Proof of The Free Fisher Information along the Free Heat Flow: Bounds, Lower Semicontinuity in Time and Integrability

Each result cited is universally quantified over the data in its own statement. Throughout, $M=M(\lambda)$, and for real $t>0$ we write $\mu_{t}=\lambda\boxplus\mathrm{sc}_{d,t}$, so $f_{\lambda}(t)=\Phi^{*}(\mu_{t})$. Since $d$ is a natural number, $d\ge1>0$. $\mathrm{Leb}$ is Lebesgue measure on $\mathcal{B}(\mathbb{R})$; for a Borel set $B$ and a function $\varphi:B\to\mathbb{R}$, $\tilde\varphi$ is the extension of $\varphi$ by $0$ to $\mathbb{R}$, as in Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §integral. A measurable $\psi:\mathbb{R}\to\mathbb{R}$ is integrable if and only if $\int_{\mathbb{R}}|\psi|\,d\mathrm{Leb}<\infty$, by Integrable Function and the Lebesgue Integral; the integrals of nonnegative measurable functions are those of Lebesgue Integral of a Nonnegative Measurable Function, with values in $[0,\infty]$. The real line $(\mathbb{R},d_{\mathbb{R}})$ is that of The Absolute Value Metric on the Real Line, and $B_{d_{\mathbb{R}}}(x,r)=\{y\in\mathbb{R}:|y-x|<r\}$ is its open ball. Every interval, and every set open in $(\mathbb{R},d_{\mathbb{R}})$, is a Borel set: the first by Borel Sigma-Algebra on the Real Line; for the second, by Euclidean, Metric and Sequential Continuity of a Real Function of a Real Variable §distance the number $|y-x|$ is the Euclidean distance of $y$ and $x$, the nonnegative square root of $(y-x)^{2}$, so for real $r>0$ the conditions $(y-x)^{2}<r^{2}$ and $|y-x|<r$ are equivalent, and a set open in $(\mathbb{R},d_{\mathbb{R}})$ is open in $\mathbb{R}^{1}$, hence Borel by Borel Sigma-Algebra on the Real Line.

Part 1 (bounds). Let $t>0$ be real. By Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information §conjugate, $\mu_{t}$ has conjugate variables, and by Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information §fisher-bound, $f_{\lambda}(t)=\Phi^{*}(\mu_{t})\le d/t$. By Free Fisher Information: the Cramer-Rao Inequality, the Semicircular Laws, and Closedness under Weak-Star Limits §cramer-rao, applied to $\mu_{t}$, we have $M(\mu_{t})>0$ and $\Phi^{*}(\mu_{t})\,M(\mu_{t})\ge d^{2}$; by The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity §moments (with $t\ge0$), $M(\mu_{t})=M+td$. Hence $M+td>0$; its inverse exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field, and multiplying $f_{\lambda}(t)(M+td)\ge d^{2}$ by it (claim 5 of Elementary Arithmetic in an Ordered Field) gives $f_{\lambda}(t)\ge d^{2}/(M+td)$.

Part 2 (lower semicontinuity). For real $a$ put $U_{a}=\{t\in(0,\infty):f_{\lambda}(t)>a\}$.

The case $a<0$. For $t>0$, $d^{2}>0$ and $(M+td)^{-1}>0$ by claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field, so by Part 1 and claim 2 of that lemma $f_{\lambda}(t)\ge d^{2}/(M+td)>0>a$. Thus $U_{a}=(0,\infty)$. For $t_{0}>0$ and $y\in B_{d_{\mathbb{R}}}(t_{0},t_{0})$ we have $-t_{0}<y-t_{0}$, hence $0<y$; so $B_{d_{\mathbb{R}}}(t_{0},t_{0})\subseteq U_{a}$, and $U_{a}$ is open in $(\mathbb{R},d_{\mathbb{R}})$.

The case $a\ge0$. Let $t_{0}\in U_{a}$; we show that some ball $B_{d_{\mathbb{R}}}(t_{0},r)$ with $r>0$ lies in $U_{a}$. Suppose not. Then for each $k\in\mathbb{N}$ the radius $r_{k}=t_{0}/k>0$ does not work, so we may choose (informally using countable choice) $t_{k}\in B_{d_{\mathbb{R}}}(t_{0},r_{k})$ with $t_{k}\notin U_{a}$. As $|t_{k}-t_{0}|<t_{0}/k\le t_{0}$, we get $t_{k}>0$; hence $t_{k}\in(0,\infty)\setminus U_{a}$, that is, $f_{\lambda}(t_{k})\le a$. The sequence $(t_{k})_{k\in\mathbb{N}}$ lies in the closed ray $[0,\infty)$ and converges to $t_{0}$ there: given real $\varepsilon>0$, claim 2 of The Archimedean Property of the Real Numbers gives $n\in\mathbb{N}$ with $t_{0}<n\varepsilon$, and for $k\ge n$, $|t_{k}-t_{0}|<t_{0}/k\le t_{0}/n<\varepsilon$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal.

Fix $p\in\mathcal{P}_{d}$. By The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity §continuity, the real function $r\mapsto\operatorname{Re}(\lambda\boxplus\mathrm{sc}_{d,r})(p)$ is continuous on $[0,\infty)$, in particular at $t_{0}$ relative to $[0,\infty)$; so by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, $\operatorname{Re}\mu_{t_{k}}(p)\to\operatorname{Re}\mu_{t_{0}}(p)$ in $(\mathbb{R},d_{\mathbb{R}})$, which, as $d_{\mathbb{R}}(s,s')=|s-s'|$, is convergence of real sequences in the sense of Limit of a Sequence of Real Numbers. The same argument applies to the imaginary parts. By Weak-Star Convergence of Noncommutative Laws §weak-star, $\mu_{t_{k}}\to\mu_{t_{0}}$ weak-star. Each $\mu_{t_{k}}$ has conjugate variables by Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information §conjugate, and $\Phi^{*}(\mu_{t_{k}})=f_{\lambda}(t_{k})\le a$ with $a\ge0$. By Free Fisher Information: the Cramer-Rao Inequality, the Semicircular Laws, and Closedness under Weak-Star Limits §closed (with $c=a$), $\Phi^{*}(\mu_{t_{0}})\le a$, that is $f_{\lambda}(t_{0})\le a$, contradicting $t_{0}\in U_{a}$. Hence $U_{a}$ is open in $(\mathbb{R},d_{\mathbb{R}})$.

Measurability. Let $\tilde f$ and $\tilde g$ be the extensions of $f_{\lambda}$ and $g_{\lambda}$ by $0$. For real $a$: if $a\ge0$ then $\{t\in\mathbb{R}:\tilde f(t)>a\}=U_{a}$, since $\tilde f(t)=0\le a$ for $t\notin(0,\infty)$; if $a<0$ then this set is $U_{a}\cup(\mathbb{R}\setminus(0,\infty))=\mathbb{R}$. In both cases it is a Borel set (an open set, or $\mathbb{R}$), so $\tilde f$ is measurable by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line. Let $u:(0,\infty)\to\mathbb{R}$, $u(t)=d/(1+t)$; note $u(t)>0$ for $t>0$. For real $a$, the set $\{t\in\mathbb{R}:\tilde u(t)>a\}$ is $\mathbb{R}$ if $a<0$ and $(0,\infty)$ if $a=0$; if $a>0$ it is $\{t\in\mathbb{R}:0<t<d/a-1\}$, because for $t>0$ the inequality $d/(1+t)>a$ is equivalent to $d>a(1+t)$ (multiply by $1+t>0$) and then to $t<d/a-1$ (multiply by $a^{-1}>0$ and subtract $1$), by claims 1, 7 and 10 of Elementary Order Arithmetic in an Ordered Field. Each of these sets is an interval, hence Borel, so $\tilde u$ is measurable by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line. Finally $\tilde g=\tilde u-\tilde f$ pointwise on $\mathbb{R}$ (on $(0,\infty)$ by the definition of $g_{\lambda}$, and elsewhere both sides vanish), so $\tilde g$ is measurable by claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

Part 3 (integrability). Two-sided bounds. Let $t>0$. From Part 1, $-d/t\le-f_{\lambda}(t)\le-d^{2}/(M+td)$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, and adding $d/(1+t)$ to each side of both inequalities (compatibility of $\le$ with addition, item 1 of Ordered Field),

$$\frac{d}{1+t}-\frac{d}{t}\le g_{\lambda}(t)\le\frac{d}{1+t}-\frac{d^{2}}{M+td}.$$

Put $m=M/d$. Then $M+td=d(m+t)$; since $d>0$ and $d(m+t)>0$ by Part 1, $m+t>0$ (otherwise $d(m+t)\le0$ by claim 5 of Elementary Arithmetic in an Ordered Field), and $d^{2}/(M+td)=d/(m+t)$. Taking $t=1$ gives $1+m>0$. Hence, for every real $t>0$,

$$d\Bigl(\frac{1}{1+t}-\frac{1}{0+t}\Bigr)\le g_{\lambda}(t)\le d\Bigl(\frac{1}{1+t}-\frac{1}{m+t}\Bigr).$$

Integrable bounds on $(1,\infty)$. Let $\ell,\upsilon:(1,\infty)\to\mathbb{R}$ be the left and right sides of the last display. By Logarithmic Integrals of Reciprocal Linear Functions over Bounded and Unbounded Intervals §half-line with $(a,b,c)=(1,1,0)$ (where $1+1>0$ and $1+0>0$) and with $(a,b,c)=(1,1,m)$ (where $1+m>0$), the functions $t\mapsto\frac{1}{1+t}-\frac{1}{0+t}$ and $t\mapsto\frac{1}{1+t}-\frac{1}{m+t}$ are integrable on $(1,\infty)$; by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, $\tilde\ell$ and $\tilde\upsilon$, which are $d$ times their extensions by $0$, are integrable, so $\int_{\mathbb{R}}|\tilde\ell|\,d\mathrm{Leb}$ and $\int_{\mathbb{R}}|\tilde\upsilon|\,d\mathrm{Leb}$ are finite.

Integrability on $[1,\infty)$. The extension by $0$ of the restriction of $g_{\lambda}$ to $[1,\infty)$ is $G=\tilde g\,\mathbf{1}_{[1,\infty)}$; it is measurable by Part 2 and claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions ($[1,\infty)$ being an interval, hence Borel), and $|G|$, $|\tilde\ell|$, $|\tilde\upsilon|$ and $|\tilde\ell|+|\tilde\upsilon|$ are measurable by claims 2 and 4 of that lemma. For $t>1$ we have $\ell(t)\le g_{\lambda}(t)\le\upsilon(t)$, so $g_{\lambda}(t)\le|\upsilon(t)|\le|\ell(t)|+|\upsilon(t)|$ and $-g_{\lambda}(t)\le-\ell(t)\le|\ell(t)|\le|\ell(t)|+|\upsilon(t)|$, whence $|G(t)|\le|\tilde\ell(t)|+|\tilde\upsilon(t)|$; for $t<1$, $|G(t)|=0\le|\tilde\ell(t)|+|\tilde\upsilon(t)|$. So this inequality holds outside the singleton $\{1\}=[1,1]$, which has $\mathrm{Leb}(\{1\})=0$ by claim 4 of Existence of Lebesgue Measure on the Real Line. By The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, in $[0,\infty]$,

$$\int_{\mathbb{R}}|G|\,d\mathrm{Leb}\le\int_{\mathbb{R}}|\tilde\ell|\,d\mathrm{Leb}+\int_{\mathbb{R}}|\tilde\upsilon|\,d\mathrm{Leb}<\infty .$$

Hence $G$ is integrable, i.e. $g_{\lambda}$ is integrable on $[1,\infty)$, by Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line §integral.

The equivalence. Let $F_{1}$, $W_{1}$ and $G_{1}$ be the extensions by $0$ of the restrictions of $f_{\lambda}$, $u$ and $g_{\lambda}$ to the open interval $(0,1)$. By Logarithmic Integrals of Reciprocal Linear Functions over Bounded and Unbounded Intervals §bounded with $(a,s,b)=(0,1,1)$ (where $0+1>0$), $t\mapsto1/(1+t)$ is integrable on $(0,1)$, so $W_{1}$, which is $d$ times its extension by $0$, is integrable by claim 2 of Linearity and Monotonicity of the Lebesgue Integral. Pointwise on $\mathbb{R}$ we have $G_{1}=W_{1}-F_{1}$, and, since $(0,1)$ and $[1,\infty)$ are disjoint with union $(0,\infty)$, $\tilde g=G_{1}+G$.

Suppose $f_{\lambda}$ is integrable on $(0,1)$, i.e. $F_{1}$ is measurable and integrable. Then $G_{1}=W_{1}-F_{1}$ is integrable, and so is $\tilde g=G_{1}+G$, both by claim 2 of Linearity and Monotonicity of the Lebesgue Integral; as $\tilde g$ is measurable by Part 2, $g_{\lambda}$ is integrable on $(0,\infty)$.

Conversely, suppose $g_{\lambda}$ is integrable on $(0,\infty)$, i.e. $\tilde g$ is integrable. Then $G_{1}=\tilde g\,\mathbf{1}_{(0,1)}$ is measurable by claims 1 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, so $|G_{1}|$ and $|\tilde g|$ are measurable by claim 4 of that lemma, and $|G_{1}|\le|\tilde g|$ pointwise, so $\int_{\mathbb{R}}|G_{1}|\,d\mathrm{Leb}\le\int_{\mathbb{R}}|\tilde g|\,d\mathrm{Leb}<\infty$ by claim 1 of Linearity and Monotonicity of the Lebesgue Integral; hence $G_{1}$ is integrable. Then $F_{1}=W_{1}-G_{1}$ is integrable (and measurable) by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, that is, $f_{\lambda}$ is integrable on $(0,1)$.