MATH Seminar
| Title: Comparison principles for stochastic heat equations |
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| Seminar: PDE Seminar |
| Speaker: Le Chen of Emory University |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2020-01-17 at 1:00PM |
| Venue: MSC E408 |
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| Abstract: The stochastic heat equation is a canonical model that is related to many models in mathematical physics, mathematical biology, particle systems, etc. It usually takes the following form: \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x)) \:\dot{M}(t,x), \qquad u(0,\cdot) =\mu, \qquad t>0, \: x\in R^d, \] where $\mu$ is the initial data, $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\rho$ is a Lipschitz continuous function. In this talk, we will study a particular set of properties of this equation --- the comparison principles, which include both {\it sample-path comparison} and {\it stochastic comparison principles}. These results are obtained for general initial data and under the weakest possible requirement on the correlation function --- Dalang's condition, namely, $\int_{ R^d}(1+|\xi|^2)^{-1}\hat{f}(d \xi)<\infty$, where $\hat{f}$ is the spectral measure of the noise. For the sample-path comparison, one can compare solutions pathwisely with respect to different initial conditions $\mu$, while for the stochastic comparison, one can compare certain functionals of the solutions either with respect to different diffusion coefficients $\rho$ or different correlation functions of the noise $f$. This talk is based on some joint works with Jingyu Huang and Kunwoo Kim. |
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