A characterisation of the quantum stochastic bounded generators of irreversible quantum state evolutions is given. This suggests the general form of quantum stochastic evolution equation with respect to the Poisson (jumps), Wiener (diffusion) or general Quantum Noise. The corresponding irreversible Heisenberg evolution in terms of stochastic completely positive (CP) cocycles is also characterized and the general form of the stochastic completely dissipative (CD) operator equation is discovered.
This paper presents a survey of maximal inequalities for stochastic convolutions in $2$-smooth Banach spaces and their applications to stochastic evolution equations.
We prove that the stochastic Burgers equation on $\mathbf{R}^{d}$, $d<4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the KPZ equation on $\mathbf{R}^{d}$ with stationary gradients. The proof works by proving tightness of the time-averaged laws of the solutions in an appropriate weighted space.
We find a wide class of Levy-Loewner evolutions for which the value of integral means beta-spectrum $β(q)$ at $q=2$ is the maximal real eigenvalue of a three-diagonal matrix. The second moments of derivatives of corresponding conformal mappings are expressed through solutions of matrix Fuchsian systems with three singular points.
In the article "Stochastic evolution equations for large portfolios of Stochastic Volatility models" (Arxiv:1701.05640) there is a mistake in the proof of Theorem 3.1. In this erratum we establish a weaker version of this Theorem and then we redevelop the regularity theory for our problem accordingly. This means that most of our regularity results are replaced by slightly weaker ones. We also clarify a point in the proof of a correct result.
Based on a recent result on characterising the path-independence of the Girsanov transformation for non-Lipschnitz stochastic differential equations (SDEs) with jumps on $R^d$, in this paper, we extend our consideration of characterising the path-indpendent property from finite-dimensional SDEs with jumps to stochastic evolution equations with jumps in Hilbert spaces. This is done via Galerkin type finite-dimensional approximations of the infinite-dimensional stochastic evolution equations with jumps in the manner that one could then link the characterisation of the path-independence for finit…
University of Copenhagen（哥本哈根大学）；Univ. Lille（里尔大学）；CNRS（法国国家科学研究中心）；Centrale Lille（里尔中央理工学院）
Modelling randomness in shape data, for example, the evolution of shapes of organisms in biology, requires stochastic models of shapes. This paper presents a new stochastic shape model based on a description of shapes as functions in a Sobolev space. Using an explicit orthonormal basis as a reference frame for the noise, the model is independent of the parameterisation of the mesh. We define the stochastic model, explore its properties, and illustrate examples of stochastic shape evolutions using the resulting numerical framework.
The main goal of the work is to study the stochastic averaging principle for two time-scales stochastic evolution equations driven by Lévy process. The solution of reduced equation with modified coefficient is derived to approximate the slow component of original equation under suitable condition. It is shown that the slow component can strongly converge to the solution of corresponding reduced equation in L^p(p>2)-strong convergence sense.Our key and novelty is how to cope with the changes caused by Lévy process and higher order moments.
When the evolution familiy is hyperbolic and satisfies the Acquistapace-Terreni conditions, the existence and uniquenness of an almost automorphic mild solution and a weighted pseudo almost automorphic mild solution in distribution of mean-filed nonautonomous stochastic evolution equations driven by fractional Brownian motion is proved. Examples illustrating the main results are included.
We present a new proof of well-posedness of stochastic evolution equations in variational form, relying solely on a (nonlinear) infinite-dimensional approximation procedure rather than on classical finite-dimensional projection arguments of Galerkin type.
We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approac…
We provide sufficient conditions on the coefficients of a stochastic evolution equation on a Hilbert space of functions driven by a cylindrical Wiener process ensuring that its mild solution is positive if the initial datum is positive. As an application, we discuss the positivity of forward rates in the Heath-Jarrow-Morton model via Musiela's stochastic PDE.
Harnack Inequalities and Applications for Multivalued Stochastic Evolution Equations
利用耦合方法和Girsanov变换，证明了Harnack不等式[F.- Y. Wang，1997]和强Feller性质，证明了Gelfand三元集上的多值随机发展方程的转移半群的性质.研究了半群的不变测度的集中性。作为Harnack不等式的应用，给出了半群的密度、压缩性、紧性和熵-代价不等式的$L^p$ -范数的显式上界.
In this article we give sufficient and necessary conditions for the existence of a weak and mild solution to stochastic evolution equations with (general) Lévy noise taking values in the dual of a nuclear space. As part of our approach we develop a theory of stochastic integration with respect to a Lévy process taking values in the dual of a nuclear space. We also derive further properties of the solution such as the existence of a solution with square moments, the Markov property and path regularity of the solution. In the final part of the paper we give sufficient conditions for the weak con…
We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provi…
