| Description |
In the first part of this work, we will show weak convergence of probability measures. The measure corresponding to the solution of the following one-dimensional nonlinear stochastic heat equation $\frac{\partial}{\partial t} u_{t}(x) = \frac{\kappa}{2} \frac{\partial^2}{\partial x^2} u_{ t}(x) + \sigma(u_{t}(x))\eta_\alpha$ with colored noise $\eta_\alpha$ will converge to the measure corresponding to the solution of the same equation but with white noise $\eta$, as $\alpha \uparrow 1$. Function $\sigma$ is taken to be Lipschitz and the Gaussian noise $\eta_\alpha$ is assumed to be colored in space and its covariance is given by $\operatorname{E} \left[ \eta_\alpha(t,x) \eta_\alpha(s,y) \right] = \delta(t-s) f_\alpha(x-y)$ where $f_\alpha$ is the Riesz kernel $f_\alpha(x) \propto 1/\left|x\right|^\alpha$. We will work with the classical notion of weak convergence of measures, that is convergence of probability measures on a space of continuous functions with compact domain and sup-norm topology. We will also state a result about continuity of measures in $\alpha$, for $\alpha \in (0,1)$. In the second part of this work, we will show existence and blow-up of the solution to $\frac{\partial}{\partial t} u_{t}(x) = \mathcal{L} u_{ t}(x) + \sigma(u_{t}(x))\eta$ on a circle with white noise $\eta$. The operator $\mathcal{L}$ is taken to be the generator of a L\'evy process and $\sigma$ is a nonlinear function of form $\sigma(x) \propto \left|x\right|^\gamma$, for $\gamma>1$. We will develop precise condition for existence or blow-up of the solution in terms of $\gamma$ and the L\'evy process corresponding to $\mathcal{L}$. |
