Diffusive heat limit for symmetric exclusion

Fix $t\in [0,T]$ and a real-valued smooth periodic $G$. For every $r\ge 0$, $P_{r}G$ is real-valued: since $G$ is real, $\widehat{G}(-k)=\overline{\widehat{G}(k)}$, and the Fourier multiplier $e^{-4{\pi }^2\mathrm{\mid }k{\mathrm{\mid }}^{2}r}$ is real and unchanged by $k\mapsto -k$; pairing the absolutely convergent $k$ and $-k$ terms (with the zero mode real) gives a real sum. By [F2], these functions are smooth, so $H_s=P_{t-s}G$ is an admissible real-valued test in [F3]. Denote its martingale by $M_s^{N,t}$. By [F2], ${\mathrm{\partial }}_s{H}_s=-\mathrm{\Delta }{H}_s$. The time-dependent martingale identity at its terminal time gives $⟨{\pi }_t^N,G⟩=⟨{\pi }_0^N,P_{t}G⟩+M_t^{N,t}+{\int }_0^t⟨{\pi }_s^N,({\mathrm{\Delta }}_N-\mathrm{\Delta })P_{t-s}G⟩ds$, and $M^{N,t}$ starts at zero with terminal second moment equal to the expected bracket.

The fourth spatial derivatives of $P_{t-s}G$ are uniformly bounded for $0\le s\le t$ by [F2]. Taylor's formula for each symmetric second difference therefore gives ${\sup }_{s,x}\mathrm{\mid }({\mathrm{\Delta }}_N-\mathrm{\Delta })P_{t-s}G(x)\mathrm{\mid }\le C_{G,T}{N}^{-2}$. Since ${\pi }_s^N({\mathbb{T}}^d)\le 1$, the drift-error integral in step 1.1 is at most $t{C}_{G,T}{N}^{-2}$. By [F3], ${\mathbb{E}}_N\mathrm{\mid }{M}_t^{N,t}{\mathrm{\mid }}^2\le C_{G,T}^{\mathrm{\prime }}{N}^{-d}$, so Chebyshev's inequality makes that martingale term tend to zero in probability. Step 1.1 establishes that $P_{t}G$ is real-valued, and [F2] gives its smoothness; hence the initial-profile assumption applies to this test. Also [F2] gives $\int {\rho }_0{P}_{t}G=\int {\rho }_{t}G$. Thus for each fixed $t$, $⟨{\pi }_t^N,G⟩\to \int {\rho }_{t}G$ in probability.

If $T=0$, the conclusion is the initial-profile assumption. Suppose $T>0$ and fix a finite time mesh $0=t_0<\cdots <t_m=T$. Step 2.1 and a finite union bound imply ${\max }_j\mathrm{\mid }⟨{\pi }_{t_j}^N,G⟩-\int {\rho }_{t_j}G\mathrm{\mid }\to 0$ in probability.

By the fundamental theorem of calculus applied twice to each centered second difference, $\mathrm{\parallel }{\mathrm{\Delta }}_{N}G{\mathrm{\parallel }}_{\mathrm{\infty }}\le K_G:={\sum }_i\mathrm{\parallel }{\mathrm{\partial }}_{ii}G{\mathrm{\parallel }}_{\mathrm{\infty }}$ for all $N$. The fixed-test martingale in [F4] gives $\mathrm{\mid }⟨{\pi }_t^N,G⟩-⟨{\pi }_s^N,G⟩\mathrm{\mid }\le K_G\mathrm{\mid }t-s\mathrm{\mid }+2{\sup }_{0\le u\le T}\mathrm{\mid }{M}_u^{N,G}\mathrm{\mid }$ for all $s,t\in [0,T]$. By [F4], $\mathbb{E}\mathrm{\mid }{M}_T^{N,G}{\mathrm{\mid }}^2\le T{N}^{-d}{\sum }_i\mathrm{\parallel }{\mathrm{\partial }}_{i}G{\mathrm{\parallel }}_{\mathrm{\infty }}^2$; [F5] therefore implies ${\sup }_{u\le T}\mathrm{\mid }{M}_u^{N,G}\mathrm{\mid }\to 0$ in probability. Also [F2] and the mass bound $\int {\rho }_t\le 1$ show $t\mapsto \int {\rho }_{t}G$ is Lipschitz with constant $\mathrm{\parallel }\mathrm{\Delta }G{\mathrm{\parallel }}_{\mathrm{\infty }}$. Given $\delta >0$, choose a finite mesh with spacing at most $\delta$; the supremum error is bounded by the maximum mesh error from step 3.1, $2\sup \mathrm{\mid }{M}^{N,G}\mathrm{\mid }$, and $(K_G+\mathrm{\parallel }\mathrm{\Delta }G{\mathrm{\parallel }}_{\mathrm{\infty }})\delta$. First let $N\to \mathrm{\infty }$ and then $\delta \downarrow 0$. This proves the stated uniform-in-time convergence in probability.