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In this work we are going to show weak convergence of a probability measurecorresponding to the solution of the following nonlinear stochastic heatequation $\frac{\partial}{\partial t} u_{t}(x) = \frac{\kappa}{2} \Delta u_{t}(x) + \sigma(u_{t}(x))\eta_\alpha$ with colored noise $\eta_\alpha$ to themeasure corresponding to the solution of the same equation but with white noise$\eta$ as $\alpha \uparrow 1$ on the space of continuous functions with compactsupport. The noise $\eta_\alpha$ is assumed to be colored in space and itscovariance 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 theRiesz kernel $f_\alpha(x) \propto 1/\left|x\right|^\alpha$. We will also statea result about continuity of measure in $\alpha$, for $\alpha \in (0,1)$. Wewill work with the classical notion of weak convergence of measures.

On weak convergence of stochastic heat equation with colored noise