Title: Noise-Induced Tipping in Stochastic Piecewise-Smooth Systems
Abstract: We present a theory for computing most probable transition paths for systems of stochastic differential equations with piecewise-smooth drift and additive noise. In particular, we consider systems with a switching manifold in the drift that forms a co-dimension one hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. For systems with a smooth drift, such paths correspond to minimizers of the Freidlin-Wentzell functional and thus are solutions to a corresponding system of Euler-Lagrange equations with appropriate boundary conditions. However, for piecewise-smooth systems the Euler-Lagrange equations are only weakly defined and can contain a continuum of solutions. To obtain a rigorous theory in the piecewise smooth limit, we mollify the drift and use gamma-converge to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin–Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of our results through case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

Title: On the almost sure exponential convergence of a stochastic process in a family of stochastic differential equation multi-population HIV/AIDS epidemic models with random delays for ART treatment
Abstract: A class of stochastic nonlinear multi-population HIV/AIDS models is studied. The multi-population structure represents behavioral changes in the population, in response to the information and education campaigns (IECs) against HIV/AIDS. The epidemic dynamics is subject to Brownian motion perturbations in the random supply of official developmental assistance (ODAs), and random poverty rates overtime. The impacts of the IECs, the supply of ODAs, early treatment and poverty rates are investigated by conducting stochastic analysis of the almost sure exponential convergence of the stochastic process in the HIV/AIDS models. The behavioral change and the noise induced basic reproduction numbers are obtained.

Title: Global attractors for a wave equation subject to nonlinear boundary dissipation and nonlinear interior/boundary sources with critical exponents
Abstract: In this talk we shall consider a wave model in 3D on a bounded domain which contains nonlinear sources with critical exponent in the interior/boundary and nonlinear feedback dissipation on the bound- ary. Similar models with simpler nonlinear boundary terms have been already studied broadly whereas the generality of our model is not only the presence of nonlinear interior and boundary damping but also nonlinear boundary source. Boundary actuators are easily accessible to external manipulations- hence feasible from the engineering point of view and practically implementable. On the other hand, the underlying mathematics is challenging. Boundary actions are represented by unbounded, unclosable operators, hence not treatable by perturbation theory(even from the point of view of well-posedness the- ory.) Our main result shows that a suitably calibrated boundary damping prevents the blow up of the waves, and allows to contain wave asymptotically (in time) in a suitable attracting set which is compact.

Title: Nonlinear Schrödinger Equation with Repulsive Potentials and Rotations
Abstract: We will introduce the nonlinear Schrödinger equation (NLS) with repulsive harmonic potentials and rotations, and we will give a transform that connects this to the classical NLS in the mass-critical case. Then we will give some results on global well-posedness and blowup rates.