We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let X1,…,Xn be independent random variables with common distribution the symmetric Jacobi measure \(d\mu (x) = C_{d} (1-x^{2})^{\frac d2 -1} dx\) with dimension d ≥ 1 on [− 1,1], and let \(\mu _{n} = \frac {1}{n} {\sum }_{i = 1}^{n} \delta _{X_{i}}\) be the associated empirical measure

We show that, under a natural scaling, the small-time behavior of the logarithmic derivatives of the subelliptic heat kernel on SU(2) converges to their analogues on the Heisenberg group at time 1. Realizing SU(2) as \(\mathbb {S}^{3}\), we then generalize these results to higher-order odd-dimensional spheres equipped with their natural subRiemannian structure, where the limiting spaces are now the

We prove a Chebyshev transform formula for a notion of (weighted) transfinite diameter that is defined using a generalized notion of polynomial degree. We also generalize Leja points to this setting. As an application of our main formula, we prove that in the unweighted case, these generalized Leja points recover the transfinite diameter.

Carbery proved that if \(u:\mathbb {R}^{n} \rightarrow \mathbb {R}\) is a positive, strictly convex function satisfying \(\det D^{2}u \geq 1\), then we have the estimate $$ \left| \left\{x \in \mathbb{R}^{n}: u(x) \leq s \right\} \right| \lesssim_{n} s^{n/2} $$ and this is optimal. We give a short proof that also implies other results. Our main result is an estimate for the sublevel set of functions

In this paper we use potential theoretic arguments to establish new results concerning the overconvergence of Dirichlet series. Let \({\sum }_{j=0}^{\infty } a_{j}e^{-\lambda _{j}s}\) converge on the half-plane {Re(s) > 0} to a holomorphic function f. Our first result gives sufficient conditions for a subsequence of partial sums of the series to converge at every regular point of f. The second result

For a hyperbolic Brownian motion in the hyperbolic space \(\mathbb {H}^{n}, n\ge 3\), we prove a representation of a Green function and a Poisson kernel for bounded and smooth sets in terms of the corresponding objects for an ordinary Euclidean Brownian motion and a conditional gauge functional. Using this representation we prove bounds for the Green functions and Poisson kernels for smooth sets. In

We show that the fractional Brownian motion (fBM) defined via the Volterra integral representation with Hurst parameter \(H\geq \frac {1}{2}\) is a quasi-surely defined Wiener functional on the classical Wiener space, and we establish the large deviation principle (LDP) for such an fBM with respect to (p,r)-capacity on the classical Wiener space in Malliavin’s sense.

We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDE $$ - {\Delta} u + b(x) \cdot \nabla u = 1 \qquad \text{in}~{\Omega} $$ subject to Dirichlet boundary conditions for \(\|b\|_{L^{\infty }}\) fixed. We show that, in any given C2 −domain Ω, the vector field maximizing the expected lifetime is (nonlinearly) coupled to the

Hunt’s hypothesis (H) and the related Getoor’s conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt’s hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes (Xt) and (Yt), if (Xt) satisfies

We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form \(\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u\) on \([0,T]\times \mathbb {R}^{d}.\) To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.

In this paper, we introduce the finite neighboring type and the finite chain length conditions for a connected self-similar set K. We show that with these two conditions, K is a finitely ramified graph directed (f.r.g.d.) fractal defined by Hambly and Nyberg (Proc. Edinb. Math. Soc. (2) 46(1), 1–34 2003). We give some nontrivial examples and compute the harmonic structures on them explicitly. Furthermore

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the Lp-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set Σ of zero Gaussian measure. To prove the equivalence we show the Wr,p(B,μ)-boundedness of certain smooth nonlinear truncation

We prove a comparison theorem for super- and sub-solutions with non-vanishing gradients to semilinear PDEs provided a nonlinearity f is Lp function with p > 1. The proof is based on a strong maximum principle for solutions of divergence type elliptic equations with VMO leading coefficients and with lower order coefficients from a Kato class. An application to estimation of periodic water waves profiles

We consider a modified Euler equation on \(\mathbb {R}^{2}\). We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are quasi-invariant for the Euler flow. Almost everywhere with respect to such measures (and, in particular, for less regular initial conditions), the flow is shown to be

This paper is concerned with higher Hölder regularity for viscosity solutions to non-translation invariant second order integro-PDEs, compared to Mou (2018). We first obtain C1,α regularity estimates for fully nonlinear integro-PDEs. We then prove the Schauder estimates for solutions if the equation is convex.

In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable

In the present paper, we consider long time behaviors of the volume of the Wiener sausage on Dirichlet spaces. We focus on the volume of the Wiener sausage for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure. We obtain the growth rate of the expectations and almost sure behaviors of the volumes of the Wiener sausages on metric measure Dirichlet

We study strongly harmonic functions in Carnot–Carathéodory groups defined via the mean value property with respect to the Lebesgue measure. For such functions we show their Sobolev regularity and smoothness. Moreover, we prove that strongly harmonic functions satisfy the sub-Laplace equation for the appropriate gauge norm and that the inclusion is sharp. We observe that appropriate spherical harmonic

The irreducibility, moderate deviation principle and ψ-uniformly exponential ergodicity with ψ(x) := 1 + ∥x∥0 are proved for stochastic Burgers equation driven by the α-stable processes for α ∈ (1, 2), where the first two are new for the present model, and the last strengthens the exponential ergodicity under total variational norm derived in Dong et al. (J. Stat. Phys. 154:929–949, 2014).

In this paper, we consider the gradient estimates for a postive solution of the nonlinear parabolic equation ∂tu = Δtu + hup on a Riemannian manifold whose metrics evolve under the geometric flow ∂tg(t) = − 2Sg(t). To obtain these estimate, we introduce a quantity \(\underline {\boldsymbol {S}}\) along the flow which measures whether the tensor Sij satisfies the second contracted Bianchi identity.

We study the mean values sets of the second order divergence form elliptic operator with principal coefficients defined as $$a^{ij}_{k}(x):= \left\{\begin{array}{llll} \alpha_{k} \delta^{ij}(x) &x_{n}>0 \\ \beta_{k} \delta^{ij}(x) &x_{n}<0. \end{array}\right. $$ In particular, we will show that the mean value sets associated to such an operator need not be convex as αk and βk converge to 1. This example

This paper studies regularity estimates in Sobolev spaces for weak solutions of a class of degenerate and singular quasi-linear elliptic problems of the form div[A(x,u,∇u)] = div[F] with non-homogeneous Dirichlet boundary conditions over bounded non-smooth domains. The coefficients A could be be singular, degenerate or both in x in the sense that they behave like some weight function μ, which is in

We prove sharp version of Riesz-Fejér inequality for functions in harmonic Hardy space \(h^{p}(\mathbb {D})\) on the unit disk \(\mathbb {D}\), for p > 1, thus extending the result from Kayumov et al. (Potential Anal. 52, 105–113, 2020) and resolving the posed conjecture.

This paper explores the limiting weak-type behaviors of certain classical operators in harmonic analysis including maximal operators, singular and fractional integral operators and maximal truncated singular integrals as well as the general convolution operators with weak-type Young’s inequalities. Some optimal limiting weak-type behaviors are given, which essentially improve and extend the previous

The aim of this paper is twofold. Firstly, we derive upper and lower non-Gaussian bounds for the densities of the marginal laws of the solutions to backward stochastic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilizing a relationship between fractional BSDEs and quasilinear partial differential equations of mixed type, together with the profound

In this paper, we prove that any W2,1 strong solution to second-order non-divergence form elliptic equations is locally \(W^{2,\infty }\) and piecewise C2 when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be in C1,Dini. We also derive global weak-type (1,1) estimates

We construct a decomposition of the identity operator on a Riemannian manifold M as a sum of smooth orthogonal projections subordinate to an open cover of M. This extends a decomposition on the real line by smooth orthogonal projection due to Coifman and Meyer (C. R. Acad. Sci. Paris, Sér. I Math., 312(3), 259–261 1991) and Auscher, Weiss, Wickerhauser (1992), and a similar decomposition when M is

In the existing language for tensor calculus on RCD spaces, tensor fields are only defined \(\mathfrak {m}\)-a.e.. In this paper we introduce the concept of tensor field defined ‘2-capacity-a.e.’ and discuss in which sense Sobolev vector fields have a 2-capacity-a.e. uniquely defined quasi-continuous representative.

In this note we will show a Calderón–Zygmund decomposition associated with a function \(f\in L^{1}(\mathbb {T}^{\omega })\). The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.

We consider the problem of geometric optimisation of the lowest eigenvalue of the Laplacian in the exterior of a compact set in any dimension, subject to attractive Robin boundary conditions. As an improvement upon our previous work (Krejčiřík and Lotoreichik J. Convex Anal. 25, 319–337, 2018), we show that under either a constraint of fixed perimeter or area, the maximiser within the class of exteriors

For spectrally negative Lévy processes, adapting an approach from Li and Palmowski (Stoch. Process. Appl. 128(10), 3273–3299 2018) we identify joint Laplace transforms involving local times evaluated at either the first passage times, or independent exponential times, or inverse local times. The Laplace transforms are expressed in terms of the associated scale functions. Connections are made with the

We study nonlinear systems of the form \(-{\Delta }_{p}u=v^{q_{1}}+\mu , -{\Delta }_{p}v=u^{q_{2}}+\eta \) and \(F_{k}[-u]=v^{s_{1}}+\mu , F_{k}[-v]=u^{s_{2}}+\eta \) in a bounded domain Ω or in \(\mathbb {R}^{N}\) where μ and η are nonnegative Radon measures, Δp and Fk are respectively the p-Laplacian and the k-Hessian operators and q1, q2, s1 and s2 positive numbers. We give necessary and sufficient

In this note we extend the classical relation between the equilibrium configurations of unit movable point charges in a plane electrostatic field created by these charges together with some fixed point charges and the polynomial solutions of a corresponding Lamé differential equation. Namely, we find similar relation between the equilibrium configurations of unit movable charges subject to a certain

We consider a subordinate Brownian motion X with Gaussian components when the scaling order of purely discontinuous part is between 0 and 2 including 2. In this paper we establish sharp two-sided bounds for transition density of X in \({\mathbb {R}}^{d}\) and C1,1-open sets. As a corollary, we obtain a sharp Green function estimates.

We provide sharp two-sided estimates on the Dirichlet heat kernel k1(t, x, y) for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel k1(t, x, y) in terms of its global two-sided sharp estimates. Such precise

We deal with the following fractional Schrödinger-Poisson equation with magnetic field $$\varepsilon^{2s}(-{\Delta})_{A/\varepsilon}^{s}u+V(x)u+\varepsilon^{-2t}(|x|^{2t-3}*|u|^{2})u=f(|u|^{2})u+|u|^{{2}_{s}^{*}-2}u \quad \text{ in } \mathbb{R}^{3}, $$ where ε > 0 is a small parameter, \(s\in (\frac {3}{4}, 1)\), t ∈ (0, 1), \({2}_{s}^{*}=\frac {6}{3-2s}\) is the fractional critical exponent, \((-{\Delta