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This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.

Partial differential equations --- Differential equations, Partial. --- Probabilities. --- Integration, Functional. --- Functional integration --- Functional analysis --- Integrals, Generalized --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- A priori estimate. --- Absolute continuity. --- Almost surely. --- Analytic continuation. --- Axiom. --- Big O notation. --- Boundary (topology). --- Boundary value problem. --- Bounded function. --- Calculation. --- Cauchy problem. --- Central limit theorem. --- Characteristic function (probability theory). --- Chebyshev's inequality. --- Coefficient. --- Comparison theorem. --- Continuous function (set theory). --- Continuous function. --- Convergence of random variables. --- Cylinder set. --- Degeneracy (mathematics). --- Derivative. --- Differential equation. --- Differential operator. --- Diffusion equation. --- Diffusion process. --- Dimension (vector space). --- Direct method in the calculus of variations. --- Dirichlet boundary condition. --- Dirichlet problem. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Exponential function. --- Feynman–Kac formula. --- Fokker–Planck equation. --- Function space. --- Functional analysis. --- Fundamental solution. --- Gaussian measure. --- Girsanov theorem. --- Hessian matrix. --- Hölder condition. --- Independence (probability theory). --- Integral curve. --- Integral equation. --- Invariant measure. --- Iterated logarithm. --- Itô's lemma. --- Joint probability distribution. --- Laplace operator. --- Laplace's equation. --- Lebesgue measure. --- Limit (mathematics). --- Limit cycle. --- Limit point. --- Linear differential equation. --- Linear map. --- Lipschitz continuity. --- Markov chain. --- Markov process. --- Markov property. --- Maximum principle. --- Mean value theorem. --- Measure (mathematics). --- Modulus of continuity. --- Moment (mathematics). --- Monotonic function. --- Navier–Stokes equations. --- Nonlinear system. --- Ordinary differential equation. --- Parameter. --- Partial differential equation. --- Periodic function. --- Poisson kernel. --- Probabilistic method. --- Probability space. --- Probability theory. --- Probability. --- Random function. --- Regularization (mathematics). --- Schrödinger equation. --- Self-adjoint operator. --- Sign (mathematics). --- Simultaneous equations. --- Smoothness. --- State-space representation. --- Stochastic calculus. --- Stochastic differential equation. --- Stochastic. --- Support (mathematics). --- Theorem. --- Theory. --- Uniqueness theorem. --- Variable (mathematics). --- Weak convergence (Hilbert space). --- Wiener process.

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Diffusive motion--displacement due to the cumulative effect of irregular fluctuations--has been a fundamental concept in mathematics and physics since Einstein's work on Brownian motion. It is also relevant to understanding various aspects of quantum theory. This book explains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. It shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis. The book's inspiration is Princeton University mathematics professor Edward Nelson's influential work in probability, functional analysis, nonstandard analysis, stochastic mechanics, and logic. The book can be used as a tutorial or reference, or read for pleasure by anyone interested in the role of mathematics in science. Because of the application of diffusive motion to quantum theory, it will interest physicists as well as mathematicians. The introductory chapter describes the interrelationships between the various themes, many of which were first brought to light by Edward Nelson. In his writing and conversation, Nelson has always emphasized and relished the human aspect of mathematical endeavor. In his intellectual world, there is no sharp boundary between the mathematical, the cultural, and the spiritual. It is fitting that the final chapter provides a mathematical perspective on musical theory, one that reveals an unexpected connection with some of the book's main themes.

Mathematical physics. --- Diffusion. --- Quantum theory. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Gases --- Liquids --- Separation (Technology) --- Solution (Chemistry) --- Solutions, Solid --- Matter --- Packed towers --- Semiconductor doping --- Physical mathematics --- Diffusion --- Properties --- Mathematics --- Affine space. --- Algebra. --- Axiom. --- Bell's theorem. --- Brownian motion. --- Central limit theorem. --- Classical mathematics. --- Classical mechanics. --- Clifford algebra. --- Combinatorial proof. --- Commutative property. --- Constructive quantum field theory. --- Continuum hypothesis. --- David Hilbert. --- Dimension (vector space). --- Discrete mathematics. --- Distribution (mathematics). --- Eigenfunction. --- Equation. --- Euclidean space. --- Experimental mathematics. --- Fermi–Dirac statistics. --- Feynman–Kac formula. --- First-order logic. --- Fokker–Planck equation. --- Foundations of mathematics. --- Fractal dimension. --- Gaussian process. --- Girsanov theorem. --- Gödel's incompleteness theorems. --- Hilbert space. --- Hilbert's program. --- Holomorphic function. --- Infinitesimal. --- Integer. --- Internal set theory. --- Interval (mathematics). --- Limit (mathematics). --- Mathematical induction. --- Mathematical optimization. --- Mathematical proof. --- Mathematician. --- Mathematics. --- Measurable function. --- Measure (mathematics). --- Minkowski space. --- Natural number. --- Neo-Riemannian theory. --- Non-standard analysis. --- Number theory. --- Operator algebra. --- Ornstein–Uhlenbeck process. --- Orthonormal basis. --- Perturbation theory (quantum mechanics). --- Philosophy of mathematics. --- Predicate (mathematical logic). --- Probability measure. --- Probability space. --- Probability theory. --- Probability. --- Projection (linear algebra). --- Pure mathematics. --- Pythagorean theorem. --- Quantum field theory. --- Quantum fluctuation. --- Quantum gravity. --- Quantum harmonic oscillator. --- Quantum mechanics. --- Quantum system. --- Quantum teleportation. --- Random variable. --- Real number. --- Renormalization group. --- Renormalization. --- Riemann mapping theorem. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Schrödinger equation. --- Scientific notation. --- Set (mathematics). --- Sign (mathematics). --- Sobolev inequality. --- Special relativity. --- Spectral theorem. --- Spin (physics). --- Statistical mechanics. --- Stochastic calculus. --- Stochastic differential equation. --- Tensor algebra. --- Theorem. --- Theoretical physics. --- Theory. --- Turing machine. --- Variable (mathematics). --- Von Neumann algebra. --- Wiener process. --- Wightman axioms. --- Zermelo–Fraenkel set theory.