Consider the Additive Stochastic Heat Equation (ASHE) given by $$ \begin{cases} \partial_t u = \Delta u + \mathcal{\dot{W}}\\ u(x,0) = \xi \end{cases} $$ where $\mathcal{\dot{W}}$ is a white noise and $\xi$ is a $L^2$ r.v. To me, it seems to be the case that people from the area already understand well this equation since before more modern techniques of the last few years (such as rough path theory).
One can proceed in a few natural ways to find a solution for such equation. For instance one could make use a Hilbert Basis $\{\phi_n\}_n$ for your $L^2$ space and solve a infinite system of independent stochastic ODE's and from there recover the original solution as the sum of those. The advantage of this method is that each of those ODE's are Ornstein–Uhlenbeck equations, in which you can get an explicit form for the solution.
So now there is the problem, it is well known that both of those methods only work in dimension $d=1$, for $d\ge2$, you get that the sum of the independent Ornstein–Uhlenbeck does not converge to a function in $L^2$, but instead as a distribution. So my main confusion:
Does a distribution-valued solution actually solve your problem?
I am sorry if the question seems superficial or too physically oriented, I am just beginning in the area. I am just confused on how we are answering the original question. I am familiar with the notion of a weak solution for a PDE, so I understand not quite having the the full original desired properties, but being close. It is just that not even having a function anymore seems like a big step.
I am looking for a more intuitive way to see this, but I would appreciate any help or references.
This will not answer your question about the ASHE, but another SPDE where we WANT a distribution valued solution. Hopefully this will offer some insight.
An important SPDE is Kushner's equation.
Kushner's equation arises in stochastic filtering theory. Stochastic filtering is when you have a noisy signal:
$$dX_t=f(X_t) dt+dW_t$$
and a noisy observation:
$$dY_t=h(X_t)dt+dV_t$$
Where $W,V$ are independent Brownian motions.
For example if you have a wireless signal that has some noise and your receiver has some noise, or something like that.
You want to find the best estimate for $h(X_t)$ given your observations. To do that, you want to find $\Bbb P(X_t\in A|\sigma(Y_s:0\le s \le t)):=\pi_t(A)$, a measure valued stochastic process called the optimal filter. $\pi_t$ satisfies Kushner's equation. $\pi_t$ is measure valued i.e. distribution valued. In this circumstance we WANT it to be distribution valued.
I am not sure about a physical interpretation of distribution valued heat distribution to be honest. However this is an example of a distribution valued solution to a SPDE with an important meaning.