Bregman variation of semimartingales
We define Bregman variation of semimartingales. We give its pathwise representation, Itô-type isometry for martingales, and applications to harmonic analysis.
math.PR↗
arXiv subjects
Katarzyna Pietruska-Pałuba
Publications and source records attributed to Katarzyna Pietruska-Pałuba.
IDS for subordinate Brownian motions in Poisson random environment on nested fractals
We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schrödinger operators \[ H^ω = ϕ(-\mathcal{L}) + V^ω \] on planar unbounded nested fractals with the Good Labeling Property. Here, $\mathcal{L}$ is the Laplacian on the fractal, $ϕ$ is an operator monotone function with mild regularity, and $V^ω$ is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of $V^ω$ can be effectively reduced to the analysis of certain alloy-type potential, where the sites are no longer lattice points as in the classical $\mathbb{Z}^d$ case, but fractal complexes. This observation enables us to apply an approach, new in the setting of Poissonian random fields, which allows us to treat a broad class of Bernstein functions $ϕ$. In particular, it covers the case $ϕ(λ)=(λ+m^{d_w/\vartheta})^{\vartheta/d_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods.
math.PR↗
Polarized Hardy--Stein identity
We prove the Hardy--Stein identity for vector functions in $L^p(\mathbb R^d;\mathbb R^n)$ with $1<p<\infty$ and for the canonical paring of two real functions in $L^p(\mathbb R^d)$ with $2\le p<\infty$. To this end we propose a notion of Bregman co-divergence and study the corresponding integral forms.
math.CA↗
Density of states for the Anderson model on nested fractals
We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians $H^ω=H_0+V^ω$ on fractal spaces of infinite diameter. The kinetic term $H_0$ is given by $ϕ(-\mathcal L),$ where $\mathcal L$ is the Laplacian on the fractal and $ϕ$ is a completely monotone function satisfying some mild regularity conditions. The random potential $V^ω$ is of alloy-type.
math.SP↗
Nonlinear nonlocal Douglas identity
We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal Lévy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev-Bregman forms.
math.AP↗
Optimal Hardy inequality for the fractional Laplacian on $L^p$
For the fractional Laplacian we give Hardy inequality which is optimal in $L^p$ for $1<p<\infty$. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on $L^p$.
math.AP↗
Lifshitz tail for continuous Anderson models driven by Lévy operators
We investigate the behavior near zero of the integrated density of states $\ell$ for random Schrödinger operators $Φ(-Δ) + V^ω$ in $L^2(\mathbb R^d)$, $d \geq 1$, where $Φ$ is a complete Bernstein function such that for some $α\in (0,2]$, one has $ Φ(λ) \asymp λ^{α/2}$, $λ\searrow 0$, and $V^ω(x) = \sum_{ \mathbf{i}\in \mathbb{Z}^d} q_{\mathbf{i}}(ω) W(x-\mathbf{i})$ is a random nonnegative alloy-type potential with compactly supported single site potential $W$. We prove that there are constants $C, \widetilde C,D, \widetilde D>0$ such that $$ -C \leq\liminf_{λ\searrow 0} \frac{λ^{d/α}}{|\log F_q(D λ)|}{\log \ell(λ)} \qquad \text{and} \qquad \limsup_{λ\searrow 0} \frac{λ^{d/α}}{|\log F_q(\widetilde D λ)|}\log \ell(λ) \leq -\widetilde C, $$ where $F_q$ is the common cumulative distribution function of the lattice random variables $q_{\mathbf i}$. In particular, we identify how the behavior of $\ell$ at zero depends on the lattice configuration. For typical examples of $F_q$ the constants $D$ and $\widetilde D$ can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, both local and non-local kinetic terms such as the Laplace operator, its fractional powers and the quasi-relativistic Hamiltonians.
math.PR↗
Lifschitz tail for alloy-type models driven by the fractional Laplacian
We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators $(-Δ)^{α/2} + V^ω$ in $L^2(\mathbb R^d)$ for the full range of $α\in(0,2]$ and a fairly large class of random nonnegative alloy-type potentials $V^ω$. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit $$\lim_{s\to 0} s^{d/α}\ln\ell([0,s]) = -C \left(λ_d^{(α)}\right)^{d/α},$$ with $C \in (0,\infty]$. The constant $C$ is is finite if and only if the common distribution of the lattice random variables charges $\left\{0\right\}$. In this case, the constant $C$ is expressed explicitly in terms of such a probability. In the limit formula, $λ_d^{(α)}$ denotes the Dirichlet ground-state eigenvalue of the operator $(-Δ)^{α/2}$ in the unit ball in $\mathbb R^d.$
math.PR↗
Extension and trace for nonlocal operators
We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the quadratic form of the Poisson extension.
math.AP↗
Reflected Brownian motion on simple nested fractals
We prove the existence of the reflected diffusion on a complex of an arbitrary size for a large class of planar simple nested fractals. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions on the fractal under which this projection can be well defined. They are illustrated by various specific examples. We first construct a proper version of the transition probability densities for reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman-Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property
math.PR↗
The quenched asymptotics for nonlocal Schrödinger operators with Poissonian potentials
We study the quenched long time behaviour of the survival probability up to time $t$, $\mathbf{E}_x\big[e^{-\int_0^t V^ω(X_s){\rm d}s}\big],$ of a symmetric Lévy process with jumps, under a sufficiently regular Poissonian random potential $V^ω$ on $\mathbb{R}^d$. Such a function is a probabilistic solution to the parabolic eq. involving the nonlocal Schrödinger operator based on the generator of $(X_t)_{t \geq 0}$ with potential $V^ω$. For a large class of processes and potentials, we determine rate functions $η(t)$ and positive constants $C_1, C_2$ such that \[-C_1 \leq \liminf_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^ω(X_s){\rm d}s}\big]}{η(t)} \leq \limsup_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^ω(X_s){\rm d}s}\big]}{η(t)} \leq -C_2, \] almost surely with respect to $ω$, for every fixed $x \in \mathbb{R}^d$. The functions $η(t)$ and the bounds $C_1, C_2$ heavily depend on the intensity of large jumps of the process. In particular, if its decay at infinity is `sufficiently fast', then we prove that $C_1=C_2$, i.e. the limit exists. Representative examples in this class are relativistic stable processes with Lévy-Khintchine exponents $ψ(ξ) = (|ξ|^2+m^{2/α})^{α/2}-m$, $α\in (0,2)$, $m>0$, for which \[\lim_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^ω(X_s)ds}\big]}{t/(\log t)^{2/d}} = \fracα{2} m^{1-\frac{2}α} \, \left(\frac{ρω_d}{d}\right)^{\frac{d}{2}} \, λ_1^{BM}(B(0,1)), \quad \mbox{for almost all $ω$,}\] where $λ_1^{BM}(B(0,1))$ is the principal eigenvalue of the Brownian motion in the unit ball, $ω_d$ is the Lebesgue measure of a unit ball and $ρ>0$ corresponds to $V^ω$. We also identify two interesting regime changes ('transitions') in the growth properties of $η(t)$
math.PR↗
Lifschitz singularity for subordinate Brownian motions in presence of the Poissonian potential on the Sierpiński gasket
We establish the Lifschitz-type singularity around the bottom of the spectrum for the integrated density of states for a class of subordinate Brownian motions in presence of the nonnegative Poissonian random potentials, possibly of infinite range, on the Sierpiński gasket. We also study the long-time behaviour for the corresponding averaged Feynman-Kac functionals.
math.PR↗
Lifschitz tail and sausage asymptotics for stable processes in the Poissonian environment on the Sierpinski gasket
We obtain the Lifschitz tail asymptotics for the integrated density of states of the subordinate $α-$stable processes on the Sierpiński gasket $\mathcal G,$ evolving among killing Poissonian obstacles. Simultaneously, we derive the large-time asymptotics for the volume of the $α-$stable sausage on the gasket.
math.PR↗
Poincaré Inequality and Hajlasz-Sobolev spaces on nested fractals
Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajlasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.
math.FA↗
Probabilistic characterisation of Besov-Lipschitz spaces on metric measure spaces
We give a probabilistic characterisation of the Besov-Lipschitz spaces $Lip(α,p,q)(X)$ on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of \cite{Jon,KPP1,KPP2,GHL} which were valid for $α=\frac{d_w}{2},p=2$, $q=\infty,$ where $d_w$ is the walk dimension of the space $X.$
math.PR↗