Affine random diffusion has a measurable Bochner solution

For each $y\in \mathrm{\Gamma }$, [F2] verifies the boundedness and coercivity hypotheses of [F3], while $F(y)=f_0+{\sum }_j{y}_j{f}_j$ is a bounded functional. Lax–Milgram gives a unique $u(y)\in V$ and $\mathrm{\parallel }u(y){\mathrm{\parallel }}_V\le F_{\max }\mathrm{/}{a}_{\ast }$, where $F_{\max }:={\sup }_{y\in \mathrm{\Gamma }}\mathrm{\parallel }F(y){\mathrm{\parallel }}_{V^{\ast }}<\mathrm{\infty }$ because $\mathrm{\Gamma }$ is bounded.

For $y,z\in \mathrm{\Gamma }$, put $d=u(y)-u(z)$ and subtract the weak equations: $a_y(d,v)=(F(y)-F(z))(v)+(a_z-a_y)(u(z),v)\mathrm{.}$ The affine formulas give $\mathrm{\parallel }F(y)-F(z){\mathrm{\parallel }}_{V^{\ast }}\le C_F\mathrm{\parallel }y-z{\mathrm{\parallel }}_1$ and $\mathrm{\mid }{a}_z(v,w)-a_y(v,w)\mathrm{\mid }\le C_a\mathrm{\parallel }y-z{\mathrm{\parallel }}_1\mathrm{\parallel }v{\mathrm{\parallel }}_V\mathrm{\parallel }w{\mathrm{\parallel }}_V$, where $C_F={\sum }_j\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_{V^{\ast }}$ and $C_a={\sum }_j\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(D)}$. Set $v=d$ and use [F2] and the bound $\mathrm{\parallel }u(z){\mathrm{\parallel }}_V\le F_{\max }\mathrm{/}{a}_{\ast }$ from step 1.1. If $d\mathrm{\ne }0$, divide the resulting coercive inequality by $\mathrm{\parallel }d{\mathrm{\parallel }}_V$; if $d=0$ the conclusion is immediate. This proves $\mathrm{\parallel }u(y)-u(z){\mathrm{\parallel }}_V\le a_{\ast }^{-1}(C_F+C_a{F}_{\max }\mathrm{/}{a}_{\ast })\mathrm{\parallel }y-z{\mathrm{\parallel }}_1$. Thus $u$ is Lipschitz and Borel measurable.

Since $V$ is separable, [F4] makes $u$ strongly measurable. The uniform bound in step 1.1 and $\rho (\mathrm{\Gamma })=1$ give ${\int }_{\mathrm{\Gamma }}\mathrm{\parallel }u(y){\mathrm{\parallel }}_V^{2}d\rho (y)\le (F_{\max }\mathrm{/}{a}_{\ast }{)}^2<\mathrm{\infty }$, so $u\in L_{\rho }^2(V)$.

For a finite-valued simple test $W(y)={\sum }_{i=1}^m{\mathbf{1}}_{E_i}(y)v_i$, integrate the pointwise weak identity for each $v_i$ over $E_i$ and sum; [F5] ensures the scalar integrands are measurable and integrable. This gives $\mathcal{A}(u,W)=\mathcal{F}(W)$.

The bounds in [F5] make both sides continuous on $L_{\rho }^2(V)$. Finite-valued simple tests are dense by [F6], so step 3.1 extends to every $W$. If two Bochner solutions existed, testing their difference against itself and using the coercivity bound in [F5] would force their difference to be zero.