Pseudo-differential equations and stochastics over non-Archimedean fields (Q2736039)

Since their discovery in 1899 by Hensel, \(p\)-adic numbers have played an important role in number theory. The set of \(p\)-adic numbers is a special case of non-Archimedean local fields, and over the years non-Archimedean counterparts have been investigated for the main concepts of the classical calculus: Fourier analysis, functional analysis, analytic theory of differential equations, and geometry. Non-Archimedean analysis has also captured the attention of physicists who were motivated by the hypothesis about a possible \(p\)-adic structure of physical space-time at sub-Planck distances. There are other potential applications of \(p\)-adic numbers in physics including spin glasses, turbulence, and quantum computing. In order to explore such applications, non-Archimedean counterparts are still to be found for many more structures of mathematical physics such as the Laplace, Schrödinger and heat equations, basic classes of random variables, and stochastic processes. The foundational tools for the investigation of such topics are discussed in this book. NEWLINENEWLINENEWLINE There are essentially two kinds of \(p\)-adic or non-Archimedean analysis. The first kind studies functions whose arguments and values are both non-Archimedean, and the second kind considers real or complex-valued functions over non-Archimedean structures. This book deals only with the second kind. Chapter 1 contains preliminary materials on non-Archimedean analysis, which are usually given without proofs. Chapter 2 describes pseudo-differential operators, especially fractional differential operators \(D^\alpha\) with \(\alpha\) being a positive rational number. It includes the discussion of a remarkable connection between \(D^\alpha\) and zeros of the Riemann zeta function. The spectral theory of \(D^\alpha\) is discussed in Chapter 3. Relations between spectral properties of \(D^\alpha\) on an open set \(G\) of a local field and the geometry of \(G\) are also discussed in the chapter. Chapter 4 is devoted to a class of equations with properties similar to those of classical parabolic equations. A fundamental solution of the Cauchy problem is constructed and studied for such an equation. Such equations are connected with Markov processes, which are studied in Chapter 5. The discussion of non-Archimedean infinite-dimensional analysis resumes in Chapter 6. The stochastic processes are parametrized by a real positive time parameter. In the non-Archimedean setting, it is natural to consider processes with the non-Archimedean time, which is the subject of Chapter 7. This chapter provides an introduction to the theory of non-Archimedean Brownian motion with both the time parameter and values from a local field and to the theory of complex-valued Wiener processes with \(p\)-adic time.