Non-archimedean generalized Bessel potentials and their applications

Torresblanca-Badillo, Anselmo

doi:10.1016/j.jmaa.2020.124874

Cited by 8 publications

(7 citation statements)

References 23 publications

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“…Unlike the Bessel potentials studied in [10,23,25], (in [10] and [23] denoted by J α ), in this article, certain classes Feller semigroups and Markov processes associated with the convolution kernel of our Bessel α-potentials are studied. Additionally, for the first time in the p-adic context, families of measures (σ α ) α>0 over Q n p associated with convolution semigroups are introduced.…”

Section: U(x T) := T (T)g(x)mentioning

confidence: 99%

“…Additionally, for the first time in the p-adic context, families of measures (σ α ) α>0 over Q n p associated with convolution semigroups are introduced. Another aspect to consider, which marks a difference between our Bessel α-potentials and those studied in [25], is that the symbol of our pseudo-differential operators is not required to be negative definite functions. Therefore, in a certain sense, our convolution semigroups do not necessarily belong to the convolution semigroups studied in [25].…”

Section: U(x T) := T (T)g(x)mentioning

confidence: 99%

“…Recently, in [10], Gutiérrez and Torresblanca study (in the p-adic context) two classes of pseudo-differential operators related to Bessel potentials introduced by Taibleson, with the aim of investigating the Cauchy problem associated with these classes of operators. Months later, in [25], Torresblanca introduces a family of padic Bessel potentials whose symbols are connected to negative definite functions on p-adic numbers.…”

Section: Introductionmentioning

confidence: 99%

“…where, F −1 ζ →x denotes the inverse Fourier transform, f is the Fourier transform of f , max{1, |φ(||ζ || p )|} −α is called symbol of the pseudo-differential operator E φ,α and φ : Q n p → C is a continuous function satisfying certain conditions. We want to emphasize that, due to the nature of the function φ, this class of operators does not necessarily belong to the family of Bessel potentials studied in [10,25]. Furthermore, they are a generalization of some of the Bessel potentials of Taibleson when considering α > 0 in [23].…”

Section: Introductionmentioning

confidence: 99%

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p-adic Bessel $$\alpha $$-potentials and some of their applications

Torresblanca-Badillo,

Ospino,

Arias

2024

J. Pseudo-Differ. Oper. Appl.

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In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel $$\alpha $$ α -potentials. These operators are denoted and defined in the form $$\begin{aligned} (\mathcal {E}_{\varvec{\phi },\alpha }f)(x)=-\mathcal {F}^{-1}_{\zeta \rightarrow x}\left( \left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\} \right] ^{-\alpha }\widehat{f}(\zeta )\right) , \text { } x\in {\mathbb {Q}}_{p}^{n}, \ \ \alpha \in \mathbb {R}, \end{aligned}$$ ( E ϕ , α f ) ( x ) = - F ζ → x - 1 max { 1 , | ϕ ( | | ζ | | p ) | } - α f ^ ( ζ ) , x ∈ Q p n , α ∈ R , where f is a p-adic distribution and $$\left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\}\right] ^{-\alpha }$$ max { 1 , | ϕ ( | | ζ | | p ) | } - α is the symbol of the operator. We will study some properties of the convolution kernel (denoted as $$K_{\alpha }$$ K α ) of the pseudo-differential operator $$\mathcal {E}_{\varvec{\phi },\alpha }$$ E ϕ , α , $$\alpha \in \mathbb {R}$$ α ∈ R ; and demonstrate that the family $$(K_{\alpha })_{\alpha >0}$$ ( K α ) α > 0 determines a convolution semigroup on $$\mathbb {Q}_{p}^{n}$$ Q p n . Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.

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Section: U(x T) := T (T)g(x)mentioning

confidence: 99%

Section: U(x T) := T (T)g(x)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

p-adic Bessel $$\alpha $$-potentials and some of their applications

Torresblanca-Badillo,

Ospino,

Arias

2024

J. Pseudo-Differ. Oper. Appl.

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“…Therefore, the ultrametric spaces (in particular the p-adic numbers) are proposed as a natural, necessary and essential structure to study the spreading of infectious diseases (say COVID-19) through a random walk on a complex energy landscape and taking into account social clusters in a situation of extreme social isolation. For more details, the reader can consult [3,19,20,23,34].…”

Section: Introductionmentioning

confidence: 99%

Energy Landscapes and Non-Archimedean Pseudo- Differential Operators as Tools for Studying the Spreading of Infectious Diseases in a Situation of Extreme Social Isolation

AGUILAR-ARTEAGA,

GUTIÉRREZ,

TORRESBLANCA-BADILLO

2024

KgJMat

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In this article, we introduce a new type of pseudo-diﬀerential equations naturally connected with non-archimedean pseudo-diﬀerential operators and whose symbols are new classes of negative deﬁnite functions in the p-adic context and in arbitrary dimension. These equations are proposed as a mathematical models to study the spreading of infectious diseases (say COVID-19) through a random walk on a complex energy landscape and taking into account social clusters in a situation of extreme social isolation.

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New classes of p-adic evolution equations and their applications

Torresblanca-Badillo

Bolaño-Benítez

2023

J. Pseudo-Differ. Oper. Appl.

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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 $${\mathfrak {X}}$$ X with state space the n-dimensional p-adic unit ball ($${\mathbb {Z}}_{p}^{n}$$ Z p n ). We introduce a family of operators $$\{T_{t}\}_{t\ge 0}$$ { T t } t ≥ 0 (obtained explicitly) that determine a Feller semigroup on $$C_{0}({\mathbb {Z}}_{p}^{n})$$ C 0 ( Z p n ) . Also, we study the asymptotic behavior of the survival probability of a strong Markov processes $${\mathfrak {X}}$$ X on a ball $$B_{-m}^{n}\subset {\mathbb {Z}}_{p}^{n}$$ B - m n ⊂ Z p n , $$m\in {\mathbb {N}}$$ m ∈ N . Moreover, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the mentioned above Feller semigroup.

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Non-archimedean generalized Bessel potentials and their applications

Cited by 8 publications

References 23 publications

p-adic Bessel $$\alpha $$-potentials and some of their applications

p-adic Bessel $$\alpha $$-potentials and some of their applications

Energy Landscapes and Non-Archimedean Pseudo- Differential Operators as Tools for Studying the Spreading of Infectious Diseases in a Situation of Extreme Social Isolation

New classes of p-adic evolution equations and their applications

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