Abstract. We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems:(i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the solution tends to the fundamental solution, (iii) optimal L 2 -decay of mild solutions in all dimensions, (iv) L 2 -decay of weak solutions via energy methods.The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain the quantitative decay rates of (ii).Next we use Fourier analysis techniques to obtain the optimal decay rate for mild solutions. Here we encounter the critical dimension phenomenon where the decay rate attains the decay rate of that in a bounded domain for large enough dimensions. Consequently, the decay rate does not anymore improve when the dimension increases. The theory is markedly different from that of the standard caloric functions and this substantially complicates the analysis.Finally, we use energy estimates and a comparison principle to prove a quantitative decay rate for weak solutions defined via a variational formulation. Our main idea is to show that the L 2 -norm is actually a subsolution to a purely time-fractional problem which allows us to use the known theory to obtain the result.

We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in R d . An important special case is the timefractional diffusion equation, which has seen much interest during the last years, mostly due to its applications in the modeling of anomalous diffusion processes. We follow three different approaches and techniques to study this particular case: (A) estimates based on the fundamental solution and Young's inequality, (B) Fourier multiplier methods, and (C) the energy method. It turns out that the decay behaviour is markedly different from the heat equation case, in particular there occurs a critical dimension phenomenon. The general subdiffusion case is treated by method (B) and relies on a careful estimation of the underlying relaxation function. Several examples of kernels, including the ultraslow diffusion case, illustrate our results.

It is known that the accuracy of the maximum likelihood based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article we propose a Ridge-type Operator for the Precision matrix Estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context.

Summary.A Bayesian sequential experimental design for fatigue testing based on D-optimality and a non-linear continuous damage model was implemented. The model has two asymptotes for the number of cycles to failure: the fatigue limit and the ultimate tensile strength. With the introduction and stochastic handling of these asymptotes, the D-optimal design accounts naturally for the whole range of reasonable testing levels. Sequential design ensures that all the available data is used efficiently while choosing the next test level.

We consider a rather general class of non-local in time Fokker-Planck equations and show by means of the entropy method that as t → ∞ the solution converges in L 1 to the unique steady state. Important special cases are the time-fractional and ultraslow diffusion case. We also prove estimates for the rate of decay. In contrast to the classical (local) case, where the usual time derivative appears in the Fokker-Planck equation, the obtained decay rate depends on the entropy, which is related to the integrability of the initial datum. It seems that higher integrability of the initial datum leads to better decay rates and that the optimal decay rate is reached, as we show, when the initial datum belongs to a certain weighted L 2 space. We also show how our estimates can be adapted to the discrete-time case thereby improving known decay rates from the literature. , GZ Za 547/4-1. R d Z(t, x − y)u 0 (y) dy satisfies the (optimal) estimates ||u(t, ·)|| L 2 (R d ) t − min{ αd 4 ,α} , t > 0, d ∈ N \ {4} and ||u(t, ·)|| L 2,∞ (R d ) t −α , t > 0, d = 4,