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We show that the Peetre K-functional between the space $$L_p$$ L p with $$0 0 < p < 1 and the corresponding smooth function space $$W_p^\psi $$ W p ψ generated by the Weyl-type differential operator $$\psi (D)$$ ψ ( D ) , where $$\psi $$ ψ is a homogeneous function of any positive order, is identically zero. The proof of the main results is based on the properties of the de la Vallée Poussin kernels and the quadrature formulas for trigonometric polynomials and entire functions of exponential type.

This paper is devoted to the study of the properties of entropy as a function of the Hurst index, which corresponds to the fractional Gaussian noise. Since the entropy of the Gaussian vector depends on the determinant of the covariance matrix, and the behavior of this determinant as a function of the Hurst index is rather difficult to study analytically at high dimensions, we also consider simple alternative entropy functionals, whose behavior, on the one hand, mimics the behavior of entropy and, on the other hand, is not difficult to study. Asymptotic behavior of the normalized entropy (so called entropy rate) is also studied for the entropy and for the alternative functionals.

We consider last passage percolation on $\mathbb Z^2$ with general weight distributions, which is expected to be a member of the Kardar-Parisi-Zhang (KPZ) universality class. In this model, an oriented path between given endpoints which maximizes the sum of the i.i.d. weight variables associated to its vertices is called a geodesic. Under natural conditions of curvature of the limiting geodesic weight profile and stretched exponential decay of both tails of the point-to-point weight, we use geometric arguments to upgrade the assumptions to prove optimal upper and lower tail behavior with the exponents of $3/2$ and $3$ for the weight of the geodesic from $(1,1)$ to $(r,r)$ for all large finite $r$. The proofs merge several ideas, including the well known super-additivity property of last passage values, concentration of measure behavior for sums of stretched exponential random variables, and geometric insights coming from the study of geodesics and more general objects called geodesic watermelons. Previously such optimal behavior was only known for exactly solvable models, with proofs relying on hard analysis of formulas from integrable probability, which are unavailable in the general setting. Our results illustrate a facet of universality in a class of KPZ stochastic growth models and provide a geometric explanation of the upper and lower tail exponents of the GUE Tracy-Widom distribution, the conjectured one point scaling limit of such models. The key arguments are based on an observation of general interest that super-additivity allows a natural iterative bootstrapping procedure to obtain improved tail estimates., 45 pages, 6 figures

We consider a sublinear fractional equation of the order in the interval (1, 2). We give conditions guaranteeing that this equation possesses asymptotically superlinear solutions. We show that all of these solutions are regularly varying and establish precise asymptotic formulae for them. Further we prove non-improvability of the conditions. In addition to the asymptotically superlinear solutions we discuss also other classes of solutions, some of them having no ODE analogy. In the very special case, when the coefficient is asymptotically equivalent to a power function and the order of the equation is 2, we get known results in their full generality. We reveal substantial differences between the integer order and non-integer order case. Among other tools, we utilize the fractional Karamata integration theorem and the fractional generalized L’Hospital rule which are proved in the paper. Several examples illustrating our results but serving also in alternative proofs are given too. We provide also numerical simulations.

We prove existence of solutions, and particularly positive solutions, of initial value problems (IVPs) for nonlinear fractional differential equations involving the Caputo differential operator of order $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) . One novelty in this paper is that it is not assumed that f is continuous but that it satisfies an $$L^{p}$$ L p -Carathéodory condition for some $$p>\frac{1}{\alpha }$$ p > 1 α (detailed definitions are given in the paper). We prove existence on an interval [0, T] in cases where T can be arbitrarily large, called global solutions. The necessary a priori bounds are found using a new version of the Bihari inequality that we prove here. We show that global solutions exist when f(t, u) grows at most linearly in u, and also in some cases when the growth is faster than linear. We give examples of the new results for some fractional differential equations with nonlinearities related to some that occur in combustion theory. We also discuss in detail the often used alternative definition of Caputo fractional derivative and we show that it has severe disadvantages which restricts its use. In particular we prove that there is a necessary condition in order that solutions of the IVP can exist with this definition, which has often been overlooked in the literature.

We study the transverse dynamics of two-dimensional traveling periodic waves for the gravity–capillary water-wave problem. The governing equations are the Euler equations for the irrotational flow of an inviscid fluid layer with free surface under the forces of gravity and surface tension. We focus on two open sets of dimensionless parameters $$(\alpha ,\beta )$$ ( α , β ) , where $$\alpha $$ α and $$\beta $$ β are the inverse square of the Froude number and the Weber number, respectively. For each arbitrary but fixed pair $$(\alpha ,\beta )$$ ( α , β ) in one of these sets, two-dimensional traveling periodic waves bifurcate from the trivial constant flow. In one open set we find a one-parameter family of periodic waves, whereas in the other open set we find two geometrically distinct one-parameter families of periodic waves. Starting from a transverse spatial dynamics formulation of the governing equations, we investigate the transverse linear instability of these periodic waves and the induced dimension-breaking bifurcation. The two results share a common analysis of the purely imaginary spectrum of the linearization at a periodic wave. We apply a simple general criterion for the transverse linear instability problem and a Lyapunov center theorem for the dimension-breaking bifurcation. For parameters $$(\alpha ,\beta )$$ ( α , β ) in the open set where there is only one family of periodic waves, we prove that these waves are linearly transversely unstable. For the other open set, we show that the waves with larger wavenumber are transversely linearly unstable. We also identify an open subset of parameters for which both families of periodic waves are transversely linearly unstable. For each of these transversely linearly unstable periodic waves, a dimension-breaking bifurcation occurs in which three-dimensional doubly periodic waves bifurcate from the two-dimensional periodic wave.

The infinitesimal generator of a one-dimensional strictly $α$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $α$ and one obtains the fractional Laplacian in the case of symmetric stable processes. Using this relationship, we compute the inverse of the infinitesimal generator on Lizorkin space, from which we can recover the potential if $α\in (0,1)$ and the recurrent potential if $α\in (1,2)$. The inverse of the infinitesimal generator is expressed in terms of a linear combination of (right and left) Riemann-Liouville fractional integrals of order $α$. One can then state a class of functions that give semimartingales when applied to strictly stable processes and state a Meyer-Itô theorem with a non-zero (occupational) local time term, providing a generalization of the Tanaka formula given by Tsukada (2019). This result is used to find a Doob-Meyer (or semimartingale) decomposition for $|X_t - x|^γ$ with $X$ a recurrent strictly stable process of index $α$ and $γ\in (α-1,α)$, generalizing the work of Engelbert and Kurenok (2019) to the asymmetric case., 26 b5 pages

Derivatives with respect to the parameters of the integral Mittag-Leffler function and the integral Wright function, recently introduced by us, are calculated. These derivatives can be expressed in the form of infinite sums of quotients of the digamma and gamma functions. In some particular cases, these infinite sums are calculated in closed-form with the help of MATHEMATICA. However, parameter differentiation reduction formulas are explicitly derived in order to check some of the results given by MATHEMATICA, as well as to provide many other new results. In addition, we present these infinite sums graphically for particular values of the parameters. Finally, new results for parameter derivatives of the Mittag-Leffler and Wright functions are reported in the Appendices.

This paper deals with the analysis of the time-fractional diffusion-wave equation as one-dimensional problem in a large plane wall, long cylinder, and sphere. The result of the analysis is the proposal of one general mathematical model that describes various geometries and different processes. Finite difference method for solving the time-fractional diffusion-wave equation using Grünwald-Letnikov definition for homogeneous or inhomogeneous material and for homogeneous or inhomogeneous boundary conditions is described. Dirichlet, Neumann and Robin boundary conditions are considered. Implementation of numerical methods for explicit, implicit, and Crank-Nicolson scheme were realised in MATLAB. Finally, illustrative examples of simulations using the developed toolbox are presented.

Using the factorization method, a method pioneered by Eskin and Mirzakhani in their groundbreaking work about measure classification over the moduli space of translation surfaces, we show that generalized $u$-Gibbs states over quantitatively non-integrable Anosov systems are absolutely continuous with respect to the whole unstable manifold., Expanded version, generalized theorem and various corrections

The concept of subordination, originally introduced in the probability and stochastic processes theories, has also appeared in analysis of evolution equations. So it is not surprising that we meet it in physics of complex systems, in particular when study equations describing diffusion and dielectric relaxation phenomena. Grace to intuitively understood decomposition of complex processes into their simpler and better known components, called parent and leading processes, subordination formalism enables us to attribute physical interpretation to integral decompositions representing plethora of solutions to anomalous diffusion and relaxation problems. Moreover, it makes investigation of properties obeyed by these solutions far easier and more effective. Using the Laplace-Fourier transform method to solve memory-dependent evolution equations we show that subordination can be naturally implemented in their solutions. The key to achieve this goal is the use of operational calculus merged with the application of the Efros theorem [1]. Adopting exclusively methods of classical mathematical analysis we are able to derive the memory-stemmed origin of subordination and build a bridge connecting functional analysis/operator calculus based methods of solving the evolution equations with well established stochastic and probabilistic approaches. With such a developed general formalism in hands we apply it to several models of anomalous diffusion and relaxation phenomena.

We study how the typical gradient and typical height of a random surface are modified by the addition of quenched disorder in the form of a random independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, in the following cases. It is shown that for real-valued disordered random surfaces of the $\nabla \phi$ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions $1\le d\le 2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $1\le d\le 4$ and localizes in dimensions $d\ge 5$. It is further shown that for the integer-valued disordered Gaussian free field: (i) The gradient of the surface delocalizes in dimensions $d=1,2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $d=1,2$. (iii) The surface localizes in dimensions $d\ge 3$ at weak disorder strength. The behavior in dimensions $d\ge 3$ at strong disorder is left open. The proofs rely on several tools: explicit identities satisfied by the expectation of the random surface, the Efron--Stein concentration inequality, a coupling argument for Langevin dynamics (originally due to Funaki and Spohn) and the Nash--Aronson estimate., Comment: Fixes a mistake in the appendix: the inequality (A.10) of arXiv v4 was incorrect, this is corrected in this version with an additional half a page. The result of Proposition 3.3 is still correct. ArXiv v4 is the version accepted for publication; 50 pages

We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size $$\rho $$ ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive $$\sqrt{n\log n}$$ n log n scaling (i) for fixed infinite horizon configurations—letting first $$n\rightarrow \infty $$ n → ∞ and then $$\rho \rightarrow 0$$ ρ → 0 —studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first $$\rho \rightarrow 0$$ ρ → 0 and then $$n\rightarrow \infty $$ n → ∞ —studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) .

We study the stochastic time-fractional stochastic heat equation $$\begin{aligned} \frac{\partial ^{\alpha }}{\partial t^{\alpha }}Y(t,x)=\lambda \varDelta Y(t,x)+\sigma W(t,x);\; (t,x)\in (0,\infty )\times \mathbb {R}^{d}, \end{aligned}$$ ∂ α ∂ t α Y ( t , x ) = λ Δ Y ( t , x ) + σ W ( t , x ) ; ( t , x ) ∈ ( 0 , ∞ ) × R d , where $$d\in \mathbb {N}=\{1,2,...\}$$ d ∈ N = { 1 , 2 , . . . } and $$\frac{\partial ^{\alpha }}{\partial t^{\alpha }}$$ ∂ α ∂ t α is the Caputo derivative of order $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , and $$\lambda >0$$ λ > 0 and $$\sigma \in \mathbb {R}$$ σ ∈ R are given constants. Here $$\varDelta $$ Δ denotes the Laplacian operator, W(t, x) is time-space white noise, defined by $$\begin{aligned} W(t,x)=\frac{\partial }{\partial t}\frac{\partial ^{d}B(t,x)}{\partial x_{1}...\partial x_{d}}, \end{aligned}$$ W ( t , x ) = ∂ ∂ t ∂ d B ( t , x ) ∂ x 1 . . . ∂ x d , $$B(t,x)=B(t,x,\omega ); t\ge 0, x \in \mathbb {R}^d, \omega \in \varOmega $$ B ( t , x ) = B ( t , x , ω ) ; t ≥ 0 , x ∈ R d , ω ∈ Ω being time-space Brownian motion with probability law $$\mathbb {P}$$ P . We consider the equation (0.1) in the sense of distribution, and we find an explicit expression for the $$\mathcal {S}'$$ S ′ -valued solution Y(t, x), where $$\mathcal {S}'$$ S ′ is the space of tempered distributions. Following the terminology of Y. Hu [11], we say that the solution is mild if $$Y(t,x) \in L^2(\mathbb {P})$$ Y ( t , x ) ∈ L 2 ( P ) for all t, x. It is well-known that in the classical case with $$\alpha = 1$$ α = 1 , the solution is mild if and only if the space dimension $$d=1$$ d = 1 . We prove that if $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) the solution is mild if $$d=1$$ d = 1 or $$d=2$$ d = 2 . If $$\alpha < 1$$ α < 1 we prove that the solution is not mild for any d.

We consider Schrödinger operators in $\ell^2(\mathbb{Z})$ whose potentials are given by the sum of an ergodic term and a random term of Anderson type. Under the assumption that the ergodic term is generated by a homeomorphism of a connected compact metric space and a continuous sampling function, we show that the almost sure spectrum arises in an explicitly described way from the unperturbed spectrum and the topological support of the single-site distribution. In particular, assuming that the latter is compact and contains at least two points, this explicit description of the almost sure spectrum shows that it will always be given by a finite union of non-degenerate compact intervals. The result can be viewed as a far reaching generalization of the well known formula for the spectrum of the classical Anderson model., 11 pages

We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given by Carnot-Carath\'eodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carath\'eodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any conjugation-invariant metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the conjugation-invariant condition is also a necessary condition in all cases where scaling limits are known to exist., Comment: 36 pages, 3 figures

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (i) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (ii) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (iii) a generalization of Gromov's non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones., 63 pages; v3: perturbative version of the Viterbo conjecture now proven for arbitrary symplectic capacities, added more properties to the normal form, added a statement on the local rigidity of Zoll contact forms

Let $X_1,X_2, \ldots $ be independent random uniform points in a bounded domain $A \subset \mathbb{R}^d$ with smooth boundary. Define the coverage threshold $R_n$ to be the smallest $r$ such that $A$ is covered by the balls of radius $r$ centred on $X_1,\ldots,X_n$. We obtain the limiting distribution of $R_n$ and also a strong law of large numbers for $R_n$ in the large-$n$ limit. For example, if $A$ has volume 1 and perimeter $|\partial A|$, if $d=3$ then $\Pr[n\pi R_n^3 - \log n - 2 \log (\log n) \leq x]$ converges to $\exp(-2^{-4}\pi^{5/3} |\partial A| e^{-2 x/3})$ and $(n \pi R_n^3)/(\log n) \to 1$ almost surely, and if $d=2$ then $\Pr[n \pi R_n^2 - \log n - \log (\log n) \leq x]$ converges to $\exp(- e^{-x}- |\partial A|\pi^{-1/2} e^{-x/2})$. We give similar results for general $d$, and also for the case where $A$ is a polytope. We also generalize to allow for multiple coverage. The analysis relies on classical results by Hall and by Janson, along with a careful treatment of boundary effects. For the strong laws of large numbers, we can relax the requirement that the underlying density on $A$ be uniform., Comment: 67 pages, 4 figures