The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

Variational Methods for Nonlocal Fractional Problems

Introduction.- 1 A probabilistic motivation.-1.1 The random walk with arbitrarily long jumps.- 1.2 A payoff model.-2 An introduction to the fractional Laplacian.-2.1 Preliminary notions.- 2.2 Fractional Sobolev Inequality and Generalized Coarea Formula.- 2.3 Maximum Principle and Harnack Inequality.- 2.4 An s-harmonic function.- 2.5 All functions are locally s-harmonic up to a small error.- 2.6 A function with constant fractional Laplacian on the ball.- 3 Extension problems.- 3.1 Water wave model.- 3.2 Crystal dislocation.- 3.3 An approach to the extension problem via the Fourier transform.- 4 Nonlocal phase transitions.- 4.1 The fractional Allen-Cahn equation.- 4.2 A nonlocal version of a conjecture by De Giorgi.- 5 Nonlocal minimal surfaces.- 5.1 Graphs and s-minimal surfaces.- 5.2 Non-existence of singular cones in dimension 2 5.3 Boundary regularity.- 6 A nonlocal nonlinear stationary Schrodinger type equation.- 6.1 From the nonlocal Uncertainty Principle to a fractional weighted inequality.- Alternative proofs of some results.- A.1 Another proof of Theorem A.2 Another proof of Lemma 2.3.- References.

Nonlocal Diffusion and Applications

Leukoencephalopathy with brain stem and spinal cord involvement and lactate elevation (LBSL) has recently been defined based on a highly characteristic constellation of abnormalities observed by magnetic resonance imaging and spectroscopy. LBSL is an autosomal recessive disease, most often manifesting in early childhood. Affected individuals develop slowly progressive cerebellar ataxia, spasticity and dorsal column dysfunction, sometimes with a mild cognitive deficit or decline. We performed linkage mapping with microsatellite markers in LBSL families and found a candidate region on chromosome 1, which we narrowed by means of shared haplotypes. Sequencing of genes in this candidate region uncovered mutations in DARS2, which encodes mitochondrial aspartyl-tRNA synthetase, in affected individuals from all 30 families. Enzyme activities of mutant proteins were decreased. We were surprised to find that activities of mitochondrial complexes from fibroblasts and lymphoblasts derived from affected individuals were normal, as determined by different assays.

Mitochondrial aspartyl-tRNA synthetase deficiency causes leukoencephalopathy with brain stem and spinal cord involvement and lactate elevation

We prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient ${a(\cdot)}$ The model case is given by the functional 

$$ w \mapsto \int (|Dw|^p + a(x)|Dw|^q) \, {\rm d}x,$$

where q > p and ${a(\cdot) \geqq 0}$

https://cvgmt.sns.it/media/doc/paper/2503/cmarma3.pdf

Regularity for Double Phase Variational Problems

We extend the De Giorgi–Nash–Moser theory to nonlocal, possibly degenerate integro-differential operators.

/pdf/local-behavior-of-fractional-p-minimizers-nkhuk2lz01.pdf

Local behavior of fractional p-minimizers

Nonlocal Harnack inequalities

We extend the De Giorgi-Nash-Moser theory to nonlocal, possibly degenerate integro-differential operators.

Local behavior of fractional $p$-minimizers

One of the basic achievements in nonlinear potential theory is that the typical linear pointwise estimates via fundamental solutions find a precise analog in the case of nonlinear equations. We give a comprehensive account of this fact and prove new unifying families of potential estimates. We also describe new fine properties of solutions to measure data problems.

/pdf/guide-to-nonlinear-potential-estimates-5bi58qvs7l.pdf

Tuomo Kuusi

Papers

Local behavior of fractional p-minimizers

Nonlocal Harnack inequalities

Local behavior of fractional $p$-minimizers

Guide to Nonlinear Potential Estimates

Universal potential estimates