The one-dimensional contact process and the KPP equation with noise

09.12.2019, 13:10  –  Campus Golm, Haus 9, Raum 2.22
Forschungsseminar Wahrscheinlichkeitstheorie

Sandra Kliem (Frankfurt)

The one-dimensional KPP-equation driven by space-time white noise,
\[ \partial_t u = \partial_{xx} u + \theta u - u^{^2} + u^{\frac{1}{2}} dW, \qquad t>0, x \in \mathbb{R}, \theta>0, \qquad \qquad u(0,x) = u_0(x) \geq 0 \]
is a stochastic partial differential equation (SPDE) that exhibits a phase transition for initial non-negative finite-mass conditions. Solutions to this SPDE arise for instance as (weak) limits of approximate densities of occupied sites in rescaled one-dimensional long range contact processes.

If $\theta$ is below a critical value $\theta_c$, solutions with initial finite mass die out to $0$ in finite time, almost surely. Above this critical value, the probability of (global) survival is strictly positive. Let $\theta>\theta_c$, then there exist stochastic wavelike solutions which travel with non-negative linear speed. For initial conditions that are ‘’uniformly distributed in space’’, the corresponding solutions are all in the domain of attraction of a unique non-zero stationary distribution.

For the (parameter-dependent) nearest-neighbor contact process on $\mathbb{\ZZ}$, an interacting particle system, more is known. A complete convergence theorem holds, that is, a full description of the limiting law of a solution is available, starting from any initial condition. Its proof relies in essence on the progression of so-called edge processes. In these models, edge speeds characterize critical values.

In my talk, I will introduce the two models in question (nearest-neighbor contact process and KPP-equation with noise). Then I explain in how far the concepts and techniques of the first model can be used to obtain new insights into the second model. In particular, the problems one encounters when changing from the discrete to the continuous (in space) setting are highlighted and approaches to resolve them are discussed.

The room can change last minute