P-adic Markov process and the problem of the first return over balls

Let $\langle x\rangle^{\alpha}=(\max\{|x|_{p},p^r\})^{\alpha}$ and $H^{\alpha}\varphi=\mathcal{F}^{-1}[(\langle \xi\rangle^{\alpha} -p^{r\alpha})\mathcal{F}\varphi]$, in this article we study the Markov process associated to this operator and the first passage time problem associated to $H^{\alpha}$.


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 (Q p ).
The problem of the first return in dimension 1 was studied in [4], and in arbitrary dimension in [6] and [10]. In these articles, pseudodifferential operators with radial symbols were considered. More recently, Chacón-Cortés [7] considers pseudodifferential operators over Q 4 p with non-radial 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 for ϕ ∈ S(Q p ), where ξ = max{|ξ| p , p r }. We also define the heat-kernel Z r as heat kernels of this type have been studied in [5], we show that function is a solution of Cauchy problem and we show that Z r (x, t) is the transition density of a time and space homegeneous Markov process, which is bounded, right-continuous and has no discontinuities other than jumps. Finally, we study the first passage time problem associated to the operator H α .

Preliminaries
In this section we fix the notation and collect some basic results on p-adic analysis that we will use through the article. For a detailed exposition on p-adic analysis the reader may consult [1,9,11].
2.1. The field of p-adic numbers. Along this article p will denote a prime number. The field of p−adic numbers Q p is defined as the completion of the field of rational numbers Q with respect to the p−adic norm | · | p , which is defined as , where a and b are integers coprime with p. The integer γ := ord(x), with ord(0) := +∞, is called the p−adic order of x.
Any p−adic number x = 0 has a unique expansion x = p ord(x) ∞ j=0 x j p j , where x j ∈ {0, 1, 2, . . . , p−1} and x 0 = 0. By using this expansion, we define the fractional part of x ∈ Q p , denoted {x} p , as the rational number For r ∈ Z, denote by B r (a) = {x ∈ Q p : |x − a| p ≤ p r } the ball of radius p r with center at a ∈ Q p , and take B r (0) := B r .

2.2.
The Bruhat-Schwartz space. A complex-valued function ϕ defined on Q p is called locally constant if for any x ∈ Q p there exists an integer l(x) ∈ Z such that The space of locally constant functions is denoted by E(Q p ). A function ϕ : Q p → C is called a Bruhat-Schwartz function (or a test function) if it is locally constant with compact support. The C-vector space of Bruhat-Schwartz functions is denoted by S(Q p ). For ϕ ∈ S(Q p ), the largest of such number l = l(ϕ) satisfying (2.1) is called the exponent of local constancy of ϕ. Let S ′ (Q p ) denote the set of all functionals (distributions) on S(Q p ). All functionals on S(Q p ) are continuous.
2.3. Fourier transform. Given ξ and x ∈ Q p , the Fourier transform of ϕ ∈ S(Q p ) is defined as where dx is the Haar measure on Q p normalized by the condition vol(B 0 ) = 1. The Fourier transform is a linear isomorphism from S(Q p ) onto itself satisfying (F (F ϕ))(ξ) = ϕ(−ξ). We will also use the notation F x→ξ ϕ and ϕ for the Fourier transform of ϕ. The The

Pseudodifferential operators
Definition 1. For all α ∈ C we define the following pseudodifferential operator It is clear that the map H α : S(Q p ) → S(Q p ) is continuous. Also it is possible to show that the pseudodifferential operator H α has the following representation integral and for α = 0 we define K 0 = δ.
After some calculations it is possible to show the following result.
Theorem 1. The Fourier transform of K α is given by ξ α for all α ∈ C.
Definition 3. For x ∈ Q p , t ∈ R the heat kernel is defined as The following properties are proved in [5].
If we set for ϕ ∈ S(Q p ) then it is easy to see that u(x, t) ∈ S(Q p ) for t ≥ 0, and also it is possible to show that for t ≥ 0, α > 0 Theorem 2. Consider the following Cauchy problem then the function u(x, t) defined in (3.4) is a solution.

p-adic Markov process over balls
The space (Q p , |·| p ) is a complete non-Archimedian metric space. Let B the Borel σ-algebra of Q p ; thus (Q p , B, dx) is a measure space. By using the terminology and results of [8, Chapters 2, 3], we set for t = 0.
Lemma 2. With the above notation the following assertions hold: (1) p(t, x, y) is a normal transition density.   Proof. (i) By Lemma 1 (6), we have Therefore, lim (ii) By using Lemma 1 (6), α > 0, we have Therefore, Theorem 3. Z r (x, t) is the transition density of a time and space homogeneous Markov process, called T(t, ω), which is bounded, right-continuous and has no discontinuities other than jumps.
Proof. The result follows from [8, Theorem 3.6] by using that (Q 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 L(B) and M(B).

The first passage time
By Proposition 2, the function Among other properties, the function u(x, t) = Z r (x, t) * Ω(|x| p ), t ≥ 0, is pointwise differentiable in t and, by using the Dominated Convergence Theorem, we can show that its derivative is given by the formula Lemma 4. If α > 0 and r < 0, then The rest of this section is dedicated to the study of the following random variable.

p-ADIC MARKOV PROCESS AND THE PROBLEM OF THE FIRST RETURN OVER BALLS 7
Definition 4. The random variable τ Ω(|x|p) (ω) : Y → R + defined by inf{t > 0 | T(t, ω) ∈ Ω(|x|p) and there exists t ′ such that 0 < t ′ < t and T(t ′ , ω) / ∈ Ω(|x|p)} is called the first passage time of a path of the random process T(t, ω) entering the domain Ω(|x| p ).
Lemma 5. The probability density function for a path of T(t, ω) to enter into Ω(|x| p ) at the instant of time t, with the condition that T(0, ω) ∈ Ω(|x| p ) is given by Proof. We first note that, for x, y ∈ Ω(|z| p ), we have i.e. u(x − y, t) − u(x, t) ≡ 0 for x, y ∈ Ω(|z| p ). The survival probability, by definition is the probability that a path of T(t, ω) remains in Ω(|x| p ) at the time t. Because there are no external or internal sources, Probability that a path of T(t, ω) goes back to Ω(|x| p ) at the time t − Probability that a path of T(t, ω) exits Ω(|x| p ) at the time t.
by using the derivative (5.3) Now if y ∈ Ω(p r |y| p ) \ Ω(|y| p ) and x ∈ Ω(|x| p ), then u(x − y, t) = u(y, t), consequently Proposition 1. The probability density function f (t) of the random variable τ Ω(|x|p) (ω) satisfies the non-homogeneous Volterra equation of second kind Proof. The result follows from Lemma 5 by using the argument given in the proof of Theorem 1 in [4].
Definition 5. We say that T(t, ω) is recurrent with respect to Ω(|x| p ) if Otherwise, we say that T(t, ω) is transient with respect to Ω(|x| p ) .
The meaning of (5.6) is that every path of T(t, ω) is sure to return to Ω(|x| p ). If (5.6) does not hold, then there exist paths of T(t, ω) that abandon Ω(|x| p ) and never go back.
Proof. By Proposition 1, the Laplace transform F (s) of f (t) equals G r (s) 1 + G r (s) , where G r (s) is the Laplace transform of g(t), and thus .
Hence in order to prove that T(t, ω) is recurrent is sufficient to show that