Consider the linear multiplicative heat equation on $\mathbb{R}^+\times \mathbb{R}^d$ given by: \begin{equation}\label{linear multiplicative heat equation} \partial_t u= \frac{1}{2}\Delta u + \xi u \quad (1) \end{equation} where the noise $\xi(t,x)$ is supposed to be smooth in time and space, and the initial condition is $u(0,x)=f(x)$ for a function $f$ on $\mathbb{R}^d$. Denote by $\cdot$ and $\ast$ respectively the convolution in space and in time-space with the heat kernel $K$: $$K\cdot f(t,x):=\int_{\mathbb{R}^d}f(y)K(t,x-y)dy$$ $$K\ast \xi(t,x):=\int_0^t\int_{\mathbb{R}^d}\xi(s,y)K(t-s,x-y)dyds$$ and let introduce the linear operator $\phi$ defined by: $$\phi:u\mapsto K\ast (\xi u)$$
The mild formulation of the equation (1) is the fixed-point equation: \begin{equation} \label{mild} u=K\cdot f + \phi(u) \quad (2) \end{equation}
The idea is to get a solution of (1) by expanding the expression (2) iteratively into a serie, that is formally speaking: \begin{equation}\label{serie} \sum_{n=0}^\infty \phi^{(n)}(K\cdot f) \quad (3) \end{equation}
where $\phi^{(0)}=id$, and $\phi^{(n)}$ is the $n$-fold composition.
Question: Can we find some nice hypothesis on $\xi$ and $f$ under which the serie (3) is well defined and a solution to the equation (1)?
Remark: The sequence of truncated series $u_n:=\sum_{k=0}^n \phi^{(k)}(K\cdot f)$ satisfy the sequence of equations: $$\partial_t u_n= \frac{1}{2}\Delta u_n + \xi u_{n-1}$$ with initial condition $u_n(0,x)=f(x)$.