Hitting Probabilities for Nonlinear Systems of Stochastic Waves
Hitting Probabilities for Nonlinear Systems of Stochastic Waves

Author: Robert C. Dalang

Publisher: American Mathematical Soc.

Published: 2015-08-21

Total Pages: 88

ISBN-13: 1470414236

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The authors consider a d-dimensional random field u={u(t,x)} that solves a non-linear system of stochastic wave equations in spatial dimensions k∈{1,2,3}, driven by a spatially homogeneous Gaussian noise that is white in time. They mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent β. Using Malliavin calculus, they establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of Rd, in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that appears in the Hausdorff measure is close to optimal, and shows that when d(2−β)>2(k+1), points are polar for u. Conversely, in low dimensions d, points are not polar. There is, however, an interval in which the question of polarity of points remains open.

Stochastic Partial Differential Equations and Related Fields
Stochastic Partial Differential Equations and Related Fields

Author: Andreas Eberle

Publisher: Springer

Published: 2018-07-03

Total Pages: 565

ISBN-13: 3319749293

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This Festschrift contains five research surveys and thirty-four shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10–14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Röckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker–Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art.

Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations
Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equations

Author: Volker Bach

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 134

ISBN-13: 1470417057

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The authors study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. They specify assumptions that ensure the global existence of its solutions and allow them to derive its asymptotics at temporal infinity. They demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.

Stability of KAM Tori for Nonlinear Schrödinger Equation
Stability of KAM Tori for Nonlinear Schrödinger Equation

Author: Hongzi Cong

Publisher: American Mathematical Soc.

Published: 2016-01-25

Total Pages: 100

ISBN-13: 1470416573

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The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation subject to Dirichlet boundary conditions , where is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier , any solution with the initial datum in the -neighborhood of a KAM torus still stays in the -neighborhood of the KAM torus for a polynomial long time such as for any given with , where is a constant depending on and as .

On the Theory of Weak Turbulence for the Nonlinear Schrodinger Equation
On the Theory of Weak Turbulence for the Nonlinear Schrodinger Equation

Author: M. Escobedo

Publisher: American Mathematical Soc.

Published: 2015-10-27

Total Pages: 120

ISBN-13: 1470414341

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The authors study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schrödinger equation. They define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. The authors also prove the existence of a family of solutions that exhibit pulsating behavior.

Classification of $E_0$-Semigroups by Product Systems
Classification of $E_0$-Semigroups by Product Systems

Author: Michael Skeide

Publisher: American Mathematical Soc.

Published: 2016-03-10

Total Pages: 138

ISBN-13: 1470417383

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In these notes the author presents a complete theory of classification of E0-semigroups by product systems of correspondences. As an application of his theory, he answers the fundamental question if a Markov semigroup admits a dilation by a cocycle perturbations of noise: It does if and only if it is spatial.

Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities
Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities

Author: Bart Bories

Publisher: American Mathematical Soc.

Published: 2016-06-21

Total Pages: 146

ISBN-13: 147041841X

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In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate surface singularity. The authors start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f∈Z[x,y,z] satisfying f(0,0,0)=0 and non-degenerate with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of f−1(0)⊂C3 close to the origin. Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f, arising from so-called B1-facets of the Newton polyhedron of f, are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerate surface singularity.

Reduced Fusion Systems over 2-Groups of Sectional Rank at Most 4
Reduced Fusion Systems over 2-Groups of Sectional Rank at Most 4

Author: Bob Oliver

Publisher: American Mathematical Soc.

Published: 2016-01-25

Total Pages: 112

ISBN-13: 1470415488

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The author classifies all reduced, indecomposable fusion systems over finite -groups of sectional rank at most . The resulting list is very similar to that by Gorenstein and Harada of all simple groups of sectional -rank at most . But this method of proof is very different from theirs, and is based on an analysis of the essential subgroups which can occur in the fusion systems.

The Local Structure Theorem for Finite Groups With a Large $p$-Subgroup
The Local Structure Theorem for Finite Groups With a Large $p$-Subgroup

Author: U. Meierfrankenfeld

Publisher: American Mathematical Soc.

Published: 2016-06-21

Total Pages: 356

ISBN-13: 1470418770

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Let p be a prime, G a finite Kp-group S a Sylow p-subgroup of G and Q a large subgroup of G in S (i.e., CG(Q)≤Q and NG(U)≤NG(Q) for 1≠U≤CG(Q)). Let L be any subgroup of G with S≤L, Op(L)≠1 and Q⋬L. In this paper the authors determine the action of L on the largest elementary abelian normal p-reduced p-subgroup YL of L.