The problem of detecting a deformation in a symmetric Gaussian random tensor is concerned about whether there exists a statistical hypothesis test that can reliably distinguish a low-rank random spike from the noise. Recently Lesieur et al. (2017) proved that there exists a critical threshold so that when the signal-to-noise ratio exceeds this critical value, one can distinguish the spiked and unspiked tensors and weakly recover the spike via the minimal mean-square-error method. In this talk, we will show that in the case of the rank-one spike with Rademacher prior, this critical value strictly separates the distinguishability and indistinguishability of the two tensors under the total variation distance. Our approach is based on a subtle analysis of the high temperature behavior of the pure p-spin model, arising initially from the field of spin glasses. In particular, the signal-to-noise criticality is identified as the critical temperature, distinguishing the high and low temperature behavior, of the pure p-spin model.

The random order of service (ROS) is a natural scheduling policy for systems where no ordering of customers can or should be established. Queueing models under ROS have been used to study molecular interactions of intracellular components in biology. However, these models often assume exponential distributions for processing and patience times, which is not realistic especially when operations such as binding, folding, transcription and translation are involved. We study a multi-class queueing model operating under ROS with reneging and generally distributed processing and patience times. We use measure-valued processes to describe the dynamic evolution of the network, and establish a fluid approximation for this representation. Obtaining a fluid limit for this network requires a multi-scale analysis of its fast and slow components, and to establish an averaging principle in the context of measure-valued process. In addition, under slightly more restrictive assumptions on the patience time distribution, we introduce a reduced, function-valued fluid model that is described by a system of non-linear Partial Differential Equations (PDEs). These PDEs, however, are non-standard and the analysis of their existence, uniqueness and stability properties requires new techniques.

In this talk, we present some results on the existence of weak-solutions of the Navier-Stokes equation perturbed by transport-type rough path noise with periodic boundary conditions in dimensions two and three. The noise is smooth and divergence free in space, but rough in time. We will also discuss the problem of uniqueness in two dimensions. The proof of these results makes use of the theory of unbounded rough drivers developed by M. Gubinelli et al.

As a consequence of our results, we obtain a pathwise interpretation of the stochastic Navier-Stokes equation with Brownian and fractional Brownian transport-type noise. A Wong-Zakai theorem and support theorem follow as an immediate corollary. This is joint work with Martina Hofmanov\'a and Torstein Nilssen.

 In this talk, we first discuss existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representation. We then talk about sharp two-  sided estimates for fundamental solutions of general time fractional equations in metric measure spaces. This is a joint work with  Zhen-Qing Chen(University of Washington, USA), Takashi Kumagai (RIMS, Kyoto University, Japan) and Jian Wang (Fujian Normal University, China).