Norm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups

Let $E$ be a real Banach space. We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$Lf(x) = \frac12 {\rm Tr} Q D^2 f(x) + .$$ Here $Q$ is a positive symmetric operator from $E^*$ to $E$ and $A$ is the generator of a $C_0$-semigroup $S(t)$ on $E$. Under the assumption that $P$ admits an invariant measure $\mu$ we prove that if $S$ is eventually compact and the spectrum of its generator is nonempty, then \n P(t)-P(s)\n_{L^1(E,\mu)} = 2 for all $t,s\ge 0$ with $t\not=s$. This result is new even when $E = \R^n$. We also study the behaviour of $P$ in the space $BUC(E)$. We show that if $A\not=0$ there exists $t_0>0$ such that \n P(t)-P(s)\n_{BUC(E)} = 2 for all $0\le t,s\le t_0$ with $t\not=s$. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either \n P(t)- P(s)\n_{BUC(E)} = 2 for all $t,s\ge 0$, \ $t\not=s$, or $S$ is the direct sum of a nilpotent semigroup and a finite-dimensional periodic semigroup. Finally we investigate the spectrum of $L$ in the spaces $L^1(E,\mu)$ and $BUC(E)$.Comment: 14 pages; to appear in J. Evolution Equation

Controllability and Qualitative properties of the solutions to SPDEs driven by boundary L\'evy noise

Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking values in an Hilbert space \HH, where $L$ is a \RR valued L\'evy process, $A:H\to H$ an infinitesimal generator of a strongly continuous semigroup, \sigma:H\to \RR bounded from below and Lipschitz continuous, and B:\RR\to H a possible unbounded operator. A typical example of such an equation is a stochastic Partial differential equation with boundary L\'evy noise. Let \CP=(\CP_t)_{t\ge 0} %{\CP_t:0\le t<\infty}$the corresponding Markovian semigroup. We show that, if the system (2) du(t) = A u(t)\: dt + B v(t),\quad t>0 u(0) = x is approximate controllable in time$T>0$, then under some additional conditions on$B$and$A$, for any$x\in H$the probability measure$\CP_T^\star \delta_x$is positive on open sets of$H$. Secondly, as an application, we investigate under which condition on$\HH$and on the L\'evy process$L$and on the operator$A$and$B$ the solution of Equation [1] is asymptotically strong Feller, respective, has a unique invariant measure. We apply these results to the damped wave equation driven by L\'evy boundary noise

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup T on a Banach space X we define its adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology (X^o,X). An application is the following: For K a Polish space we consider operator semigroups on the space C(K) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(K) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(K) are precisely those that are adjoints of a bi-continuous semigroups on C(K). We also prove that the class of bi-continuous semigroups on C(K) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if K is not Polish space this is not the case

DEDALO: PHASE II STUDY OF DARATUMUMAB PLUS POMALIDOMIDE AND DEXAMETHASONE (DPD) IN PATIENTS WITH RELAPSED/REFRACTORY MULTIPLE MYELOMA AND 17P DELETION

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Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $y_0 \in H$ we associate the minimal "energy" needed to transfer $y_0$ to $0$ in a time $T \ge T_0$ ("energy" of a control being the square of its $L^2$ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $0$ for $T\to+\infty$. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality.Comment: In this version we have also added a section on examples and applications of our main results. This version is similar to the one which will be published on "SIAM Journal on Control and Optimization" (SIAM

Exponential Ergodicity of stochastic Burgers equations driven by α\alpha-stable processes

In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by $\alpha/2$-subordinated cylindrical Brownian motions with $\alpha\in(1,2)$. To prove the results, we truncate the nonlinearity and use the derivative formula for SDEs driven by $\alpha$-stable noises established in Zhang (arXiv:1204.2630v2).Comment: 17p

Anti-prion drug mPPIg5 inhibits PrP(C) conversion to PrP(Sc).

Prion diseases, also known as transmissible spongiform encephalopathies, are a group of fatal neurodegenerative diseases that include scrapie in sheep, bovine spongiform encephalopathy (BSE) in cattle and Creutzfeldt-Jakob disease (CJD) in humans. The 'protein only hypothesis' advocates that PrP(Sc), an abnormal isoform of the cellular protein PrP(C), is the main and possibly sole component of prion infectious agents. Currently, no effective therapy exists for these diseases at the symptomatic phase for either humans or animals, though a number of compounds have demonstrated the ability to eliminate PrPSc in cell culture models. Of particular interest are synthetic polymers known as dendrimers which possess the unique ability to eliminate PrP(Sc) in both an intracellular and in vitro setting. The efficacy and mode of action of the novel anti-prion dendrimer mPPIg5 was investigated through the creation of a number of innovative bio-assays based upon the scrapie cell assay. These assays were used to demonstrate that mPPIg5 is a highly effective anti-prion drug which acts, at least in part, through the inhibition of PrP(C) to PrP(Sc) conversion. Understanding how a drug works is a vital component in maximising its performance. By establishing the efficacy and method of action of mPPIg5, this study will help determine which drugs are most likely to enhance this effect and also aid the design of dendrimers with anti-prion capabilities for the future