Stochastic Processes and Their Applications : In Mathematics and Physics

Et moi, ... , si j'avait su comment en revenIT, je n'y serais point allt\. ' Jules Verne The series is divergent; therefore we may be able to do something with it. O. Heaviside One service mathematics has rendered the human race. It has put common sense back where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded nonsense'. ## Eric T. Bell Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered computer science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series. This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experimental mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics." By and large, all this still applies today. It is still true that at first sight mathematics seems rather fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available. If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability, and so on. In addition, the applied scientist needs to cope increasingly with the nonlinear world and the The shortest path between two truths in the real domain passes through the complex domain. J. Hadamard La physique ne nons donne pas seulement I' occasion de resoudre des probIemes ... elle nons fait pressentir \a solution. H. Poincare Bussum, July 1990 Never lend books, for no one ever returns them; the only books I have in my library are books that other folk have lent me. Anatole France The function of an expert is not to be more right than other people, but to be wrong for more sophisticated reasons. Front Matter....Pages i-xiii Stochastic Stability for Vector Fields with a Manifold of Singular Points, and an Application to Lattice Gauge Theory....Pages 1-20 Ricci Curvature and Dimension for Diffusion Semigroups....Pages 21-31 The Zitterbewegung of a Dirac Electron in a Central Field....Pages 33-51 Maximum Entropy Principles for Markov Processes....Pages 53-69 An Optimal Carleman-Type Inequality for the Dirac Operator....Pages 71-94 Toeplitz Operators — An Asymptotic Quantization of Symplectic Cones....Pages 95-106 Perturbation Theory for Random Disordered Systems....Pages 107-122 On rigorous Hydrodynamics, Self-diffusion and the Green-Kubo formulae....Pages 123-147 A Stochastic Model for Plasma Dynamics....Pages 149-152 Macroscopic Potentials of Dissipative Dynamical Systems....Pages 153-175 Random-Path Intersections in Geometry, Probability and Physics....Pages 177-186 Noncommutative Version of The Central Limit Theorem and of Cramér’s Theorem....Pages 187-202 Distributions, Sobolev Spaces on Gaussian Vector Spaces and Ito’s Calculus....Pages 203-225 On Problems in Stochastic Differential Equations Connected with Some Particular Type of Interacting Particles....Pages 227-244 Asymptotic behaviors of moments for one-dimensional generalized diffusion processes....Pages 245-273 Langevin Equation and Fluctuation-Dissipation Theorem....Pages 275-299 The Dirichlet Problem for Quasi-Linear Partial Differential Operators with Boundary Data Given by a Distribution....Pages 301-326 Stationary Stochastic Perturbation of a Linear Delay Equation....Pages 327-332 Random Lattice Models....Pages 333-356 Interactions Galileennes Aimant-Charge....Pages 357-373 The Polaron Functional Integral....Pages 375-387 The Bosonic String....Pages 389-402 'et Moi ..., Si J'avait Su Comment En Revenit, One Service Mathematics Has Rendered The Je N'y Serais Point Allt\.' Human Race. It Has Put Common Sense Back Where It Belongs, On The Topmost Shelf Next Jules Verne To The Dusty Canister Labelled 'discarded Non- The Series Is Divergent; Therefore We May Be Sense'. Able To Do Something With It. Eric T. Bell O. Heaviside Mathematics Is A Tool For Thought. A Highly Necessary Tool In A World Where Both Feedback And Non Linearities Abound. Similarly, All Kinds Of Parts Of Mathematics Serve As Tools For Other Parts And For Other Sciences. Applying A Simple Rewriting Rule To The Quote On The Right Above One Finds Such Statements As: 'one Service Topology Has Rendered Mathematical Physics .. :; 'one Service Logic Has Rendered Com Puter Science .. :; 'one Service Category Theory Has Rendered Mathematics .. :. All Arguably True. And All Statements Obtainable This Way Form Part Of The Raison D'etre Of This Series.