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I would greatly appreciate any references were they study the stochastic equation in higher dimensions: $u_{t}=\Delta u+f$ in great detail, especially in dimension 2.

In Hairer's Spde notes , he mentions that the solutions will not be function-valued, and I am curious what the particular regularity will be with ,say, $u_{0}\equiv 0$.

Q: Over which space they will be distributions?

For $u_{0}\equiv 0$ the covariance for $u(x,t)$ is

$$E[u(x,t)u(y,s)]=2^{-(n+1)}\int_{[|s-t|,s+t]}l^{-n/2}exp(-|x-y|^{2}/4l)dl=:C(t,s,x,y)$$

and one idea is to find a Gelfand-triple so that using Gross results, we can define a Gaussian process with this covariance. To obtain some space of measures s.t. $<u,\rho>$ is a Gaussian process with covariance

$$E[<u,\rho><u,\mu>]=\int \int C(t,s,x,y)d\rho(x,t) d\mu(y,s). $$

This is in spirit of the mathematical construction for white noise and Gaussian free field.

Update: You can find more information as described in the answer below in "Stochastic PDEs, Regularity Structures, and Interacting Particle Systems" Theorem 2.8.

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I hope this is an answer to your question. In any case it is too long for a comment.

With which space of distributions you are working depends on where you are solving the equation. Two natural choices are $[0,T]\times \mathbb{R}^d$ or $[0,T]\times \mathbb{T}^d$, the latter being a torus (i.e. periodic boundary conditions). Let us call $\mathcal{X}$ any of the above two, then white noise has a modification that is a Schwartz distribution: $$ \xi \in \mathcal{D}(\mathcal{X}). $$ Meaning that it is a distribution both in space and in time. Indeed the noise does not see the difference, and it can be proven that it lies in a space of negative Holder regularity. $$ \xi \in C^{-(d+1)/2-\epsilon} (\mathcal{X}), $$ for any $\epsilon >0.$ Now the rule of thumb is that solving the heat equation (i.e. convolution with the heat operator) makes you gain $+2$ in parabolic regularity, where parabolic stands for the fact that time regularity should be counted as double of the space regularity. In this sense $\xi$ should be of parabolic regularity: $$ \xi \in C^{-(d+2)/2-\epsilon} (\mathcal{X}) $$ so by Schauder estimates (which I believe you can find in some of M. Hairers papers) integrating against the heat kernel (which we denote wich $\mathcal{I}$) gives you parabolic regularity: $$ u = \mathcal{I}(\xi) \in C^{-d/2 +1-\epsilon} (\mathcal{X}). $$ so that in dimension $1$ this is still function valued, while in dimension $2$ or greater it is distribution valued.

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