Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time H\"older regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is \F_0-measurable and bounded, then this problem has a unique mild solution, which has trajectories in C^\l([0,T];\D((-A)^\theta) provided $\lambda\ge 0$ and $\theta\ge 0$ satisfy \l+\theta<\frac12. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations.Comment: Accepted for publication in Journal of Functional Analysi

Stochastic integration in UMD Banach spaces

In this paper we construct a theory of stochastic integration of processes with values in $\mathcal{L}(H,E)$, where $H$ is a separable Hilbert space and $E$ is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an $H$-cylindrical Brownian motion. Our approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of $\mathcal{L}(H,E)$-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the It\^{o} isometry, the Burkholder--Davis--Gundy inequalities, and the representation theorem for Brownian martingales.Comment: Published at http://dx.doi.org/10.1214/009117906000001006 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

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.Comment: Accepted for publication in Journal of Differential Equation

Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations

In this paper we study the following non-autonomous stochastic evolution equation on a UMD Banach space $E$ with type 2, {equation}\label{eq:SEab}\tag{SE} {{aligned} dU(t) & = (A(t)U(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), \quad t\in [0,T], U(0) & = u_0. {aligned}. {equation} Here $(A(t))_{t\in [0,T]}$ are unbounded operators with domains $(D(A(t)))_{t\in [0,T]}$ which may be time dependent. We assume that $(A(t))_{t\in [0,T]}$ satisfies the conditions of Acquistapace and Terreni. The functions $F$ and $B$ are nonlinear functions defined on certain interpolation spaces and $u_0\in E$ is the initial value. $W_H$ is a cylindrical Brownian motion on a separable Hilbert space $H$. Under Lipschitz and linear growth conditions we show that there exists a unique mild solution of \eqref{eq:SEab}. Under assumptions on the interpolation spaces we extend the factorization method of Da Prato, Kwapie\'n, and Zabczyk, to obtain space-time regularity results for the solution $U$ of \eqref{eq:SEab}. For Hilbert spaces $E$ we obtain a maximal regularity result. The results improve several previous results from the literature. The theory is applied to a second order stochastic partial differential equation which has been studied by Sanz-Sol\'e and Vuillermot. This leads to several improvements of their result.Comment: Accepted for publication in Journal of Evolution Equation

Maximal regularity for non-autonomous equations with measurable dependence on time

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.Comment: Application to a quasilinear equation added. Accepted for publication in Potential Analysi

Dysphagia in Intensive Care Evaluation (DICE): An International Cross-Sectional Survey.

Dysphagia occurs commonly in the intensive care unit (ICU). Despite the clinical relevance, there is little worldwide research on prevention, assessment, evaluation, and/or treatment of dysphagia for ICU patients. We aimed to gain insight into this international knowledge gap. We conducted a multi-center, international online cross-sectional survey of adult ICUs. Local survey distribution champions were recruited through professional and personal networks. The survey was administered from November 2017 to June 2019 with three emails and a final telephone reminder. Responses were received from 746 ICUs (26 countries). In patients intubated > 48 h, 17% expected a > 50% chance that dysphagia would develop. This proportion increased to 43% in patients intubated > 7 days, and to 52% in tracheotomized patients. Speech-language pathologist (SLP) consultation was available in 66% of ICUs, only 4% reported a dedicated SLP. Although 66% considered a routine post-extubation dysphagia protocol important, most (67%) did not have a protocol. Few ICUs routinely assessed for dysphagia after 48 h of intubation (30%) or tracheostomy (41%). A large proportion (46%) used water swallow screening tests to determine aspiration, few (8%) used instrumental assessments (i.e., flexible endoscopic evaluation of swallowing). Swallowing exercises were used for dysphagia management by 30% of ICUs. There seems to be limited awareness among ICU practitioners that patients are at risk of dysphagia, particularly as ventilation persists, protocols, routine assessment, and instrumental assessments are generally not used. We recommend the development of a research agenda to increase the quality of evidence and ameliorate the implementation of evidence-based dysphagia protocols by dedicated SLPs

Ito’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Abstract. Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô 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