The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\star$ be the free product (for $m=n=d$), $\boxplus$ the free additive convolution, and $\mathrm{sc}_{d}$ and $\mathrm{sc}_{d,t}$ the semicircular law and the semicircular law of variance $t$. For real $t\ge0$, $\sqrt{t}$ is its nonnegative square root, and ¶$\Gamma_{t}=(A^{t},0)$ is the affine datum from $2d$ to $d$ variables with $A^{t}_{ii}=1$, $A^{t}_{i,d+i}=\sqrt{t}$ and $A^{t}_{ij}=0$ otherwise ($i\in[d]$, $j\in[2d]$), whose tuple is $(x_{1}+\sqrt{t}\,x_{d+1},\dots,x_{d}+\sqrt{t}\,x_{2d})$. The first and quadratic moments $\mathrm{m}_{i}$, $\mathrm{m}_{ij}$ are those of Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data, and $\delta_{ij}$ is $1$ if $i=j$ and $0$ otherwise. Let $\lambda,\mu,\nu\in\Sigma_{d}$ and let $s,t\ge0$ be real.

1. (Realisation)¶ $\lambda\boxplus\mathrm{sc}_{d,t}=(\lambda\star\mathrm{sc}_{d})\circ\sigma_{\Gamma_{t}}$.

2. (Norm bound)¶ $\lambda\boxplus\mathrm{sc}_{d,0}=\lambda$, and if $R>0$ is real with $\lambda\in\Sigma_{d,R}$, then $\lambda\boxplus\mathrm{sc}_{d,t}\in\Sigma_{d,R+2\sqrt{t}}$.

3. (Moments)¶ For all $i,j\in[d]$, $\mathrm{m}_{i}(\lambda\boxplus\mathrm{sc}_{d,t})=\mathrm{m}_{i}(\lambda)$ and $\mathrm{m}_{ij}(\lambda\boxplus\mathrm{sc}_{d,t})=\mathrm{m}_{ij}(\lambda)+t\,\delta_{ij}$; in particular $M(\lambda\boxplus\mathrm{sc}_{d,t})=M(\lambda)+td$.

4. (Distance to the initial law)¶ $W_{2}(\lambda\boxplus\mathrm{sc}_{d,t},\lambda)\le\sqrt{td}$.

5. (Contraction)¶ $W_{2}(\mu\boxplus\mathrm{sc}_{d,t},\nu\boxplus\mathrm{sc}_{d,t})\le W_{2}(\mu,\nu)$.

6. (Semigroup)¶ $(\lambda\boxplus\mathrm{sc}_{d,t})\boxplus\mathrm{sc}_{d,s}=\lambda\boxplus\mathrm{sc}_{d,t+s}$.

7. (Continuity)¶ ¶For every $p\in\mathcal{P}_{d}$ the real functions $r\mapsto\operatorname{Re}(\lambda\boxplus\mathrm{sc}_{d,r})(p)$ and $r\mapsto\operatorname{Im}(\lambda\boxplus\mathrm{sc}_{d,r})(p)$ are continuous on the closed ray $[0,\infty)$, the reals carrying the absolute value metric. ¶If $R>0$ is real and $(\mu_{k})_{k\in\mathbb{N}}$ is a sequence in $\Sigma_{d,R}$ that converges weak-star to $\mu$, then $\mu_{k}\boxplus\mathrm{sc}_{d,t}\to\mu\boxplus\mathrm{sc}_{d,t}$ weak-star.