We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

IN two previous notes1, Prof. Max Born and I have shown that one can obtain a theory of superconductivity by taking account of the fact that the interaction of the electrons with the ionic lattice is appreciable only near the boundaries of Brillouin zones, and particularly strong near the corners of these. This leads to the criterion that the metal should be superconductive if a set of corners of a Brillouin zone is lying very near the Fermi surface, considered as a sphere, which limits the region in the momentum space completely filled with electrons.

On the theory of superconductivity

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

The development of wave optics for light brought many new insights into our understanding of physics, driven by fundamental experiments like the ones by Young, Fizeau, Michelson-Morley and others. Quantum mechanics, and especially the de Broglie’s postulate relating the momentum p of a particle to the wave vector k of an matter wave: k = 2 λ = p/ℏ, suggested that wave optical experiments should be also possible with massive particles (see table 1), and over the last 40 years electron and neutron interferometers have demonstrated many fundamental aspects of quantum mechanics [1].

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Optics and interferometry with atoms and molecules

G-protein–coupled receptors (GPCRs) constitute the largest family of receptors and major pharmacological targets. Whereas many GPCRs have been shown to form di-/oligomers, the size and stability of such complexes under physiological conditions are largely unknown. Here, we used direct receptor labeling with SNAP-tags and total internal reflection fluorescence microscopy to dynamically monitor single receptors on intact cells and thus compare the spatial arrangement, mobility, and supramolecular organization of three prototypical GPCRs: the β1-adrenergic receptor (β1AR), the β2-adrenergic receptor (β2AR), and the γ-aminobutyric acid (GABAB) receptor. These GPCRs showed very different degrees of di-/oligomerization, lowest for β1ARs (monomers/dimers) and highest for GABAB receptors (prevalently dimers/tetramers of heterodimers). The size of receptor complexes increased with receptor density as a result of transient receptor–receptor interactions. Whereas β1-/β2ARs were apparently freely diffusing on the cell surface, GABAB receptors were prevalently organized into ordered arrays, via interaction with the actin cytoskeleton. Agonist stimulation did not alter receptor di-/oligomerization, but increased the mobility of GABAB receptor complexes. These data provide a spatiotemporal characterization of β1-/β2ARs and GABAB receptors at single-molecule resolution. The results suggest that GPCRs are present on the cell surface in a dynamic equilibrium, with constant formation and dissociation of new receptor complexes that can be targeted, in a ligand-regulated manner, to different cell-surface microdomains.

https://www.pnas.org/content/pnas/110/2/743.full.pdf

Single-molecule analysis of fluorescently labeled G-protein–coupled receptors reveals complexes with distinct dynamics and organization

This book fills a gap between theory-oriented investigations in PDE-constrained optimization and the practical demands made by numerical solutions of PDE optimization problems. The authors discuss computational techniques representing recent developments that result from a combination of modern techniques for the numerical solution of PDEs and for sophisticated optimization schemes. Computational Optimization of Systems Governed by Partial Differential Equations offers readers a combined treatment of PDE-constrained optimization and uncertainties and an extensive discussion of multigrid optimization. It provides a bridge between continuous optimization and PDE modeling and focuses on the numerical solution of the corresponding problems. Audience: This book is intended for graduate students working in PDE-constrained optimization and students taking a seminar on numerical PDE-constrained optimization. It is also suitable as an introduction for researchers in scientific computing with PDEs who want to work in the field of optimization and for those in optimization who want to consider methodologies from the field of numerical PDEs. It will help researchers in the natural sciences and engineering to formulate and solve optimization problems.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611972054.fm

Computational Optimization of Systems Governed by Partial Differential Equations

Research on multigrid methods for optimization problems is reviewed. Optimization problems considered include shape design, parameter optimization, and optimal control problems governed by partial differential equations of elliptic, parabolic, and hyperbolic type.

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Multigrid Methods for PDE Optimization

A Fokker-Planck control framework for multidimensional stochastic processes

An optimal control formulation for determining optical flow is presented. The new framework differs from preceding approaches in that it does not require differentiation of the data and does combine optical flow with image reconstruction. It can be considered as a control-in-the-coefficients problem with a cost functional of tracking type. A numerical algorithm that solves the optimality system consisting of hyperbolic and elliptic partial differential equations is presented. Numerical experiments demonstrate the effectiveness of the optimal control approach.

Alfio Borzì

Papers

Single-molecule analysis of fluorescently labeled G-protein–coupled receptors reveals complexes with distinct dynamics and organization

Computational Optimization of Systems Governed by Partial Differential Equations

Multigrid Methods for PDE Optimization

A Fokker-Planck control framework for multidimensional stochastic processes

Optimal Control Formulation for Determining Optical Flow