xi )}^{\infty }J(p^{ord_{p}(\xi )-j})p^{-nj}&{}\quad \text {if }\xi \in {\mathbb {Z}}_{p}^{n}, \\ (1-p^{-n})||\xi ||_{p}^{-n}\sum \nolimits _{j=0}^{\infty }J(p^{ord_{p}(\xi )-j})p^{-nj}&{}\quad \text {if} \ \ \xi \in {\mathbb {Q}}_{p}^{n}\backslash {\mathbb {Z}}_{p}^{n}. \end{array} \right. \end{aligned}$$

  Therefore, $1-{\widehat{J}}(||\xi ||_{p})$ is an increasing function with respect to $||\cdot ||_{p}$.

Remark 2

The function $1-{\widehat{J}}(||\xi ||_{p})$ is locally constant on ${\mathbb {Q}}_{p}^{n}$. Indeed, by Remark 1-(ii) we have that $1-{\widehat{J}}(||\xi ||_{p})=0$ for all $\xi \in {\mathbb {Z}}_{p}^{n}$. On the other hand, by [34, Lemma 1] we have that the function $1-{\widehat{J}}(||\xi ||_{p})$ is locally constant on ${\mathbb {Q}}_{p}^{n}{\setminus } {\mathbb {Z}}_{p}^{n}$.

Remark 3

By Remark 1-(i) we have $0\le 1-{\widehat{J}}(||\xi ||_{p})\le 2$ for all $\xi \in {\mathbb {Q}}_{p}^{n}$. Then, for $t\ge 0$ we have that

$$\begin{aligned} \int _{{{\mathbb {Q}}}_{p}^{n}}e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi\ge & {} 1+(1-p^{-n})e^{-2t}\sum \limits _{j=1}^{\infty }p^{nj}\longrightarrow +\infty , \end{aligned}$$

i.e., $e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}\notin L^{1}({\mathbb {Q}}_{p}^{n})$ for $t\ge 0$. Since $e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}$ is a locally integrable function, then by [3, Subsection 4.4.1] we have that $e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}$, $t\ge 0$, defines a regular distribution on ${\mathbb {Q}}_{p}^{n}$.

We define the operator

$$\begin{aligned} ({\mathcal {A}}\varphi )(x)=(J*\varphi )(x)-\varphi (x), \ \ \text { } \varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}). \end{aligned}$$

Lemma 1

The operator ${\mathcal {A}}:{\mathcal {D}}({\mathbb {Q}}_{p}^{n})\rightarrow {\mathcal {D}}({\mathbb {Q}}_{p}^{n})$ is a well-defined p-adic pseudo-differential operator.

Proof

Let $\varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})$ fixed. By [3, Sections 4.3, 4.8 and 4.9] we have that

$$\begin{aligned} ({\mathcal {A}}\varphi )(x)= & {} (J*\varphi )(x)-\varphi (x) \\= & {} \int \nolimits _{{\mathbb {Q}}_{p}^{n}}\chi _{p}(-x\cdot \xi ){\widehat{J}}(||\xi ||_{p}){\widehat{\varphi }} (\xi )d^{n}\xi -\int \nolimits _{{\mathbb {Q}}_{p}^{n}}\chi _{p}(-x\cdot \xi ){\widehat{\varphi }}(\xi )d^{n}\xi \\= & {} -{\mathcal {F}}_{\xi \rightarrow x}^{-1}\left[ (1-{\widehat{J}}(||\xi ||_{p})){\widehat{\varphi }}(\xi )\right] . \end{aligned}$$

Moreover, by Remark 1-(ii) we have that $(1-{\widehat{J}}){\widehat{\varphi }}$ is a compactly supported function.

Therefore, Remark 2 and [3, Section 4.8] implies that $({\mathcal {A}}\varphi ) \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})$. $\square$

We define the heat Kernel attached to operator ${\mathcal {A}}$ as

$$\begin{aligned} Z(x,t):=Z_{t}(x)=\int _{{{\mathbb {Q}}}_{p}^{n}}\chi _{p}(-x\cdot \xi )e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi , \text { } x\in {\mathbb {Q}}_{p}^{n}, \ \ t\ge 0. \end{aligned}$$

(3.1)

By [3, Section 4.9] and Remark 3 we have that

$$\begin{aligned} Z(x,t)\in {\mathcal {D}}'({\mathbb {Q}}_{p}^{n}), \ \ x\in {\mathbb {Q}}_{p}^{n}, \ \ t\ge 0. \end{aligned}$$

(3.2)

Theorem 1

The heat Kernel $Z_{t}(x)$ has the following properties:

• (i) For any $t>0$ and $x\in {\mathbb {Q}}_{p}^{n}$,

  $$\begin{aligned} Z(x,t)= & {} \left[ (1-e^{-t(1-{\widehat{J}}(p))})+\sum _{i=1}^{ord(x)}p^{in}(e^{-t(1-{\widehat{J}}(p^{i}))}-e^{-t(1-{\widehat{J}}(p^{i+1}))})\right] \\{} & {} \Omega \left( ||x||_{p}\right) . \end{aligned}$$

  Consequently, for any fixed $t>0$ we have that $supp(Z_{t})={\mathbb {Z}}_{p}^{n}$.

• (ii) $Z(x,t)=||x||_{p}^{-1}\left\{ (1-p^{-n})\sum _{j=0}^{\infty }p^{-nj}e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p^{-j}))}-e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p))} \right\}$, for any $t>0$ and $x\in {\mathbb {Q}}_{p}^{n}\backslash \left\{ 0\right\}$.

• (iii) $\int _{{\mathbb {Z}}_{p}^{n}}Z(x,t)d^{n}x=1$, for all $t\ge 0$, i.e., $Z(\cdot ,t)$ is a probability measure on ${\mathbb {Z}}_{p}^{n}$.

Proof

• (i) By (3.1) and Remark 1-(ii) we have that

  $$\begin{aligned} Z(x,t)=\int _{{\mathbb {Z}}_p^n}\chi _{p}(-x\cdot \xi )d^{n}\xi +\int _{{\mathbb {Q}}_{p}^{n}\backslash {\mathbb {Z}}_{p}^{n}}\chi _{p}(-x\cdot \xi )e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi . \end{aligned}$$

  (3.3)

  Consider the following cases for x: If $||x||_{p}=p^{\gamma }$, with $-ord(x)=\gamma \ge 1$, then by (3.3) and the formula

  $$\begin{aligned} \int _{{\mathbb {Z}}_p^n}\chi _{p}(-x\cdot \xi )d^{n}\xi =\left\{ \begin{array}{lll} 1, &{}\quad \text {if} &{}\quad ||x||_{p}\le 1 , \\ 0, &{}\quad \text {if} &{}\quad ||x||_{p}\ge p, \end{array} \right. \end{aligned}$$

  (3.4)

  we have that

  $$\begin{aligned} Z(x,t)&= \int _{{\mathbb {Q}}_{p}^{n}\backslash {\mathbb {Z}}_{p}^{n}}\chi _{p}(-x\cdot \xi )e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi \\&=\sum _{j=1}^{\infty }e^{-t(1-{\widehat{J}}(p^{j}))}\int _{||w||_{p}=1}\chi _{p}(-p^{-j}x\cdot w)d^{n}w{ \ }(\text {taking }w=p^{j}\xi \text { }). \end{aligned}$$

  By using the formula

  $$\begin{aligned} \int _{||w||_{p}=1}\chi _{p}\left( -p^{-j}x\cdot w\right) d^{n}w=\left\{ \begin{array}{lll} 1-p^{-n}, &{}\quad \text {if} &{}\quad j\le -\gamma , \\ -p^{-n}, &{}\quad \text {if } &{}\quad j=-\gamma +1, \\ 0, &{}\quad \text {if} &{}\quad \ j\ge -\gamma +2, \end{array} \right. \end{aligned}$$

  we get that $Z(x,t)=0.$

  If $||x||_{p}=p^{-\gamma }$, with $ord(x)=\gamma \ge 0$, then by (3.3) and (3.4), we have that

  $$\begin{aligned} Z(x,t)= & {} 1+\sum _{j=1}^{\infty }e^{-t(1-{\widehat{J}}(p^{j}))}\int _{||p^{j}\xi ||_{p}=1}\chi _{p}(-x\cdot \xi )d^{n}\xi \nonumber \\= & {} 1+\sum _{j=1}^{\infty }e^{-t(1-{\widehat{J}}(p^{j}))}\nonumber \\{} & {} p^{nj}\int _{||w||_{p}=1}\chi _{p}(-p^{-j}x\cdot w)d^{n}w{ \ }(\text {taking }w=p^{j}\xi \text { }). \end{aligned}$$

  (3.5)

  Now, we have that

  $$\begin{aligned} \int _{||w||_{p}=1}\chi _{p}\left( -p^{-j}x\cdot w\right) d^{n}w=\left\{ \begin{array}{lll} 1-p^{-n}, &{}\quad \text {if} &{}\quad j\le \gamma , \\ -p^{-n}, &{}\quad \text {if } &{}\quad j=\gamma +1, \\ 0, &{}\quad \text {if} &{}\quad \ j\ge \gamma +2. \end{array} \right. \end{aligned}$$

  (3.6)

  We now proceed by induction on $\gamma$.

  If $\gamma =0$, then by (3.5) and (3.6) we have that $Z(x,t)=1-e^{-t(1-{\widehat{J}}(p))}$.

  If $\gamma =1$, then by (3.5) and (3.6) we have that

  $$\begin{aligned} Z(x,t)&=1+(p^{n}-1)e^{-t(1-{\widehat{J}}(p))}-p^{n}e^{-t(1-{\widehat{J}}(p^{2}))} \\&=(1-e^{-t(1-{\widehat{J}}(p))})+p^{n}(e^{-t(1-{\widehat{J}}(p))}-e^{-t(1-{\widehat{J}}(p^{2}))}). \end{aligned}$$

  Suppose that for a certain $\gamma >1$ the hypothesis $Z(x,t)=(1-e^{-t(1-{\widehat{J}}(p))})+\sum _{i=1}^{ord(x)}p^{in}(e^{-t(1-{\widehat{J}}(p^{i}))}-e^{-t(1-{\widehat{J}}(p^{i+1}))})$ is satisfied. Then, by a direct calculation one verifies that the statement is true for $\gamma +1$.

• (ii) For $t>0$, $x=p^{\gamma }x_{0}\ne 0$ such that $\gamma \in {\mathbb {Z}}$ and $||x_{0}||_{p}=1$, and making the change of variables $z=p^{\gamma }\xi$ and $w=p^{j}z$, respectively, we have that

  $$\begin{aligned} Z(x,t)= & {} \int _{{\mathbb {Q}}_{p}^{n}}\chi _{p}\left( -p^{\gamma }\xi \cdot x_{0}\right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi \\= & {} ||x||_{p}^{-n}\int _{{\mathbb {Q}}_{p}^{n}}\chi _{p}\left( -z\cdot x_{0}\right) e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}||z||_{p}))}d^{n}z\\= & {} ||x||_{p}^{-n}\sum _{-\infty<j<\infty }p^{nj}e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p^{j}))}\int _{||w||_{p}=1}\chi _{p}\left( -p^{-j}x_{0}\cdot w\right) d^{n}w \end{aligned}$$

  By using the formula

  $$\begin{aligned} \int _{||w||_{p}=1}\chi _{p}\left( -p^{-j}x_{0}\cdot w\right) d^{n}w=\left\{ \begin{array}{lll} 1-p^{-n}, &{}\quad \text {if} &{}\quad \text { }j\le 0 , \\ -p^{-n}, &{}\quad \text {if } &{}\quad {\ }j=1, \\ 0, &{}\quad \text {if} &{}\quad \ j\ge 2, \end{array} \right. \end{aligned}$$

  we get $Z(x,t)=||x||_{p}^{-1}\left\{ (1-p^{-n})\sum _{j=0}^{\infty }p^{-nj}e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p^{-j}))}-e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p))} \right\}$.

• (iii) If $t=0$, then by (3.1) and [3, Example 4.7.1] we have that

  $$\begin{aligned} \int _{{{\mathbb {Q}}}_{p}^{n}}Z(x,t)d^{n}x=\int _{{{\mathbb {Q}}}_{p}^{n}}\delta (x)d^{n}x=\delta *1=1. \end{aligned}$$

  If $t>0$, then by (3.2) and [3, Section 4.9] we have that

  $$\begin{aligned} \left<{\mathcal {F}}(Z(x,t)),\varphi \right>&=\left<Z(x,t),{\widehat{\varphi }}\right>\\&=\left<e^{-t(1-{\widehat{J}}(||\xi ||_{p}))},\varphi \right>, \end{aligned}$$

  for any $\varphi \in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})$. Therefore, ${\mathcal {F}}(Z(x,t))=e^{-t(1-{\widehat{J}}(||x||_{p}))}$.

  On the other hand, ${\mathcal {F}}(Z(x,t))=\int _{{\mathbb {Q}}_{p}^{n}}\chi _{p}(\xi \cdot x)Z(x,t)d^{n}x$ and ${\mathcal {F}}(Z(0,t))=\int _{{\mathbb {Q}}_{p}^{n}}Z(x,t)d^{n}x$. Moreover, by Remark 1-(i) we have that ${\mathcal {F}}(Z(0,t))=1$.

  Therefore, by (i) we have that $\int _{{\mathbb {Z}}_{p}^{n}}Z(x,t)d^{n}x=1$, for all $t>0$.

$\square$

Corollary 1

If $1-{\widehat{J}}$ is an increasing function with respect to $||\cdot ||_{p}$, then

• (i) $Z(x,t)\ge 0$, for all $x\in {\mathbb {Q}}_{p}^{n}$ and $t\ge 0$.

• (ii) $Z(x,t)\le 2t||x||_{p}^{-n}$, for any $t>0$ and $x\in {\mathbb {Q}}_{p}^{n}\backslash \left\{ 0\right\}$.

Proof

• (i) It follows from Theorem 1-(i).

• (ii) By Theorem 1-(ii) and the fact that $e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p^{-j}))}, e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p))}\le 1$, we have that

  $$\begin{aligned} Z(x,t)&\le ||x||_{p}^{-1}\left\{ \sum _{j=0}^{\infty }(1-p^{-n})p^{-nj}-e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p))}\right\} \\&\le ||x||_{p}^{-n}(1-e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p))}). \end{aligned}$$

  Now, by applying the mean value theorem to the real function $e^{-u(1-{\widehat{J}}(||x||_{p}^{-1}p))}$ on [0, t] with $t>0$, we have

  $$\begin{aligned} 1-e^{-t(1-{\widehat{J}}(||x||_{p}^{-1}p))}=t(1-{\widehat{J}}(||x||_{p}^{-1}p))e^{-\tau (1-{\widehat{J}}(||x||_{p}^{-1}p))} \end{aligned}$$

  for some $\tau \in (0,t)$. Therfore, by Remark 1-(ii) we have that $Z(x,t)\le 2t||x||_{p}^{-n}$.

$\square$

4 Feller semigroups

From now on, we assume that $1-{\widehat{J}}$ is an increasing function with respect to $||\cdot ||_{p}$. Given ${\mathbb {Q}}_{p}^{n}$ is a locally compact topological space, see [3, Sections 1.10 and 3.2], we have that ${\mathbb {Z}}_{p}^{n}$ is locally compact. We will denote by $C_{0}({\mathbb {Z}}_{p}^{n})$ the Banach space of continuous functions on ${\mathbb {Z}}_{p}^{n}$ vanishing at infinity with norm $||f||:=\sup _{x\in {\mathbb {Z}}_{p}^{n}}|f(x)|$. Note that $C_{0}({\mathbb {Z}}_{p}^{n})=C({\mathbb {Z}}_{p}^{n})$ since ${\mathbb {Z}}_{p}^{n}$ is compact. For further details the reader may consult [13, Chapter 4-Section 2].

Definition 1

A one-parameter family $\{T_{t}\}_{t\ge 0}$ of bounded linear operators on $C_{0}({\mathbb {Z}}_{p}^{n})$ into itself is called a contraction semigroup if it satisfies the following conditions:

• (i) $T_{t+s}=T_{t}\cdot T_{s}$ for all $t,s\ge 0.$

• (ii) $\lim _{t\rightarrow 0^{+}}||T_{t}u-u||_{L^{\infty }}=0$ for every $u\in C_{0}({{\mathbb {Z}}}_{p}^{n})$ (strongly continuous)

• (iii) $||T_{t}||_{L^{\infty }}\le 1$ for all $t\ge 0.$

Definition 2

A strongly continuous contraction semigroups $\{T_{t}\}_{t\ge 0}$ on $C_{0}({{\mathbb {Z}}}_{p}^{n}),$ for which all the operators $T_{t}$ are positive, i.e. such that for all $t>0$

$$\begin{aligned} u\in C_{0}({{\mathbb {Z}}}_{p}^{n})\text { with }u\ge 0\text { implies }T_{t}u\ge 0, \end{aligned}$$

is called a Feller semigroup on ${{\mathbb {Z}}}_{p}^{n}$.

We are interested in obtaining a Feller semigroup on $C_{0}({\mathbb {Z}}_{p}^{n})$ in an explicit way and naturally associated with the heat Kernel.

For $u\in C_{0}({{\mathbb {Z}}}_{p}^{n}),$ $x\in {{\mathbb {Q}}}_{p}^{n}$ and $t\ge 0$, we define

$$\begin{aligned} T_{t}u(x):=\left\{ \begin{array}{ll} u(x) &{}\quad \text {if }t=0\text {, } \\ &{} \\ (Z_{t}*u)(x) &{}\quad \text {if }t>0, \end{array} \right. \end{aligned}$$

(4.1)

where $Z_{t}$ is the heat Kernel given at (3.1).

Lemma 2

For all $t\ge 0$,

$$\begin{aligned} T_{t}:C_{0}({\mathbb {Z}}_{p}^{n})\rightarrow C_{0}({\mathbb {Z}}_{p}^{n}) \end{aligned}$$

is a bounded linear operator with $||T_{t}||_{L^{\infty }}\le 1$.

Proof

If $t=0,$ the assertion is clear. Let $t>0$, $u\in C_{0}({{\mathbb {Z}}}_{p}^{n})$ and $x\in {{\mathbb {Q}}}_{p}^{n}$.

By Theorem 1-(i), (3.1) and [3, Subsections 4.9 and 5.2] we have that $T_{t}u(x)$ is a continuous function. Moreover, by Theorem 1-(iii) we have

$$\begin{aligned} |T_{t}u(x)|=\left| \int _{{{\mathbb {Z}}}_{p}^{n}}Z_{t}(x-y)u(y)d^{n}y\right| \le ||u||_{L^{\infty }}\int _{{{\mathbb {Z}}}_{p}^{n}}Z_{t}(x-y)d^{n}y=||u||_{L^{\infty }}. \nonumber \\ \end{aligned}$$

(4.2)

On the other hand, by Theorem 1 and Corollary 1, we have for $||x||_{p}\gg 0$ that

$$\begin{aligned} 0\le & {} \left| T_{t}u(x)\right| \le ||u||_{L^{\infty }}\int _{{\mathbb {Z}}_{p}^{n}}Z_{t}(x-y)d^{n}y\le 2t||u||_{L^{\infty }}\int _{{\mathbb {Z}}_{p}^{n}}||x-y||_{p}^{-n}d^{n}y \\= & {} 2t||u||_{L^{\infty }}||x||_{p}^{-n}Vol({\mathbb {Z}}_{p}^{n})=0. \end{aligned}$$

Therefore, $T_{t}:C_{0}({\mathbb {Z}}_{p}^{n})\rightarrow C_{0}({\mathbb {Z}}_{p}^{n})$ is a well-defined bounded linear operator. $\square$

Remark 4

By a direct calculation one verifies that $T_{t}(T_{s}u)(x)=T_{t+s}u(x)$, for all $u\in C_{0}({\mathbb {Z}}_{p}^{n})$, i.e., the family of operators $\{T_{t}\}_{t\ge 0}$ defined in (4.1) determine a semigroup over the space $C_{0}({\mathbb {Z}}_{p}^{n})$.

Lemma 3

The family of operators $T_{t}$ is uniformly continuous, i.e. for all $u\in C_{0}({{\mathbb {Z}}}_{p}^{n})$ we have that

$$\begin{aligned} \lim _{t\rightarrow 0^{+}}||T_{t}u-u||_{L^{\infty }}=0. \end{aligned}$$

Proof

Let fixed $x\in {\mathbb {Q}}_{p}^{n}$. Note that

$$\begin{aligned} (T_{t}u-u)(x)=\int _{{\mathbb {Q}}_{p}^{n}}Z_{t}(x-y)\left[ u(y)-u(x)\right] d^{n}y. \end{aligned}$$

(4.3)

Since that $u\in C_{0}({{\mathbb {Z}}}_{p}^{n})$, given any number $\epsilon >0$, however small, there exists some number $s:=s(x,\epsilon )\in {\mathbb {Z}}$ such that if $||x-y||_{p}<p^{s}$ then $||u(y)-u(x)||_{L^{\infty }}<\epsilon$. Then, by (4.3), Theorem 1-(iii) and Corollary 1, we have that

$$\begin{aligned} \begin{aligned} \left| (T_{t}u-u)(x)\right|&\le \int _{||x-y||_{p}<p^{s}}Z_{t}(x-y)\left| u(y)-u(x)\right| d^{n}y\\&\quad +\int _{||x-y||_{p}\ge p^{s}}Z_{t}(x-y)\left| u(y)-u(x)\right| d^{n}y\\&\le \epsilon +2||u||_{L^{\infty }}\int _{||x-y||_{p}\ge p^{s}}Z_{t}(x-y)d^{n}y\\&=\epsilon +2||u||_{L^{\infty }}\int _{||w||_{p}\ge p^{s}}Z_{t}(w)d^{n}w\\&\le \epsilon +4t||u||_{L^{\infty }}\int _{||w||_{p}\ge p^{s}}||w||_{p}^{-n}d^{n}w \end{aligned} \end{aligned}$$

Now, since $\int _{||w||_{p}\ge p^{s}}||w||_{p}^{-n}d^{n}w=C<\infty$, we have that

$$\begin{aligned} \left| (T_{t}u-u)(x)\right| \le \epsilon +4Ct||u||_{L^{\infty }}. \end{aligned}$$

Therefore, given any $\epsilon >0$ we have that $\lim _{t\rightarrow 0^{+}}\sup |(T_{t}u-u)(x)|\le \epsilon$, for all $x\in {{\mathbb {Q}}}_{p}^{n}$. $\square$

Theorem 2

The family of operators $\{T_{t}\}_{t\ge 0}$ defined in (4.1) determine a Feller semigroup on ${{\mathbb {Z}}}_{p}^{n}$.

Proof

The result follows from Lemma 2, Remark 4, Lemma 3 and Corollary 1-(ii), taking into account that if $u\in C_{0}({{\mathbb {Z}}}_{p}^{n})$ and $u\ge 0$, then $(T_{t}u)(x)\ge 0$, $t\ge 0.$ $\square$

Remark 5

By Lemmas 2 and 3 we have that the semigroup $\{T_{t}\}_{t\ge 0}$ is strongly continuous in t for all $t\ge 0:$

$$\begin{aligned} \lim _{s\rightarrow 0^{+}}||T_{t+s}f-T_{t}f||_{L^{\infty }}=0,\text { }f\in C_{0}({{\mathbb {Z}}}_{p}^{n}). \end{aligned}$$

Moreover, by Theorem 1-(iii) and Lemma 3 we have that the Feller semigroup $\{T_{t}\}_{t\ge 0}$ is non-negative and contractive on $C_{0}({{\mathbb {Z}}}_{p}^{n})$:

$$\begin{aligned} f\in C_{0}({{\mathbb {Z}}}_{p}^{n}),\text { }0\le f(x)\le 1\text { on }K\Longrightarrow \text { }0\le T_{t}f(x)\le 1\text { on }{{\mathbb {Z}}}_{p}^{n}. \end{aligned}$$

5 The abstract Cauchy problem and their applications

Consider the following homogeneous Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}(x,t)={\mathcal {A}}u(x,t), \ \ t\in [0,\infty ), \ \ x\in {\mathbb {Q}}_{p}^{n} \\ &{} \\ u(x,0)=u_{0}(x)\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\text {, } &{} \end{array} \right. \end{aligned}$$

(5.1)

where ${\mathcal {A}}$ is the pseudo-differential operator above defined.

Then, we have that Z(x, t), $x\in {\mathbb {Q}}_{p}^{n}$, $t\ge 0$, given by (3.1) is the fundamental solution of the Cauchy problem (5.1). Moreover, by proceeding as in the proof of [36, Proposition 1], we have that

$$\begin{aligned} u(x,t)= & {} \int _{{{\mathbb {Q}}}_{p}^{n}}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}\widehat{u_{0}}(\xi )d^{n}\xi ,\text { }u_{0}(x)\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n}),\text { }x\in {\mathbb {Q}}_{p}^{n}\text {, }t\ge 0,\text { }\\= & {} Z(t,x)*u_{0}(x) \end{aligned}$$

is the unique (classical) solution of the Cauchy problem (5.1).

In the following lemma we will list some results about the solution u(x, t), $x\in {\mathbb {Q}}_{p}^{n}$, $t\in [0,\infty )$, of the Cauchy problem (5.1).

Lemma 4

The classical solution u(x, t) of the Cauchy problem (5.1) satisfies the following properties:

• (i) $u(x,t)\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})$;

• (ii) (Principle of mass conservation)

  $$\begin{aligned} \int _{{{\mathbb {Q}}}_{p}^{n}}u(x,t)d^{n}x=\int _{{{\mathbb {Q}}}_{p}^{n}}u_{0}(x)d^{n}x; \end{aligned}$$

• (iii) (Comparison principle) Let u(x, t) and v(x, t) be classical solutions of the Cauchy problem (5.1) with initial data $u_{0}$ and $v_{0}$ respectively.

  If $u_{0}(x)\ge v_{0}(x)$ for all $x\in {{\mathbb {Q}}}_{p}^{n}$ and $1-{\widehat{J}}$ is an increasing function with respect to $||\cdot ||_{p}$, then

  $$\begin{aligned} u(x,t)\ge v(x,t) \ \ \text {for all} \ \ (x,t)\in {{\mathbb {Q}}}_{p}^{n}\times [0,\infty ). \end{aligned}$$

Proof

• (i) This result follow from [3, Section 4.8], Remark 2 and Lemma 1.

• (ii)
  $$\begin{aligned} u(x,t)= & {} \int _{{{\mathbb {Q}}}_{p}^{n}}\chi _{p}\left( -x \cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}\widehat{u_{0}}(\xi )d^{n}\xi \nonumber \\= & {} {\mathcal {F}}^{-1}_{\xi \rightarrow x }\left( e^{-t(1-{\widehat{J}}(||\xi ||_{p}))} \right) *{\mathcal {F}}^{-1}_{\xi \rightarrow x }\left( \widehat{u_{0}}(\xi )\right) \nonumber \\= & {} Z_{t}(x)*u_{0}(x) \nonumber \\= & {} \int _{{{\mathbb {Q}}}_p^n}Z_{t}(x-y)u_{0}(y)d^{n}y. \end{aligned}$$

  (5.2)

  Therefore, by Fubini’s Theorem and Theorem 1 we have that

  $$\begin{aligned} \int _{{{\mathbb {Q}}}_{p}^{n}}u(x,t)d^{n}x= & {} \int _{{{\mathbb {Q}}}_p^n}\int _{{{\mathbb {Q}}}_{p}^{n}}Z_{t}(x-y)u_{0}(y)d^{n}yd^{n}x\\= & {} \int _{{{\mathbb {Q}}}_p^n}u_{0}(y)\int _{{{\mathbb {Q}}}_{p}^{n}}Z_{t}(x-y)d^{n}xd^{n}y\\= & {} \int _{{{\mathbb {Q}}}_p^n}u_{0}(y)d^{n}y. \end{aligned}$$

• (iii) By Corollary 1-(i) and using Fubini’s Theorem we have that

  $$\begin{aligned}&u(x,t)-v(x,t)\\&\quad =\int _{{{\mathbb {Q}}}_p^n}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}\left\{ ({\mathcal {F}}u_{0})(\xi )-({\mathcal {F}}v_{0})(\xi )\right\} d^{n}\xi \\&\quad =\int _{{{\mathbb {Q}}}_p^n}\left( u_{0}(y)-v_{0}(y)\right) \left\{ {\mathcal {F}}^{-1}_{\xi \rightarrow -(x-y)}\left( e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}\right) \right\} d^{n}y\\&\quad =\int _{{{\mathbb {Q}}}_{p}^{n}}\left( u_{0}(y)-v_{0}(y)\right) Z_{t}(x-y)d^{n}y\ge 0. \end{aligned}$$

$\square$

We denote by ${\mathcal {B}}({\mathbb {Z}}_{p}^{n})$ the family of Borel sets of ${\mathbb {Z}}_{p}^{n}$. Next, we will show that the fundamental solutions Z(x, t) determine a transition function of some strong Markov processes with state space ${\mathbb {Z}}_{p}^{n}$. For the basic concepts on Markov transition function, the condition (L), uniformly stochastically continuous $C_{0}-$transition function the reader may consult [33, Section 2.2].

Definition 3

For $E\in {\mathcal {B}}({\mathbb {Z}}_{p}^{n})$, we define

$$\begin{aligned} p_{t}(x,E)=\left\{ \begin{array}{ll} Z_{t}(x)*1_{E}(x)\text {,} &{}\quad \text {for }t>0, x\in {{\mathbb {Q}}}_{p}^{n} \\ &{} \\ 1_{E}(x), &{}\quad \text {for }t=0, x\in {{\mathbb {Q}}}_{p}^{n}. \end{array} \right. \end{aligned}$$

(5.3)

Theorem 3

$p_{t}(x,\cdot )$ is a uniformly stochastically continuous $C_{0}-$transition function on ${{\mathbb {Z}}}_{p}^{n}$, satisfy condition (L) and the formula

$$\begin{aligned} T_{t}f(x):=\int _{{{\mathbb {Q}}}_{p}^{n}}p_{t}(x,d^{n}y)f(y) \end{aligned}$$

holds. Moreover, it is the transition function of some strong Markov processes ${\mathfrak {X}}$ with state space ${\mathbb {Z}}_{p}^{n}$ and transition function $p_{t}(x,\cdot )$ whose paths are right continuous and have no discontinuities other than jumps.

Proof

The results are a consequence of [33, Subsections 2.2.4 and 2.2.5], Remark 5 and Theorem 2. $\square$

Next, we will study the asymptotic behavior of the survival probability of a strong Markov processes ${\mathfrak {X}}$ on a ball $B_{-m}^{n}\subset {\mathbb {Z}}_{p}^{n}$, $m\in {\mathbb {N}}$, at the time t.

Set $\Delta _{-k}(x):=\Omega (p^{k}||x||_{p}),$ $k\in {\mathbb {N}}.$

Consider the following Cauchy problem:

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}(x,t)={\mathcal {A}}u(x,t), &{}\quad t\in \left[ 0,\infty \right) , {\ }x\in {\mathbb {Q}}_{p}^{n} \\ &{} \\ u(x,0)=p^{nm}\Omega (p^{m}||x||_{p}), &{}\quad m\in {\mathbb {N}}. \end{array} \right. \end{aligned}$$

(5.4)

We have that $u(x,t)=Z_{t}(x)*p^{nm}\Omega (p^{m}||x||_{p})$ is the unique classical solution of Cauchy problem (5.4). Now, since

$$\begin{aligned} {\widehat{\Delta }}_{-k}(\xi )=p^{-kn}\left\{ \begin{array}{lll} 1 &{}\quad \text {if} &{}\quad ||\xi ||_{p}\le p^{k} \\ 0 &{}\quad \text {if} &{}\quad ||\xi ||_{p}>p^{k}, \end{array} \right. \end{aligned}$$

we have that

$$\begin{aligned} u(x,t)= & {} p^{mn}\int _{{{\mathbb {Q}}}_{p}^{n}}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}{\widehat{\Delta }}_{-m}(\xi )d^{n}\xi \nonumber \\= & {} \int _{B_{m}^{n}}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}} (||\xi ||_{p}))}d^{n}\xi . \end{aligned}$$

(5.5)

Definition 4

The survival probability, denoted and defined by

$$\begin{aligned} S(t):=S_{p^{m}{\mathbb {Z}}_{p}^{n}}(t)=\int _{B_{-m}^{n}}u(x,t)d^{n}x, \text { }m\in {\mathbb {N}},\text { } \end{aligned}$$

is the probability that a path of ${\mathfrak {X}}$ remains on the ball $B_{-m}^{n}\subset {\mathbb {Z}}_{p}^{n}$, $m\in {\mathbb {N}}$, at the time t.

Remark 6

If take $m=0$ in the Cauchy problem (5.4), then $S(t)=1$ for all $t\ge 0$.

The asymptotic behavior of the function S(t) for all $t\ge 0$ is described by the following theorem.

Theorem 4

There exist positive real constants, $C_{1},$ $C_{2}$ such that for all $t\ge 0$ and $m\in {\mathbb {N}}$,

$$\begin{aligned} p^{-nm}+C_{1}e^{-t(1-{\widehat{J}}(p))}\le S(t)\le p^{-nm}+C_{2}e^{-t(1-{\widehat{J}}(p))}. \end{aligned}$$

Proof

By definition of S(t) and (5.5),

$$\begin{aligned} S(t)= & {} \int _{B_{-m}^{n}}u(x,t)d^{n}x \\= & {} \int _{B_{-m}^{n}}\int _{{\mathbb {Z}}_{p}^{n}}d^{n}\xi d^{n}x+\int _{B_{-m}^{n}}\int _{B_{m}^{n}\backslash {\mathbb {Z}}_{p}^{n}}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi d^{n}x \\= & {} p^{-mn}+\int _{B_{-m}^{n}}\int _{B_{m}^{n}\backslash {\mathbb {Z}}_{p}^{n}}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi d^{n}x. \end{aligned}$$

For $||x||_{p}=p^{-M}\le p^{-m}$, with $M\ge m$, and taking $w=p^{j}\xi$, we have that

$$\begin{aligned}{} & {} \int _{B_{m}^{n}\backslash {\mathbb {Z}}_{p}^{n}}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi \\{} & {} \qquad = \sum _{j=1}^{m}e^{-t(1-{\widehat{J}}(p^{j}))}p^{nj} \int _{||w||_{p}=1}\chi _{p}\left( -p^{-j}x\cdot w\right) d^{n}w. \end{aligned}$$

By using the formula

$$\begin{aligned} \int _{||w||_{p}=1}\chi _{p}\left( -p^{-j}x\cdot w\right) d^{n}w=\left\{ \begin{array}{lll} 1-p^{-n}, &{}\quad \text {if} &{}\quad j\le M , \\ -p^{-n}, &{}\quad \text {if } &{}\quad j=M +1, \\ 0, &{}\quad \text {if} &{}\quad j\ge M +2, \end{array} \right. \end{aligned}$$

we have that

$$\begin{aligned} \int _{B_{m}^{n}\backslash {\mathbb {Z}}_{p}^{n}}\chi _{p}\left( -x\cdot \xi \right) e^{-t(1-{\widehat{J}}(||\xi ||_{p}))}d^{n}\xi =(1-p^{-n})\sum \nolimits _{j=1}^{m}e^{-t(1-{\widehat{J}} (p^{j}))}p^{nj}. \end{aligned}$$

Now, since $1-{\widehat{J}}$ is an increasing function with respect to $||\cdot ||_{p}$, we have that

$$\begin{aligned} S(t)= & {} p^{-nm}+p^{-nm}(1-p^{-n})\sum \nolimits _{j=1}^{m}e^{-t(1-{\widehat{J}} (p^{j}))}p^{nj}\\\le & {} p^{-nm}+C_{2}e^{-t(1-{\widehat{J}}(p))}. \end{aligned}$$

Moreover, we have that

$$\begin{aligned} S(t)\ge & {} p^{-nm}+p^{-nm}(1-p^{-n})p^{n}e^{-t(1-{\widehat{J}}(p))}\\= & {} p^{-nm}+C_{1}e^{-t(1-{\widehat{J}}(p))}. \end{aligned}$$

$\square$

Let X be a Banach space, and let T(t) be a uniformly continuous semigroup.

The linear operator A defined by

$$\begin{aligned} D(A)=\left\{ x\in X: \lim _{t\rightarrow 0^{+}}\frac{T(t)x-x}{t} \ \ \text { exists }\right\} \end{aligned}$$

and

$$\begin{aligned} Ax=\lim _{t\rightarrow 0^{+}}\frac{T(t)x-x}{t} \ \ \text { for } \ \ x\in D(A), \end{aligned}$$

is the infinitesimal generator of the semigroup T(t), D(A) is the domain of A.

Consider the following inhomogeneous Cauchy problem given by

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}(x,t)={\mathcal {A}}u(x,t)+f(t), \ \ t\in (0,\infty ), \ \ x\in {\mathbb {Q}}_{p}^{n} \\ &{} \\ u(x,0)=u_{0}(x)\in {\mathcal {D}}({\mathbb {Q}}_{p}^{n})\text {, } &{} \end{array} \right. \end{aligned}$$

(5.6)

where $f:[0,T)\rightarrow C_{0}({{\mathbb {Z}}}_{p}^{n})$, $T>0$.

By [28, Sections 1.1 and 1.2] and the above Section we have that the pseudo-differential operator ${\mathcal {A}}$ is the infinitesimal generator of the Feller semigroup, $\{T_{t}\}_{t\ge 0}$, obtained in Theorem 2.

Theorem 5

Let $g\in C_{0}({{\mathbb {Z}}}_{p}^{n})$, $f\in L^{1}\left( [0,T]:C_{0}({{\mathbb {Z}}}_{p}^{n})\right)$, $T>0$, and $\{T_{t}\}_{t\ge 0}$ the Feller semigroup obtained in Theorem 2. Then, the function $u\in C([0,T]:C_{0}({{\mathbb {Z}}}_{p}^{n}))$ given by

$$\begin{aligned} u(t)=T_{t}g+\int _{0}^{t}T_{t-s}f(s)ds, \ \ 0\le t\le T, \end{aligned}$$

is the mild solution of the initial value problem (5.6) on [0, T].