A Stochastic Wave Equation in Two Space Dimension
Author: Annie Millet
Publisher:
Published: 1997
Total Pages: 54
ISBN-13:
DOWNLOAD EBOOKPosts
Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three
Author: Robert C. Dalang
Publisher: American Mathematical Soc.
Published: 2009-04-10
Total Pages: 83
ISBN-13: 0821842889
DOWNLOAD EBOOKThe authors study the sample path regularity of the solution of a stochastic wave equation in spatial dimension $d=3$. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. The authors prove that at any fixed time, a.s., the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any fixed $x\in\mathbb{R}^3$, the sample paths in time are Holder continuous functions. Further, the authors obtain joint Holder continuity in the time and space variables. Their results rely on a detailed analysis of properties of the stochastic integral used in the rigourous formulation of the s.p.d.e., as introduced by Dalang and Mueller (2003). Sharp results on one- and two-dimensional space and time increments of generalized Riesz potentials are a crucial ingredient in the analysis of the problem. For spatial covariances given by Riesz kernels, the authors show that the Holder exponents that they obtain are optimal.
Stochastic Partial Differential Equations With Additive Gaussian Noise - Analysis And Inference
Author: Ciprian A Tudor
Publisher: World Scientific
Published: 2022-10-11
Total Pages: 205
ISBN-13: 9811264473
DOWNLOAD EBOOKThe stochastic partial differential equations (SPDEs) arise in many applications of the probability theory. This monograph will focus on two particular (and probably the most known) equations: the stochastic heat equation and the stochastic wave equation.The focus is on the relationship between the solutions to the SPDEs and the fractional Brownian motion (and related processes). An important point of the analysis is the study of the asymptotic behavior of the p-variations of the solutions to the heat or wave equations driven by space-time Gaussian noise or by a Gaussian noise with a non-trivial correlation in space.The book is addressed to public with a reasonable background in probability theory. The idea is to keep it self-contained and avoid using of complex techniques. We also chose to insist on the basic properties of the random noise and to detail the construction of the Wiener integration with respect to them. The intention is to present the proofs complete and detailed.
Stochastic Wave Equations with Cubic Nonlinearities in Two Dimensions
Author: Haziem Mohammad Hazaimeh
Publisher:
Published: 2012
Total Pages: 288
ISBN-13:
DOWNLOAD EBOOKThe main focus of my dissertation is the qualitative and quantative behavior of stochastic Wave equations with cubic nonlinearities in two dimensions. The author evaluated the stochastic nonlinear wave equation in terms of its Fourier coecients. The author proved that the strong solution of that equation exists and is unique on an appropriate Hilbert space. Also, the author studied the stability of N -dimensional truncations and give conclusions in three cases: stability in probability, estimates of [Special characters omitted.] -growth, and almost sure exponential stability. The main tool is the study of related Lyapunov-type functionals which admits to control the total energy of randomly vibrating membranes. Finally, the author studied numerical methods for the Fourier coecients. The author focussed on the linear-implicit Euler method and the linear-implicit mid-point method. Their schemes have explicit representations. Eventually, the author investigated their mean consistency and mean square consistency.
A Minicourse on Stochastic Partial Differential Equations
Author: Robert C. Dalang
Publisher: Springer Science & Business Media
Published: 2009
Total Pages: 230
ISBN-13: 3540859934
DOWNLOAD EBOOKThis title contains lectures that offer an introduction to modern topics in stochastic partial differential equations and bring together experts whose research is centered on the interface between Gaussian analysis, stochastic analysis, and stochastic PDEs.
Stochastic Partial Differential Equations, Second Edition
Author: Pao-Liu Chow
Publisher: CRC Press
Published: 2014-12-10
Total Pages: 336
ISBN-13: 1466579552
DOWNLOAD EBOOKExplore Theory and Techniques to Solve Physical, Biological, and Financial Problems Since the first edition was published, there has been a surge of interest in stochastic partial differential equations (PDEs) driven by the Lévy type of noise. Stochastic Partial Differential Equations, Second Edition incorporates these recent developments and improves the presentation of material. New to the Second Edition Two sections on the Lévy type of stochastic integrals and the related stochastic differential equations in finite dimensions Discussions of Poisson random fields and related stochastic integrals, the solution of a stochastic heat equation with Poisson noise, and mild solutions to linear and nonlinear parabolic equations with Poisson noises Two sections on linear and semilinear wave equations driven by the Poisson type of noises Treatment of the Poisson stochastic integral in a Hilbert space and mild solutions of stochastic evolutions with Poisson noises Revised proofs and new theorems, such as explosive solutions of stochastic reaction diffusion equations Additional applications of stochastic PDEs to population biology and finance Updated section on parabolic equations and related elliptic problems in Gauss–Sobolev spaces The book covers basic theory as well as computational and analytical techniques to solve physical, biological, and financial problems. It first presents classical concrete problems before proceeding to a unified theory of stochastic evolution equations and describing applications, such as turbulence in fluid dynamics, a spatial population growth model in a random environment, and a stochastic model in bond market theory. The author also explores the connection of stochastic PDEs to infinite-dimensional stochastic analysis.
Stochastic Processes, Physics and Geometry: New Interplays. I
Author: Sergio Albeverio
Publisher: American Mathematical Soc.
Published: 2000
Total Pages: 348
ISBN-13: 9780821819593
DOWNLOAD EBOOKThis volume and "IStochastic Processes, Physics and Geometry: New Interplays II" present state-of-the-art research currently unfolding at the interface between mathematics and physics. Included are select articles from the international conference held in Leipzig (Germany) in honor of Sergio Albeverio's sixtieth birthday. The theme of the conference, "Infinite Dimensional (Stochastic) Analysis and Quantum Physics", was chosen to reflect Albeverio's wide-ranging scientific interests. The articles in these books reflect that broad range of interests and provide a detailed overview highlighting the deep interplay among stochastic processes, mathematical physics, and geometry. The contributions are written by internationally recognized experts in the fields of stochastic analysis, linear and nonlinear (deterministic and stochastic) PDEs, infinite dimensional analysis, functional analysis, commutative and noncommutative probability theory, integrable systems, quantum and statistical mechanics, geometric quantization, and neural networks. Also included are applications in biology and other areas. Most of the contributions are high-level research papers. However, there are also some overviews on topics of general interest. The articles selected for publication in these volumes were specifically chosen to introduce readers to advanced topics, to emphasize interdisciplinary connections, and to stress future research directions. Volume I contains contributions from invited speakers; Volume II contains additional contributed papers. Members of the Canadian Mathematical Society may order at the AMS member price.
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
DOWNLOAD EBOOKThe 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 Analysis and Applications 2014
Author: Dan Crisan
Publisher: Springer
Published: 2014-12-13
Total Pages: 520
ISBN-13: 3319112929
DOWNLOAD EBOOKArticles from many of the main contributors to recent progress in stochastic analysis are included in this volume, which provides a snapshot of the current state of the area and its ongoing developments. It constitutes the proceedings of the conference on "Stochastic Analysis and Applications" held at the University of Oxford and the Oxford-Man Institute during 23-27 September, 2013. The conference honored the 60th birthday of Professor Terry Lyons FLSW FRSE FRS, Wallis Professor of Mathematics, University of Oxford. Terry Lyons is one of the leaders in the field of stochastic analysis. His introduction of the notion of rough paths has revolutionized the field, both in theory and in practice. Stochastic Analysis is the branch of mathematics that deals with the analysis of dynamical systems affected by noise. It emerged as a core area of mathematics in the late 20th century and has subsequently developed into an important theory with a wide range of powerful and novel tools, and with impressive applications within and beyond mathematics. Many systems are profoundly affected by stochastic fluctuations and it is not surprising that the array of applications of Stochastic Analysis is vast and touches on many aspects of life. The present volume is intended for researchers and Ph.D. students in stochastic analysis and its applications, stochastic optimization and financial mathematics, as well as financial engineers and quantitative analysts.
Malliavin Calculus and Stochastic Analysis
Author: Frederi Viens
Publisher: Springer Science & Business Media
Published: 2013-02-15
Total Pages: 580
ISBN-13: 1461459060
DOWNLOAD EBOOKThe stochastic calculus of variations of Paul Malliavin (1925 - 2010), known today as the Malliavin Calculus, has found many applications, within and beyond the core mathematical discipline. Stochastic analysis provides a fruitful interpretation of this calculus, particularly as described by David Nualart and the scores of mathematicians he influences and with whom he collaborates. Many of these, including leading stochastic analysts and junior researchers, presented their cutting-edge research at an international conference in honor of David Nualart's career, on March 19-21, 2011, at the University of Kansas, USA. These scholars and other top-level mathematicians have kindly contributed research articles for this refereed volume.
