We study a system of hard rods of finite size in one space dimension, which move by Brownian noise while avoiding overlap. We consider a scaling in which the number of particles tends to infinity while the volume fraction of the rods remains constant; in this limit the empirical measure of the rod positions converges almost surely to a deterministic limit evolution. We prove a large-deviation principle

We study the higher order q- Poincaré and other coercive inequalities for a class probability measures satisfying Adam’s regularity condition.

We first derive a double-weighted Hardy-Sobolev inequality with monomial weights on Euclidean space, then we establish the Gross’ type logarithmic Sobolev inequality with monomial weights using product structure. We will also study Moser-Onofri-Beckner inequality with monomial weights. Some best constants and extremals are also explicitly given.

Probability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration

In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted Lq-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the

Lions in Lions (1969) solved the anisotropic p-Laplace equation in deterministic setting by considering the anisotropic p-Laplace operator in d-dimensions as a sum of d monotone coercive operators each defined on a different space. Motivated by this example, we prove existence and uniqueness results for a large class of stochastic partial differential equations(SPDEs) driven by Lévy noise when the

It is well known that some important Markov semi-groups have a “regularization effect” – as for example th hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in Lp for p > 1. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property

We prove a uniform large deviations principle for the law of the solutions to a class of Burgers-type stochastic partial differential equations in any space dimension. The equation has nonlinearities of polynomial growth of any order, the driving noise is a finite dimensional Wiener process, and the proof is based on variational principle methods. We prove the uniform large deviations principle for

In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author’s previous results under integral

In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define

This paper proves two-sided estimates for the Dirichlet heat kernel on inner uniform domains in metric measure Dirichlet spaces satisfying the volume doubling condition, the Poincaré inequality, and a cutoff Sobolev inequality. More generally, we obtain local upper and lower bounds for the Dirichlet heat kernel on locally inner uniform domains under local geometric assumptions on the underlying space

Let G be a compact connected Lie group of dimension m. Once a bi-invariant metric on G is fixed, left-invariant metrics on G are in correspondence with m × m positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on G in terms of the eigenvalues of the corresponding positive definite

Let \(\mathcal {X}\) be a separable Hilbert space with norm \(\left \|{\cdot }\right \|\) and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on \(\mathcal {X}\), let \(F:\mathcal {X}\rightarrow \mathcal {X}\) be a (smooth enough) function and let W(t) be a \(\mathcal {X}\)-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator \(A:=-(1/2)Q^{2\alpha

We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also

Let L be a non-negative self-adjoint operator on L2(X) where X is a metric space with a doubling measure. Assume that the kernels of the semigroup generated by − L satisfy a suitable upper bound related to a critical function ρ but these kernels are not assumed to satisfy any regularity conditions on spacial variables. In this paper, we prove the quantitative weighted estimates for some square functions

We consider the torsion function for the Dirichlet Laplacian −Δ, and for the Schrödinger operator −Δ + V on an open set \({\Omega }\subset \mathbb {R}^{m}\) of finite Lebesgue measure \(0<|{\Omega }|<\infty \) with a real-valued, non-negative, measurable potential V. We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the

The paper deals with the theory of potentials with respect to the α-Riesz kernel |x − y|α−n of order α ∈ (0,2] on \(\mathbb R^{n}\), \(n\geqslant 3\). Focusing first on the inner α-harmonic measure \({\varepsilon _{y}^{A}}\) (εy being the unit Dirac measure at \(y\in \mathbb R^{n}\), and μA the inner α-Riesz balayage of a Radon measure μ to \(A\subset \mathbb R^{n}\) arbitrary), we describe its Euclidean

The Girsanov transform and Kolmogorov equations are two useful methods for studying SPDEs. It is shown that, under suitable conditions, the series expansion obtained from the Girsanov transform coincides with the one generated by an iteration scheme for Kolmogorov equations. We also apply the iteration approach to extend the well posedness theory for Kolmogorov equations beyond the boundedness condition

The present work is the continuation of our study (Arenas et al. J. Math. Anal. Appl. 490(123996), 21, 2020) on discrete harmonic analysis related to Jacobi expansions. The role of a Laplacian is played by the operator \(\mathcal {J}^{(\alpha ,\beta )}\) defined by the three-term recurrence relation for the normalised Jacobi polynomials. The main interest is to establish weighted inequalities for the

We consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially non-homogeneous. Despite these crucial differences, we compute the asymptotics of the Martin kernel, deduce the Martin boundary and show that it is minimal. Similar results on the grid \(\mathbb {Z}^{d}\) are derived

In this work we search for boundedness results for operators related to the semigroup generated by the degenerate Schrödinger operator \({{\mathscr{L}}} u = -\frac {1}{\omega } \text {div} A\cdot \nabla u +V u\), where ω is a weight, A is a matrix depending on x and satisfying λω(x)|ξ|2 ≤ A(x)ξ ⋅ ξ ≤Λω(x)|ξ|2 for some positive constants λ, Λ and all x, ξ in \(\mathbb {R}^{d}\), assuming further suitable

We prove two versions of a boundary Harnack principle in which the constants do not depend on the domain by using probabilistic methods.

In this paper, we obtain some interesting reproducing kernel estimates and some Carleson properties that play an important role. We characterize the bounded and compact Toeplitz operators on the weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, we estimate the essential norm of them assuming that they are bounded.

We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2(G), where G is an open bounded domain in \(\mathbb {R}^{d}\) with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in \(\mathbb {R}^{n}\) and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case.

We study properties of \(\mathcal {A}\)-harmonic and \(\mathcal {A}\)-superharmonic functions involving an operator having generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. In particular, Harnack’s Principle and Minimum Principle are provided for \(\mathcal {A}\)-superharmonic functions.

Motivated by the recent work of Galindo et al. (J. Funct. Anal. 265, 629–643, 2013), in this paper, we give an elegant compactness criterion for any finite linear combination of composition operators on the weighted Bergman space in terms of power type characterization. More precisely, let T be any finite linear combination of composition operators, then $$ T \ \text{is compact on} \ {A}^{p}_{\alpha}(\mathbf{D})

In this paper, we establish C1,α regularity up to the boundary for a class of degenerate fully nonlinear elliptic equations with Neumann boundary conditions. Our main result Theorem 2.1 constitutes the boundary analogue of the interior C1,α regularity result established in Imbert and Silvestre (Adv. Math. 233: 196–206, 2013) for equations with similar structural assumptions. The proof of our main result

We investigate the 3rd term of the spectral heat content for killed subordinate and subordinate killed Brownian motions on a bounded open interval D = (a,b) in a real line when the underlying subordinators are stable subordinators with index α ∈ (1, 2) or α = 1. We prove that in the 3rd term of the spectral heat content, one can observe the length b − a of the interval D.

In this article we prove mixed inequalities for maximal operators associated to Young functions, which are an improvement of a conjecture established in Berra (Proc. Am. Math. Soc. 147(10), 4259–4273, 2019). Concretely, given r ≥ 1, u ∈ A1, \(v^{r}\in A_{\infty }\) and a Young function Φ with certain properties, we have that inequality $$ uv^{r}\left( \left\{x\in \mathbb{R}^{n}: \frac{M_{\Phi}(fv)(x)}{

We investigate the validity, as well as the failure, of Sobolev-type inequalities on Cartan-Hadamard manifolds under suitable bounds on the sectional and the Ricci curvatures. We prove that if the sectional curvatures are bounded from above by a negative power of the distance from a fixed pole (times a negative constant), then all the Lp inequalities that interpolate between Poincaré and Sobolev hold

The boundary of a regular tree can be viewed as a Cantor-type set. We equip our tree with a weighted distance and a weighted measure via the Euclidean arc-length and consider the associated first-order Sobolev spaces. We give characterizations for the existence of traces and for the density of compactly supported functions.

We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the Lévy system formula and estimates on the heat kernel.

In this paper, we get an Lp boundedness of Fourier integral operators with rough amplitude \(a\in L^{\infty } S^{m}_{\varrho },~\) and phase \(\varphi \in L^{\infty }{\Phi }^{2}\) for \(1\leq p\leq +\infty \). This is an improvement of the corresponding results in Dos Santos Ferreira and Staubach (Mem. Amer. Math. Soc. 229, 1074, 2014).

By studying parabolic equations in mixed-norm spaces, we prove the existence and uniqueness of strong solutions to stochastic differential equations driven by Brownian motion with coefficients in spaces with mixed-norm, which extends Krylov and Röckner’s result in Krylov (Probab. Theory Rel. Fields. 131(2)154–196, 2005) and Zhang’s result in Zhang (Electron. J. Probab. 16,1096–1116, 2011).

We consider a nonlinear parametric Neumann problem driven by the anisotropic (p, q)-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result

Using an operator approach, we discuss stationary solutions to Fokker-Planck equations and systems with nonlinear reaction terms. The existence of solutions is obtained by using Banach, Schauder and Schaefer fixed point theorems, and for systems by means of Perov’s fixed point theorem. Using the Ekeland variational principle, it is proved that the unique solution of the problem minimizes the energy

By classical results of G.C. Evans and G. Choquet on “good” kernels G in potential theory, for every polar Kσ-set P, there exists a finite measure μ on P such that its potential Gμ is infinite on P, and a set P admits a finite measure μ on P such that Gμ is infinite exactly on P if and only if P is a polar Gδ-set. A known application of Evans’ theorem yields the solutions of the generalized Dirichlet

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then

We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by $$ \left \{\begin {array}{ll} \displaystyle -{\Delta }_{p} u= \frac {f}{u^{\gamma }} + g u^{q} & \text { in } {\Omega }, \\ u = 0 & \text {on } \partial {\Omega }, \end {array}\right . $$ where Ω is an open bounded subset of \(\mathbb {R}^{N}\) where Ω is an open bounded subset of \(\mathbb

In this paper we deal with the notion of the effective impedance of alternating current (AC) networks consisting of resistances, coils and capacitors. Mathematically such a network is a locally finite graph whose edges are endowed with complex-valued weights depending on a complex parameter λ (by the physical meaning, λ = iω, where ω is the frequency of the AC). For finite networks, we prove some estimates

In this paper, we give a probabilistic interpretation for solutions to the Neumann boundary problems for a class of semi-linear parabolic partial differential equations (PDEs for short) with singular non-linear divergence terms. This probabilistic approach leads to the study on a new class of backward stochastic differential equations (BSDEs for short). A connection between this class of BSDEs and

This note studies the asymptotic behavior of global solutions to the fourth-order Schrödinger equation $$ i\dot u+{\Delta}^{2} u-F(x,u)=0 . $$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the defocusing mass super-critical and energy sub-critical regimes, with radial setting.

In this paper, we prove a reverse Hölder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of Gamara et al. (Open Math. 13(1), 557–570, 2015) and Colladay et al. (J. Geom. Anal. 28(4)

We give sufficient conditions for the essential self-adjointness of perturbed biharmonic operators acting on sections of a Hermitian vector bundle over a Riemannian manifold with additional assumptions, such as lower semi-bounded Ricci curvature or bounded sectional curvature. In the case of lower semi-bounded Ricci curvature, we formulate our results in terms of the completeness of the metric that

We study the local regularity of solutions f to the integro-differential equation $$ Af=g \quad \text{in } U $$ for open sets \(U \subseteq \mathbb {R}^{d}\), where A is the infinitesimal generator of a Lévy process (Xt)t≥ 0. Under the assumption that the transition density of (Xt)t≥ 0 satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions

In this article we study stability properties of \(g_{_{\mathcal {O}}}\!\), the standard Green kernel for \(\mathcal {O}\) an open regular set in \(\mathbb {R}^{d}\). In d ≥ 3 we show that \( g^{\beta }_{_{\mathcal {O}}}\) is again a Green kernel of a Markov Feller process, for any power β ∈ [1,d/(d − 2)). In dimension d = 2, we show the same result for \( g^{\beta }_{_{\mathcal {O}}}\), for any β

In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane—resolution of a functional equation via boundary value problem using a conformal mapping—to

We consider a family of dense Gδ subsets of [0, 1], defined as intersections of unions of small uniformly distributed intervals, and study their logarithmic capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a Gδ set can be considered as a toy model for the set of exceptional energies in the parametric

We investigate contraction of the Wasserstein distances on \(\mathbb {R}^{d}\) under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space—where the heat semigroup corresponds to smoothing the measures by Gaussian convolution—the situation is more subtle

We introduce generalized local and global Herz spaces where all their characteristics are variable. As one of the main results we show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized variable Herz spaces with application

This work provides explicit conditions to establish the dimension-free Harnack inequalities respectively for state-independent and state-dependent regime-switching processes. Then such Harnack inequalities are applied to establish the hypercontractivity of the associated semigroup, and further to study the long time asymptotics of unbounded additive functionals of regime-switching processes. Much effort

Furstenberg (1971) and Lyons and Sullivan (1984) have shown how to discretize harmonic functions on a Riemannian manifold M whose Brownian motion satisfies a certain recurrence property called ∗-recurrence. We study analogues of this discretization for tensor fields which are harmonic in the sense of the covariant Laplacian. We show that, under certain restrictions on the holonomy of the connection

It is well-known that for a harmonic function u defined on the unit ball of the d-dimensional Euclidean space, d ≥ 2, the tangential and normal component of the gradient ∇u on the sphere are comparable by means of the Lp-norms, \(p\in (1,\infty )\), up to multiplicative constants that depend only on d,p. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions

In this paper, motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature lower bound, we first introduce the notion of Li-Yau multiplier set and show that it can be computed by heat kernel of the manifold. Then, an optimal Li-Yau-type gradient estimate is obtained on hyperbolic spaces by using recurrence

We present a refined Green’s function estimate of the time measurable parabolic operators on conic domains that involves mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary.

According to the well-known Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact

Using coupling by change of measure and an approximation technique, Wang’s Harnack inequalities are established for a class of functional SDEs driven by subordinate Brownian motions. The results cover the corresponding ones in the case without delay.

Firstly we consider a finite dimensional Markov semigroup generated by Dunkl Laplacian with drift terms. For this semigroup we prove gradient bounds involving a symmetrised carré-du-champ operator, and we show that for small coefficients this semigroup has a unique invariant measure which satisfies ergodicity properties. We then extend this analysis to an infinite dimensional model on \((\mathbb {R}^{N})^{\mathbb

We consider parabolic equations of the form $$ u_{t}-\text{div} \left( |\nabla u|^{p-2}\nabla u+ a(x,t)|\nabla u|^{q-2}\nabla u\right)= 0, a(x,t)\geq 0. $$ In the range \(\frac {2n}{n+1}