I am confused by the idea that when the data is stochastic, the solution u($\omega$ ,x) of the SPDE can be expressed as the conditional expectation of some functional solution to the SDE
Take Darcy equation related to the modeling of groundwater flow as an example.
PDE: -$\nabla (k(x)\nabla u(x)) = f(x), x\in D$, $u(x) = g(x), x \in \partial D$.
Here, u is the hydraulic head, k denotes the conductivity coefficient describing the porosity of the medium f is the source term and the Dirichlet boundary data is defined by g.
And data of the PDE can be modeled as being stochastic. When stochastic data is assumed, choose Gaussian fields as stochastic fields.
Then the stochastic field k with finite variance can be represented as $k(\omega , x) = E[k] + \sum a_m (x) \zeta _m ( \omega)$ through KL expansion. The basis $a_m$ consists of eigenfunctions os the covariance integral operator weighted by the eigenvalues of this operator.
SDE: $ dX_t = b(X_t)dt + \sigma (X_t) dW_t$
where b and $\sigma$ are coefficients, W Brownian motion.
Then we can have u($\omega$ ,x) = E[$\phi ^x$ | k,f,g], where k, f, g are deterministic data.
I think $\phi ^x$ has something to do with Feynman-Kac formula.
I am really new to the field. I want to know how to get this.
Does this caused by properties of SPDE, SDE, or something else?
Thank you so much