arXiv subjects
Publications and source records attributed to Ivana Gómez.
In this paper we obtain the fundamental solution kernel of dyadic diffusions in $\mathbb{R}^+$ as a Central Limit of dyadic mollification of iterations of stable Markov kernels. The main tool is provided by the substitution of classical Fourier analysis by Haar wavelet analysis.
In this note we consider the pointwise convergence to the initial data for the solutions of some nonlocal dyadic Schrödinger equations on spaces of homogeneous type. We prove the a.e. convergence when the initial data belongs to a dyadic version of an $L^2$ based Besov space.
Let $X$ be a (data) set. Let $K(x,y)>0$ be a measure of the affinity between the data points $x$ and $y$. We prove that $K$ has the structure of a Newtonian potential $K(x,y)=φ(d(x,y))$ with $φ$ decreasing and $d$ a quasi-metric on $X$ under two mild conditions on $K$. The first is that the affinity of each $x$ to itself is infinite and that for $x\neq y$ the affinity is positive and finite. The second is a quantitative transitivity; if the affinity between $x$ and $y$ is larger than $λ>0$ and the affinity of $y$ and $z$ is also larger than $λ$, then the affinity between $x$ and $z$ is larger than $ν(λ)$. The function $ν$ is concave, increasing, continuous from $\mathbb{R}^+$ onto $\mathbb{R}^+$ with $ν(λ)<λ$ for every $λ>0$.
We obtain a new general extension theorem in Banach spaces for operators which are not required to be symmetric, and apply it to obtain Harnack estimates and a priori regularity for solutions of fractional powers of several second order differential operators. These include weighted elliptic and subellitptic operators in divergence form (nonnecessarily self-adjoint), and nondivergence form operators with rough coefficients. We utilize the reflection extension technique introduced by Caffarelli and Silvestre.
In this paper we deal with anomalous diffusions induced by Continuous Time Random Walks - CTRW in $\mathbb{R}^n$. A particle moves in $\mathbb{R}^n$ in such a way that the probability density function $u(\cdot,t)$ of finding it in region $Ω$ of $\mathbb{R}^n$ is given by $\int_Ωu(x,t) dx$. The dynamics of the diffusion is provided by a space time probability density $J(x,t)$ compactly supported in $\{t\geq 0\}$. For $t$ large enough, $u$ must satisfy the equation $u(x,t)=[(J-δ)\ast u](x,t)$ where $δ$ is the Dirac delta in space time. We give a sense to a Cauchy type problem for a given initial density distribution $f$. We use Banach fixed point method to solve it, and we prove that under parabolic rescaling of $J$ the equation tends weakly to the heat equation and that for particular kernels $J$ the solutions tend to the corresponding temperatures when the scaling parameter approaches to zero.
We prove a mean value formula for weak solutions of $div(|y|^{a}\grad u)=0$ in $\mathbb{R}^{n+1}=\{(x,y): x\in\mathbb{R}^{n}, y\in\mathbb{R}\}$, $-1<a<1$ and balls centered at points of the form $(x,0)$. We obtain an explicit nonlocal kernel for the mean value formula for solutions of $(-\triangle)^{s}f=0$ on a domain $D$ of $\mathbb{R}^{n}$. When $D$ is Lipschitz we prove a Besov type regularity improvement for the solutions of $(-\triangle)^{s}f=0$.
In this note we consider the initial boundary value problem for the heat equation on cylinders based on Lipschitz domains with Besov data. We obtain a regularity exponent for the solution that improves the rate of convergence of nonlinear approximation methods.
In this paper we consider the pointwise convergence to the initial data for the Schrödinger-Dirac equation $i\tfrac{\partial u}{\partial t}=D^βu$ with $u(x,0)=u^0$ in a dyadic Besov space. Here $D^β$ denotes the fractional derivative of order $β$ associated to the dyadic distance $δ$ on $\mathbb{R}^+$. The main tools are a sumability formula for the kernel of $D^β$ and pointwise estimates of the corresponding maximal operator in terms of the dyadic Hardy-Littlewood function and the Calderón sharp maximal operator.