We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution to the Fokker–Planck equation in Omega Subset mathbb {R}^d with general—strictly positive and temporally constant—Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where Omega is an interval in mathbb {R}^1, we prove that such a solution is a gradient flow—curve of maximal slope—within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107–130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41–88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary partial Omega throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure {{overline{Omega }}}. The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when Omega is an interval in mathbb {R}^1, we find a formula for the descending slope of this geodesically nonconvex functional.

Abstract: We show that, under certain specific hypotheses, the Taylor-Wiles method can be applied to the cohomology of a Shimura variety $S$ of PEL type attached to a unitary similitude group $G$, with coefficients in the coherent sheaf attached to an automorphic vector bundle $\mathcal{F}$, when $S$ has a smooth model over a $p$-adic integer ring. As an application, we show that, when the hypotheses are satisfied, the congruence ideal attached to a coherent cohomological realization of an automorphic Galois representation is independent of the signatures of the hermitian form to which $G$ is attached. We also show that the Gorenstein hypothesis used to construct $p$-adic $L$-functions in the second author's article with Eischen, Li, and Skinner, as elements of Hida's ordinary Hecke algebra, is valid rather generally. The present paper generalizes the main results of an earlier paper by the second author, which treated the case when $S$ is compact. As in the previous article, the starting point is a theorem of Lan and Suh that proves the vanishing of torsion in the cohomology under certain conditions on the parameters of the bundle $\mathcal{F}$ and the prime $p$. Most of the additional difficulty in the non-compact case is related to showing that the contributions of boundary cohomology are all of Eisenstein type. We also need to show that the coverings giving rise to the diamond operators can be extended to \'etale coverings of appropriate toroidal compactifications.

Elliptic Harnack inequality and its applications on Finsler metric measure spaces

Abstract This article presents new gradient estimates for positive solutions to the nonlinear porous medium equation in the context of smooth metric measure spaces. The diffusion operator here is the f -Laplacian and the gradient estimates of interest are mainly of Hamilton-Souplet-Zhang types. These estimates are established using a variety of methods and techniques and several implications, most notably, to parabolic Liouville-type results and characterisation of ancient solutions are given. The problem is posed in the general framework where the metric and potential evolve with time and the proofs make use of natural lower bounds on the time derivative of the metric and the Bakry-Émery m -Ricci curvature tensors. Our results extend and improve various existing ones in the literature.

We study geodesics in the Brownian map ( S , d , ν ) (\mathcal {S},d,\nu ) , the random metric measure space which arises as the Gromov-Hausdorff scaling limit of uniformly random planar maps. Our results apply to all geodesics including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5 5 , and the maximal number of geodesics which connect any pair of points is 9 9 . For each 1 ≤ k ≤ 9 1\le k \le 9 , we obtain the Hausdorff dimension of the pairs of points connected by exactly k k geodesics. For k = 7 , 8 , 9 k=7,8,9 , such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points in S \mathcal {S} , up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting ν \nu -typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame of S \mathcal {S} , the union of all of the geodesics in S \mathcal {S} minus their endpoints, has dimension one, the dimension of a single geodesic.

This study considers general Markov chains (MCs) with discrete time in an arbitrary phase space. The transition function of the MC generates two operators: T, which acts on the space of measurable functions, and A, which acts on the space of bounded countably additive measures. The operator T*, which is adjoint to T and acts on the space of finitely additive measures, is also constructed. A number of theorems are proved for the operator T*, including the ergodic theorem. Under certain conditions it is proved that if the MC has a finite number of basic invariant finitely additive measures then all of them are countably additive and the MC is quasi-compact. We demonstrate a methodology that applies finitely additive measures for the analysis of MCs, using examples with detailed proofs of their non-simple properties. Some of these proofs in the examples are more complicated than the proofs in our theorems.

Abstract In this paper, we discuss the boundedness of generalized fractional integral operators on Musielak–Orlicz–Morrey spaces of an integral form over bounded non‐doubling metric measure spaces , where both and depend on . As an application, we give Sobolev‐type inequalities for multiphase functions where , , and are log‐Hölder continuous, for , and and are nonnegative, bounded, and Hölder continuous.

Let $B$ be a complete Boolean algebra, let $Q(B)$ be the Stone compact of $B$, let $C_\infty (Q(B))$ be the commutative unital algebra of all continuous functions $x:Q(B) \rightarrow [-\infty,+\infty],$ assuming possibly the values $\pm\infty$ on nowhere-dense subsets of $Q(B)$. We consider Maharam measure $m$ defined on $B$, which takes on value in the algebra $L^0(\Omega)$ of all real measurable functions on the measurable space $(\Omega, \Sigma, \mu)$ with a $\sigma$-finite numerical measure $\mu$. The decreasing rearrangements of functions from $C_\infty (Q(B))$, associated with such a measure $m$ and taking values in the algebra $L^0(\Omega)$ are determined. The basic properties of such rearrangements are established, which are similar to the properties of classical decreasing rearrangements of measurable functions.As an application, with the help of the property of equimeasurablity of elements from $ C_\infty (Q(B))$, associated with such a measure $m$, the notion of a symmetric Banach-Kantorovich space $(E,\|\cdot\|_{E})$ over $L^0(\Omega)$ is introduced and studied in detail. Here $E\subset C_\infty (Q(B)),$ and \ $\|\cdot\|_{E}$ -- $L^0(\Omega)$-valued norm in $E$, endowing it with the structure of the Banach-Kantorovich space. Examples of symmetric Banach-Kantorovich spaces are given, which are vector-valued analogues of classical $L^p$-spaces, $ 1\leq p \leq \infty$, associated with a numerical $\sigma$-finite measure.

Li-Yau estimates and Harnack inequalities for nonlinear slow diffusion equations on a smooth metric measure space

On mappings generating embedding operators in Sobolev classes on metric measure spaces

On vanishing results for smooth metric measure spaces with weighted curvature tensors

This paper investigates stochastic differential equations with interaction, introduced by Dorogovtsev the model of the evolution of large systems of interacting particles in random environments. The study emphasizes the difference approximation scheme for these equations, which involve approximating solutions in an infinite-dimensional, nonlinear space of measures. The key contributions include the formulation of approximation schemes for compactly supported initial measures, the derivation of Wasserstein distance-based estimates, and spatial discretization techniques.

Effect of Different Trend Functions with Stochastic Terms and Measurable Spaces for Capital Markets

Neuronal activity is organized in collective patterns that are critical for information coding, generation, and communication between neural populations. These patterns are often described in terms of synchrony, oscillations, and phase relationships. Many methods have been proposed for the quantification of these collective states of dynamic neuronal organization. However, it is difficult to determine which method is best suited for which experimental setting and research question. This choice is further complicated by the fact that most methods are sensitive to a combination of synchrony, oscillations, and other factors; in addition, some of them display systematic biases that can complicate their interpretation. To address these challenges, we adopt a highly comparative approach, whereby spike trains are represented by a diverse library of measures. This enables unsupervised or supervised analysis in the space of measures, or in that of spike trains. We compile a battery of 122 measures of synchrony, oscillations, and phase relationships, complemented with 9 measures of spiking intensity and variability. We first apply them to sets of synthetic spike trains with known statistical properties, and show that all measures are confounded by extraneous factors such as firing rate or population frequency, but to different extents. Then, we analyze spike trains recorded in different species-rat, mouse, and monkey-and brain areas-primary sensory cortices and hippocampus-and show that our highly comparative approach provides a high-dimensional quantification of collective network activity that can be leveraged for both unsupervised and supervised characterization of firing patterns. Overall, the highly comparative approach provides a detailed description of the empirical properties of multineuron spike train analysis methods, including practical guidelines for their use in experimental settings, and advances our understanding of neuronal coordination and coding.

A metric measure space equipped with a Dirichlet form is called strongly recurrent if its Hausdorff dimension is less than its walk dimension. In bounded domains of such spaces we study the parabolic Anderson models ∂ t u ( t , x ) = Δ u ( t , x ) + β u ( t , x ) W ˙ α ( t , x ) , \begin{equation*} \partial _{t} u(t,x) = \Delta u(t,x) + \beta u(t,x) \, \dot {W}_\alpha (t,x), \end{equation*} where the noise W α W_\alpha is white in time and colored in space when α > 0 \alpha >0 while for α = 0 \alpha =0 it is also white in space. Both Dirichlet and Neumann boundary conditions are considered. Besides proving existence and uniqueness in the Itô sense we also get precise L p L^p estimates for the moments and intermittency properties of the solution as a consequence. Our study reveals new exponents which are intrinsically associated to the geometry of the underlying space and the results for instance apply in metric graphs or fractals like the Sierpiński gasket for which we prove scaling invariance properties of the models.

Abstract It is a major problem in analysis on metric spaces to understand when a metric space is quasisymmetric to a space with strong analytic structure, a so‐called Loewner space. A conjecture of Kleiner, recently disproven by Anttila and the second author, proposes a combinatorial sufficient condition. The counterexamples constructed are all topologically one‐dimensional, and the sufficiency of Kleiner's condition remains open for most other examples. A separate question of Kleiner and Schioppa, apparently unrelated to the problem above, asks about the existence of ‘analytically one‐dimensional planes’: metric measure spaces quasisymmetric to the Euclidean plane but supporting a one‐dimensional analytic structure in the sense of Cheeger. In this paper, we construct an example for which the conclusion of Kleiner's conjecture is not known to hold. We show that either this conclusion fails in our example or there exists an ‘analytically one‐dimensional plane’. Thus, our construction either yields a new counterexample to Kleiner's conjecture, different in kind from those of Anttila and the second author, or a resolution to the problem of Kleiner–Schioppa.

We establish Trudinger-type inequalities for variable Riesz potentials of functions in Musielak-Orlicz-Morrey spaces of an integral form over bounded metric measure spaces, as an extension of the previous paper (Math. Scand. 128 (2022), no. 1, 78–108). As a corollary, we deal with Trudinger-type inequalities for double phase functionals with variable exponents.

Let $(\mathcal{X}, d, \mu)$ be a non-homogeneous metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, we study weighted inequalities of the Calder\'{o}n-Zygmund operator on $(\mathcal{X}, d, \mu)$. Specifically, for $1 < p < \infty$, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted $L^p$ inequality holds. We deal with this problem by developing a vector-valued theory for Calder\'{o}n-Zygmund operators on the non-homogeneous metric measure spaces which is interesting in its own right.

Abstract In the article, we study variational eigenvalues on doubling metric measure spaces. We prove the existence of minimizers of variational Neumann (p, q)-eigenvalues on metric measure spaces, and on this base, we obtain estimates of Neumann eigenvalues.

Application of Matrix Calculus and Measurable Spaces in Predicting Stock Market Price Changes