We show that a continuous local semiflow of $C^k$-maps on a finite-dimensional $C^k$-manifold M can be embedded into a local $C^k$-flow on M under some weak (necessary) assumptions. This result is applied to an open problem in [fil/tei:01]. We prove that finite-dimensional realizations for interest rate models are highly regular objects, namely given by submanifolds M of $D(A^{\infty})$, where A is the generator of a strongly continuous semigroup.
This paper gives further regularity properties of the evolution family associated with a non-autonomous evolution equation \begin{equation*}\label{Abstract equation} \dot u(t)+A(t)u(t)=f(t),\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ arise from non-autonomous sesquilinear forms $\mathfrak a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ Results on norm continuity, compactness and results on the \textit{Gibbs} character of the evolution family are established. The abstract results are applied to the Laplacian operator with time dependent Rob…
We study continuity and Hölder continuity of $t\mapsto P_tf$, where $P_t$ is a generalized Mehler semigroup in $C_b(X)$, the space of the continuous and bounded functions from a Banach space $X$ to $R$, and $f\in C_b(X)$. The generators $L$ of such semigroups are realizations of a class of differential and pseudo-differential operators, both in finite and in infinite dimension. Examples of operators $L$ to which this theory is applicable include Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and Ornstein-Uhlenbeck operators with associated strong-Feller semigroups,…
In this paper we prove maximal $L^p$-regularity for a system of parabolic PDEs, where the elliptic operator $A$ has coefficients which depend on time in a measurable way and are continuous in the space variable. The proof is based on operator-theoretic methods and one of the main ingredients in the proof is the construction of an evolution family on weighted $L^q$-spaces.
We consider the maximal regularity problem for non-autonomous evolution equations of the form $u(t) + A(t) u(t) = f(t)$ with initial data $u(0) = u\_0$ . Each operator $A(t)$ is associated with a sesquilinear form $a(t; *, *)$ on a Hilbert space $H$ . We assume that these forms all have the same domain and satisfy some regularity assumption with respect to t (e.g., piecewise $α$-H{ö}lder continuous for some $α\textgreater{} 1/2$). We prove maximal Lp-regularity for all initial values in the real-interpolation space $(H, D(A(0)))\_{1/p,p}$ . The particular case where $p = 2$ improves previously…
We study the Cauchy problem for an abstract quasilinear stochastic parabolic evolution equation on a Banach space driven by a cylindrical Brownian motion. We prove existence and uniqueness of a local strong solution up to a maximal stopping time, that is characterised by a blow-up alternative. The key idea is an iterative application of the theory about maximal $ L^p $- regularity for semilinear stochastic evolution equations by Van Neerven, Veraar and Weis. We apply our local well-posedness result to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary …
In this paper we generalize classical results regarding minimal realizations of non-commutative (nc) rational functions using nc Fornasini-Marchesini realizations which are centred at an arbitrary matrix point. We prove the existence and uniqueness of a minimal realization for every nc rational function, centred at an arbitrary matrix point in its domain of regularity. Moreover, we show that using this realization we can evaluate the function on all of its domain (of matrices of all sizes) and also with respect to any stably finite algebra. As a corollary we obtain a new proof of the theorem b…
We prove existence of strongly continuous evolution systems in L^2 for Schroedinger-type equations with non-Lipschitz coefficients in the principal part. The underlying operator structure is motivated from models of paraxial approximations of wave propagation in geophysics. Thus, the evolution direction is a spatial coordinate (depth) with additional pseudodifferential terms in time and low regularity in the lateral variables. We formulate and analyze the Cauchy problem in distribution spaces with mixed regularity. The key point in the evolution system construction is an elliptic regularity re…
In this paper we consider $L^p$-regularity estimates for solutions to stochastic evolution equations, which is called stochastic maximal $L^p$-regularity. Our aim is to find a theory which is analogously to Dore's theory for deterministic evolution equations. He has shown that maximal $L^p$-regularity is independent of the length of the time interval, implies analyticity and exponential stability of the semigroup, is stable under perturbation and many more properties. We show that the stochastic versions of these results hold.
The main purpose of this paper is to investigate the concept of maximal $L^p$-regularity for perturbed evolution equations in Banach spaces. We mainly consider three classes of perturbations: Miyadera-Voigt perturbations, Desch-Schappacher perturbations, and more general Staffans-Weiss perturbations. We introduce conditions for which the maximal $L^p$-regularity can be preserved under these kind of perturbations. We give examples for a boundary perturbed heat equation in $L^r$-spaces and a perturbed boundary integro-differential equation. We mention that our results mainly extend those in the …
We discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier--Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-pose…
We solve the regularity problem for Milnor's infinite dimensional Lie groups in the $C^0$-topological context, and provide necessary and sufficient regularity conditions for the (standard) $C^k$-topological setting. We prove that the evolution map is $C^0$-continuous on its domain $\textit{iff}\hspace{1pt}$ the Lie group $G$ is locally $μ$-convex. We furthermore show that if the evolution map is defined on all smooth curves, then $G$ is Mackey complete. Under the assumption that $G$ is locally $μ$-convex, we show that each $C^k$-curve for $k\in \mathbb{N}_{\geq 1}\sqcup\{\mathrm{lip},\infty\}$…
Let A be a positive self-adjoint linear operator acting on a real Hilbert space H and $α$, c be positive constants. We show that all solutions of the evolution equation u + Au + cA $α$ u = 0 with u(0) $\in$ D(A 1 2), u (0) $\in$ H belong for all t > 0 to the Gevrey space G(A, $σ$) with $σ$ = min{ 1 $α$ , 1 1--$α$ }. This result is optimal in the sense that $σ$ can not be reduced in general. For the damped wave equation (SDW) $α$ corresponding to the case where A = --$Δ$ with domain D(A) = {w $\in$ H 1 0 ($Ω$), $Δ$w $\in$ L 2 ($Ω$)} with $Ω$ any open subset of R N and (u(0), u (0)) $\in$ H 1 0 …
In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an $L^p$-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete pr…
The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent Hölder estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and hi…
We prove maximal $L^p$-regularity for the stochastic evolution equation \[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial operator with a bounded $H^\infty$-calculus of angle less than $\frac12π$ on a space $L^q(\mathcal{O},μ)$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\in (2,\infty)$ and $q\in [2,\infty)$ and initial conditions $u_0$ in the real interpolation space $\XAp $ we prove existence of unique strong solution with t…
On Matrix Valued Schrödinger Operators on the Discrete Real Line: Resolvent Boundary Values, Limiting Absorption Principle, Hölder Regularity and Dispersive Estimates
Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autónoma de México (UNAM)（墨西哥国立自治大学应用数学与系统研究所）；Institute of Analysis and Algebra, Technische Universität Braunschweig（布伦瑞克工业大学分析与代数研究所）
Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.
