The asymptotics of stochastically perturbed reaction-diffusion equations and front propagation

We study the asymptotics of Allen-Cahn-type bistable reaction-diffusion equations which are additively perturbed by a stochastic forcing (time white noise). The conclusion is that the long time, large space behavior of the solutions is governed by an interface moving with curvature dependent normal velocity which is additively perturbed by time white noise. The result is global in time and does not require any regularity assumptions on the evolving front. The main tools are (i)~the notion of stochastic (pathwise) solution for nonlinear degenerate parabolic equations with multiplicative rough (stochastic) time dependence, which has been developed by the authors, and (ii)~the theory of generalized front propagation put forward by the second author and collaborators to establish the onset of moving fronts in the asymptotics of reaction-diffusion equations.Comment: arXiv admin note: substantial text overlap with arXiv:1809.0174

Delayed effects and critical transitions in climate models

There is a continuous demand for new and improved methods of understanding our climate system. The work in this thesis focuses on the study of delayed feedback and critical transitions. There is much room to develop upon these concepts in their application to the climate system. We explore the two concepts independently, but also note that the two are not mutually exclusive. The thesis begins with a review of delay differential equation (DDE) theory and the use of delay models in climate, followed by a review of the literature on critical transitions and examples of critical transitions in climate. We introduce various methods of deriving delay models from more complex systems. Our main results center around the Saltzman and Maasch (1988) model for the Pleistocene climate (`Carbon cycle instability as a cause of the late Pleistocene ice age oscillations: modelling the asymmetric response.' Global biogeochemical cycles, 2(2):177-185, 1988). We observe that the model contains a chain of first-order reactions. Feedback chains of this type limits to a discrete delay for long chains. We can then approximate the chain by a delay, resulting in scalar DDE for ice mass. Through bifurcation analysis under varying the delay, we discover a previously unexplored bistable region and consider solutions in this parameter region when subjected to periodic and astronomical forcing. The astronomical forcing is highly quasiperiodic, containing many overlapping frequencies from variations in the Earth's orbit. We find that under the astronomical forcing, the model exhibits a transition in time that resembles what is seen in paleoclimate records, known as the Mid-Pleistocene Transition. This transition is a distinct feature of the quasiperiodic forcing, as confi rmed by the change in sign of the leading nite-time Lyapunov exponent. Additional results involve a box model of the Atlantic meridional overturning circulation under a future climate scenario and time-dependent freshwater forcing. We find that the model exhibits multiple types of critical transitions, as well as recovery from potential critical transitions. We conclude with an outlook on how the work presented in this thesis can be utilised for further studies of the climate system and beyond.European Commissio

Forcing a countable structure to belong to the ground model

Suppose that $P$ is a forcing notion, $L$ is a language (in $V$), $\dot{\tau}$ a $P$-name such that $P\Vdash$ "$\dot{\tau}$ is a countable $L$-structure". In the product $P\times P$, there are names $\dot{\tau_{1}},\dot{\tau_{2}}$ such that for any generic filter $G=G_{1}\times G_{2}$ over $P\times P$, $\dot{\tau}_{1}[G]=\dot{\tau}[G_{1}]$ and $\dot{\tau}_{2}[G]=\dot{\tau}[G_{2}]$. Zapletal asked whether or not $P \times P \Vdash \dot{\tau}_{1}\cong\dot{\tau}_{2}$ implies that there is some $M\in V$ such that $P \Vdash \dot{\tau}\cong\check{M}$. We answer this negatively and discuss related issues

Strolling through Paradise

With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A \cap [S] = \emptyset, where [S] denotes the set of branches through S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem j^0 \neq \emptyset. E.g., for I=S and J=R this gives a Marczewski null set which is not Ramsey, extending earlier partial results by Aniszczyk, Frankiewicz, Plewik, Brown and Corazza and answering a question of the latter. In case I=M and J=L this gives a Miller null set which is not Laver null; this answers a question addressed by Spinas. We also investigate the question which pairs of the ideals considered are orthogonal and which are not. Furthermore we include Mycielski's ideal P_2 in our discussion