Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
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Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...