Stochastic Heat Equation: Violated Conservation Law?

Question

Solution of stochastic heat equation $u_t=\frac{1}{2}u_{xx}+dW$ with initial condition $u(x,0)=\delta_{x_0}$ is proven to be $$u(x,t)=\int_0^t\int_{\mathbb{R}^d}G((t-s,x-y)dW(s,y)$$ where $G$ is the fundamental solution of non-stochastic heat equation. It seems to me that with space time white noise the conservation is violated. i.e. $u(x,t)$ is not a time-varying distribution anymore. I'm wondering if there's any theory that "normalizes" $u(x,t)$ so that we can associate it with a distribution-valued process. i.e. $$\int_{\mathbb{R}^d}u(x,t)dx=1$$