Mass, Hellinger, and Transport Convergence#
TLDR. An alive measure carries two kinds of information: how much mass survives and where that mass is distributed. The square-root mass difference and the normalized distribution separate exactly in Hellinger distance. Combine that identity with the entropy estimates of Logarithmic Sobolev inequalities and entropy convergence and the entropy-to-transport inequality of Quantitative Error Bounds, and one obtains a quantitative bound for mass, shape, and transport together.
Think of two clouds of walkers. One cloud may contain fewer alive walkers even when both clouds have the same spatial profile. Or their alive fractions may agree while the clouds occupy different regions. A convergence statement should tell us which difference it controls. The first three sections make those questions precise and assemble the main estimate. The density and linearization sections give additional tools for local comparisons and perturbations.
Choose the underlying space before choosing the distance. An alive one-particle measure can have mass below one; its normalization describes where an alive particle lies. A QSD is instead a probability law on living swarm configurations. Its mass is one, while the fraction of alive slots is a random observable of the swarm. The same measure inequalities apply on either space, with the mass and reference law appropriate to that space.
Metrics for Alive Measures#
Hellinger distance compares square roots of mass density. Wasserstein distance compares the cost of moving a normalized distribution. The additive distance below keeps both measurements. Canonical Hellinger–Kantorovich distance solves a different problem: it chooses transport and creation or removal of mass along a path, charging for both. Its normalization determines the factors in the reaction and transport bounds.
Definition 591 (Hellinger distance and the additive transport distance)
For finite measures \(\mu,\nu\), define \(H^2(\mu,\nu)=\int(\sqrt{d\mu/d\lambda}-\sqrt{d\nu/d\lambda})^2d\lambda\), where \(\lambda\) dominates both. For positive masses \(m=\mu(E)\) and \(n=\nu(E)\), put \(p=\mu/m\), \(q=\nu/n\) and
This additive distance is distinct from canonical Hellinger–Kantorovich (HK) distance. In the normalization
pure reaction gives \(\mathrm{HK}(\mu,\nu)\leq H(\mu,\nu)\), and pure transport gives \(\mathrm{HK}^2(mp,mq)\leq mW_2^2(p,q)\). The dynamic metric and its measure-valued extension are developed by Liero, Mielke, and Savaré.
Proof
For pure reaction use \(\rho_t=((1-t)\sqrt{d\mu/d\lambda}+t\sqrt{d\nu/d\lambda})^2\lambda\), \(v_t=0\), and \(r_t=\partial_t\rho_t/\rho_t\). Its action is \(\int(\sqrt{d\nu/d\lambda}-\sqrt{d\mu/d\lambda})^2d\lambda\); approximation handles zeros. For equal-mass transport, multiply a probability transport curve by \(m\) and take \(r_t=0\). Its action is \(m\) times the probability transport action. Infimizing proves the second bound. The Hellinger formula is independent of \(\lambda\) by the Radon–Nikodym chain rule. The additive \(D\) is a metric on positive finite measures with finite normalized second moments: apply the triangle inequality to its two components and then the Euclidean norm on \(\mathbb R^2\).
First hold the shape fixed. If both clouds have the same normalized profile \(p\), Hellinger distance becomes \(|\sqrt m-\sqrt n|\). Now let the profiles change as well. Expanding the two squares produces exactly one extra term: the Hellinger distance between the normalized profiles, weighted by \(\sqrt{mn}\). That algebra is the main organizing identity of the chapter.
Lemma 182 (Exact separation of mass and shape)
For \(\mu=mp\), \(\nu=nq\) with probability laws \(p,q\),
Moreover \(H^2(p,q)\leq D_{\mathrm{KL}}(p\Vert q)\), and if \(m,n\geq m_0>0\), then \((\sqrt m-\sqrt n)^2\leq(m-n)^2/(4m_0)\).
Proof
Write the affinity \(A=\int\sqrt{dp\,dq}\). Expanding gives \(H^2(mp,nq)=m+n-2\sqrt{mn}A\) and \(H^2(p,q)=2-2A\). If the entropy is finite, Jensen applied under \(p\) gives \(\log A\geq-\tfrac12D_{\mathrm{KL}}(p\Vert q)\). Thus \(H^2(p,q)\leq2(1-e^{-D_{\mathrm{KL}}/2})\leq D_{\mathrm{KL}}\). The infinite-entropy case is immediate. Rationalize the difference of square roots for the last inequality.
There is a useful warning about what constitutes a cloud. The probability law of a particle can have a smooth density, while a recorded cloud consists of finitely many points. A continuous density assigns zero mass to that finite set; the empirical measure puts all its mass there. Hellinger distance detects this distinction even when the points form an excellent transport or numerical approximation to the density.
Remark 197 (Laws and finite empirical measures)
A finite atomic empirical probability measure and an absolutely continuous probability law are mutually singular, so their squared Hellinger distance is exactly \(2\). Entropy and Hellinger convergence below concern evolving probability laws or explicitly smoothed empirical measures. Empirical transport and observable convergence use Quantitative Error Bounds. This distinction applies even if the individual walkers have smooth marginal laws: a realization of their empirical measure remains atomic.
Surviving mass and measured conditional shape
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Alive-Mass Balance and Fluctuations#
Follow one update in the order the algorithm performs it. Revival fills dead slots when its required alive companions are available. The next kinetic and boundary stage determines which slots survive. Once the shared choices are fixed, independent survival indicators give a variance of order \(1/N\) for their average. This is the source of the sampling term in the first estimate.
The drift of the conditional mean and the fluctuation around that mean are separate quantities. A contracting mean can coexist with a nonzero variance floor, and an alive replacement preserves the number of alive slots.
Lemma 183 (Revival and death in a fixed-size population)
Suppose a successful revival stage leaves \(N\) alive walkers. Conditional on the resulting configuration \(S^+\), let their next kinetic survival indicators be independent with probabilities \(p_i(S^+)\). For the next alive fraction \(m'=N^{-1}\sum_iI_i\),
If \(|b(S^+)-m_*|\leq r|m-m_*|+\delta(S^+)\), then for every \(\eta>0\),
Thus \((1+\eta)r^2<1\) and a uniform defect bound imply a geometric approach to the corresponding variance floor. Replacing an alive walker by a clone preserves the alive count; it is not an additional birth.
Proof
Conditional independence gives variance \(N^{-2}\sum_ip_i(1-p_i)\leq1/(4N)\). Apply the conditional second-moment identity around \(m_*\), followed by \((a+b)^2\leq(1+\eta)a^2+(1+\eta^{-1})b^2\). Iterating \(u_{j+1}\leq Ru_j+B\), \(R<1\), gives \(u_j\leq R^ju_0+B(1-R^j)/(1-R)\). The assumptions specify the update stage and its available alive population; extinction or an unsuccessful revival stage is included as an extra defect or treated by the conditioned-kernel estimate below.
Lemma 184 (Structural drift retains its fluctuation floor)
If the combined transition satisfies \(\mathbb E[V_{j+1}\mid S_j]\leq rV_j+b\) for \(V\geq0\), \(r<1\), then
The component-matrix estimates in Convergence, Survival, and Parameter Dependence establish this form for their specified weighted structural observables. If \(V\) controls a squared transport distance, the same upper bound controls that distance.
Proof
Take expectations and iterate the affine inequality. A positive \(b\) produces a positive floor; the displayed inequality by itself does not show \(\mathbb EV_j\to0\) or identify a stationary distribution.
In the continuous alive equation, death removes existing mass and revival supplies mass from the inactive population. Integrating the equation exposes that balance directly. The cancellation of cloning gain and loss uses a probability-valued proposal; a proposal that can leave the valid domain has its own loss term.
Proposition 170 (Killing rates and alive mass in the continuous model)
For bounded interior killing \(0\leq c\leq C\), probability-valued revival at rate \(\lambda>0\), and a mass-neutral cloning kernel, the continuous alive fraction obeys
For boundary absorption or subprobability cloning proposals, the flux and proposal-loss terms from The Mean-Field Law of the Euclidean Gas enter the same balance.
Proof
Integrate the alive equation and cancel the cloning gain and loss. Since
\(\int cf\leq Cm_a\), multiply the resulting differential inequality by
\(e^{(C+\lambda)t}\) and integrate. The boundary and failed-proposal terms are
exactly those in Proof and
rem-mean-field-cloning-boundary-loss.
A small chance of extinction in one step gives a useful bound over a specified number of steps. The bound pays for every opportunity for extinction. It therefore retains the horizon \(T\); it does not create an event of survival forever with positive probability.
Theorem 251 (Finite-horizon survival from safe-walker estimates)
Suppose at each successful step, conditional on its history, at least \(aN\) walkers have independent death probabilities at most \(q<1\). Then \(\mathbb P(\tau_\dagger\leq T)\leq Tq^{aN}\) for integer \(T\). If the safe-walker condition fails with conditional probability at most \(\delta_N\), replace the right side by \(T(q^{aN}+\delta_N)\).
Proof
On the stated stage condition, total extinction requires all \(aN\) designated walkers to die, which has probability at most \(q^{aN}\). Add the condition’s failure probability, then sum conditional one-step probabilities over the first \(T\) steps. Corollary 60 supplies the same counting argument with its explicit Gaussian and barrier inputs.
Lemma 185 (Lower alive-mass concentration)
Conditional on a post-revival configuration, suppose independent indicators \(I_i\) have mean alive fraction \(\bar p\geq p_0>0\). For \(0<b<p_0\),
Proof
For a variable in \([0,1]\), the log moment generating function of its centered version has second derivative at most \(1/4\) (the variance under each exponential tilt), and vanishing value and derivative at zero. Thus its log moment generating function is at most \(t^2/8\). Multiply the independent bounds and apply Markov’s inequality to \(\exp[-t\sum_i(I_i-\mathbb EI_i)]\). Minimizing at \(t=4(\bar p-b)\) gives the result.
The mass concentration estimate makes a positive lower bound available with high probability. That event must be carried into subsequent estimates: a positive mean alive fraction is not a pathwise lower bound. Conditioning on survival also changes probabilities, through the denominator in the following elementary calculation.
Corollary 82 (Conditioning on finite-time survival)
If \(\mathbb P(A^c)\leq\varepsilon\) and \(\mathbb P(\tau_\dagger>t)\geq s>0\), then \(\mathbb P(A^c\mid\tau_\dagger>t)\leq\varepsilon/s\). Likewise, for \(Y\geq0\), \(\mathbb E[Y\mid\tau_\dagger>t]\leq\mathbb EY/s\).
Proof
Bound the numerator of each conditional probability or expectation by its unconditional counterpart and divide by the survival probability.
Proposition 171 (Alive-mass variance under stationary concentration)
For an exchangeable marked-state law, let \(I_i\) be its alive indicators. If \(\operatorname{Var}(I_i)\leq1/4\) and \(|\operatorname{Cov}(I_i,I_j)|\leq C/N\), then \(\operatorname{Var}(N^{-1}\sum_iI_i)\leq(1/4+C)/N\). The bounded-observable entropy estimate Theorem 245 also applies directly to these indicators when its joint relative-entropy hypothesis holds.
Proof
Expand the variance of the sum. There are \(N\) diagonal and \(N(N-1)\) off-diagonal terms. Divide by \(N^2\) and use the stated bounds.
Hellinger and Transport Bounds from Entropy#
Here the entropy calculation does most of the work. Relative entropy bounds Hellinger distance directly. A logarithmic Sobolev inequality for the specified reference law also converts entropy into a squared transport bound. The two conversions preserve the time dependence and any residual in the original entropy estimate.
For a kinetic process, that entropy estimate comes from the modified Fisher information and hypocoercive calculation. For a killed process, it includes the normalization of the evolving law. Those proofs are in Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm and Logarithmic Sobolev inequalities and entropy convergence; here we use their conclusions for the law to which they apply.
Lemma 186 (Hellinger convergence from the recovered kinetic estimate)
Suppose the normalized evolving law \(p_t\) and its specified reference \(q\) satisfy the recovered entropy estimate
Then \(H^2(p_t,q)\leq Ae^{-\lambda t}+E_t\). If \(q\) has LSI constant \(C\) in the convention \(\operatorname{Ent}_q(g^2)\leq2C\int|\nabla g|^2dq\), then also \(W_2^2(p_t,q)\leq2C(Ae^{-\lambda t}+E_t)\).
Proof
Use Lemma 182 for Hellinger distance and Lemma 264 for transport. The complete kinetic and conditioned-QSD entropy proofs, including their distinct reference laws and normalization terms, are in Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm and Logarithmic Sobolev inequalities and entropy convergence. Their population-independent constants pass to these inequalities without a global upper bound on \(dp_t/dq\).
Now assemble the pieces. The lower mass bound controls the difference of square roots by the difference of masses. The upper mass bound controls the factor multiplying the shape distance. The transport contribution adds its LSI constant. Every term has a visible origin, so a fluctuation or approximation floor in an input remains visible in the result.
Theorem 252 (Joint convergence of mass and shape)
Let \(\mu_t=m_tp_t\) and \(\mu_*=m_*q\), with \(0<m_0\leq m_t,m_*\leq m_1\). Suppose
and \(q\) has LSI constant \(C\). Then
The same right side bounds \(\mathbb E\mathrm{HK}^2(\mu_t,\mu_*)\).
Proof
The exact mass-shape identity and the lower mass bound control the squared root-mass difference by \((m_t-m_*)^2/(4m_0)\). Since \(\sqrt{m_tm_*}\leq m_1\), its remaining term is at most \(m_1H^2(p_t,q)\). Apply the preceding entropy-to-Hellinger and transport inequalities, then take expectations. Finally \(\mathrm{HK}\leq H\leq D\). All defects remain in the estimate; if both vanish the squared distances decay with rate \(\min(\lambda_m,\lambda_h)\).
For two normalized laws the mass term vanishes. In particular, compare the swarm’s law conditioned on survival to its QSD at each finite time. Both are probabilities on the same configuration space. The survival factor belongs in the evolution that produced the entropy estimate, not in a fictitious conditioning event at infinite time.
Theorem 253 (QSD convergence at finite conditioning horizons)
Let \(p_t=\mathcal L(S_t\mid\tau_\dagger>t)\) and let \(\pi\) be its QSD. Whenever the normalized QSD entropy estimate of Theorem 287 supplies \(D_{\mathrm{KL}}(p_t\Vert\pi)\leq Ae^{-\lambda t}\),
If the specified full reference law also has LSI constant \(C\), then \(D^2(p_t,\pi)\leq(1+2C)Ae^{-\lambda t}\).
Proof
Both laws have mass one. Apply the entropy inequalities and the pure-reaction HK bound. These are estimates between conditional probability laws at time \(t\). They do not condition on survival forever: under a QSD with killing rate \(\lambda_\dagger>0\), survival to \(t\) has probability \(e^{-\lambda_\dagger t}\) and infinite survival has probability zero.
Density Estimates and Their Domains#
The entropy argument already supplies the preceding convergence bounds. Density estimates answer additional questions: whether a law has a density, where it is positive, and how its values compare with a reference density. Keep track of the set on which each estimate holds.
A positive Gaussian kernel gives a positive contribution everywhere, but its lower bound becomes small far from its source. On a bounded observation set, source mass and distance give an explicit lower bound. A global comparison to a reference measure instead uses an order relation that the transition kernel can propagate.
Definition 592 (Domination by a reference measure)
For probability measures \(\mu,\pi\), write \(\mu\leq M\pi\) if \(\mu(A)\leq M\pi(A)\) for every measurable set \(A\). This is equivalent to \(\mu\ll\pi\) and \(d\mu/d\pi\leq M\) almost everywhere. A pointwise bound on a density relative to Lebesgue measure is a different assertion.
Theorem 254 (Propagation of density domination)
If \(P\) is a conservative kernel with \(\pi P=\pi\) and \(\mu\leq M\pi\), then \(\mu P^j\leq M\pi\) for every \(j\). For a killed kernel \(Q\) with \(\pi Q=\alpha\pi\) and \(\mu Q^j1>0\),
If additionally \(\mu\geq m\pi\), \(m>0\), the normalized bound is \(M/m\).
Proof
Positive kernels preserve order. Iterate \(\mu Q^j\leq M\alpha^j\pi\) and normalize. The lower order bound gives \(\mu Q^j1\geq m\alpha^j\). For \(P\), take \(\alpha=1\) and use conservation.
Velocity noise reaches position through transport. Differentiating the transport coefficient \(v\) in a noisy velocity direction produces a position direction. The bracket calculation records this mechanism exactly. Smoothness from that local mechanism still has to be distinguished from a quantitative kernel comparison on a prescribed region.
Lemma 187 (Kinetic bracket calculation)
For constant nondegenerate velocity noise and smooth drift, put \(X_i=\partial_{v_i}\) and \(Y=v\cdot\nabla_x-(\gamma v+\nabla U)\cdot\nabla_v\). Then \([X_i,Y]=\partial_{x_i}-\gamma\partial_{v_i}\); the noise fields and these brackets span the phase tangent space.
Proof
Differentiate the coefficients of \(Y\) with respect to \(v_i\);
\(X_i\) has constant coefficients. The resulting \(2d\) fields span because the
\(x\) components of the brackets form the identity matrix. The local
hypoelliptic consequence under smooth coefficients is stated and applied in
thm-uniqueness-hypoelliptic-regularity.
Theorem 255 (A kernel comparison estimate)
If transition densities at two specified points and positive times satisfy \(k_s(z_0,y)\leq C k_t(z_1,y)\) for almost every \(y\), then for every nonnegative datum \(f\), \(\int k_s(z_0,y)f(y)dy\leq C\int k_t(z_1,y)f(y)dy\). For a compact set of arguments, such a constant follows from a finite upper bound on the first kernel and a strictly positive lower bound on the second.
Proof
Multiply the kernel inequality by \(f\geq0\) and integrate. The compact-set statement follows by dividing the two extrema. Local hypoellipticity alone does not supply a global comparison constant on an unbounded phase space or across an absorbing boundary.
Lemma 188 (Upper density bound from a transition kernel)
If a sub-Markov kernel has density \(k(z,y)\leq K(y)\) and its survival mass from \(\mu\) is at least \(s>0\), then the normalized output has density at most \(K(y)/s\). A mixture containing a singular unsmoothed branch need not have a Lebesgue density.
Proof
Integrate \(k(z,y)\) against the probability measure \(\mu(dz)\) and divide by the output mass. For the second assertion, a positive coefficient of a point mass remains singular under addition of an absolutely continuous measure.
To get a numerical lower bound, reserve some source mass in a specified set \(C\). If the observation point remains at a controlled distance from \(C\), every piece of that reserved mass contributes at least the same Gaussian value. The source mass and distance remain in the bound because either can make the observed density small.
Lemma 189 (Gaussian lower bound on specified bounded sets)
For \(g_\sigma(y-x)=(2\pi\sigma^2)^{-D/2}e^{-|y-x|^2/(2\sigma^2)}\) and sets \(K,C\) with \(\sup_{y\in K,x\in C}|y-x|\leq R\), every positive finite measure \(\mu\) satisfies
Proof
Restrict the convolution integral to \(C\) and use the displayed pointwise Gaussian lower bound. The constant keeps both the source mass \(\mu(C)\) and the distance \(R\).
Lemma 190 (Positivity of a Gaussian cloning contribution)
If a transition has a positive Gaussian cloning contribution of total mass \(w>0\), that contribution has a strictly positive smooth density everywhere in the ambient Euclidean space. Its derivatives are bounded by \(w\) times the corresponding Gaussian derivative suprema. This statement describes the contribution, not an unsmoothed remainder of the full transition.
Proof
Gaussian values are positive, so their integral against a nonzero positive measure is positive. Each derivative of a fixed Gaussian is a polynomial times a Gaussian and is bounded. Differentiate under the integral using this bound.
Lemma 191 (Positive QSD density from a positive transition density)
If \(Q^j\) has positive density \(k_j(z,y)\) and \(\pi Q^j=\alpha^j\pi\), then \(\pi\) has density \(\alpha^{-j}\int k_j(z,y)\pi(dz)>0\) almost everywhere. A continuous positive version has a positive minimum on each compact set.
Proof
Use the eigenmeasure identity and Tonelli. Continuity plus positivity on a compact set gives a positive minimum. A probability density cannot have a strictly positive global lower bound on a set of infinite Lebesgue volume, since its integral would then be infinite.
Theorem 256 (Local density ratios and global reference domination)
If \(p_t(y)\leq B_K\) and \(q(y)\geq b_K>0\) on a specified set \(K\), then \(p_t/q\leq B_K/b_K\) there. A global time-uniform ratio follows instead from the reference-domination hypotheses of Theorem 254, or from separately proved global weighted density estimates with matching tails.
Proof
Divide the local bounds. For the global order statement apply the cited kernel proof. Compact-set bounds cannot be extended globally by replacing their positive minimum with an infimum that may be zero. The entropy proof of Theorem 252 does not require this ratio estimate.
Density estimates in a declared region
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Linearization and Stability Estimates#
Near an equilibrium, split the equation into its first-order response and a quadratic remainder. For a normalized killed equation, normalization itself contributes to both pieces. Omitting it would mean linearizing a different evolution.
The remaining estimates separate three tasks. A semigroup bound controls the linear dynamics, a perturbation bound shows how much of that decay survives, and a smoothing estimate converts convergence in an integral norm into pointwise control. Each task uses a different hypothesis.
Lemma 192 (Linearization on the correct mass space)
Let \(F\) be twice Fréchet differentiable on a neighborhood of an equilibrium \(u_*\) in a Banach space, with \(F(u_*)=0\) and \(\|D^2F\|\leq M\) there. Then
For a normalized killed equation with bounded killing \(c\), \(F(f)=A^*f-cf+(\int cf)f\), its derivative on zero-mass perturbations is
and its remainder is \((\int ch)h\).
Proof
Twice integrate the derivative along \(u_*+sh\), \(0\leq s\leq1\): \(R(h)=\int_0^1(1-s)D^2F(u_*+sh)[h,h]ds\). For the normalized killed equation, multiply out its quadratic normalization term. The QSD solves \(A^*q-cq=-(\int cq)q\), so \(F(q)=0\). For an alive measure of varying mass, a normalization expansion uses the signed increment \(\int h\), rather than \(\|h\|_1\).
Lemma 193 (Linearized decay with an explicit bounded perturbation)
Suppose the zero-mass semigroup of \(A\) satisfies \(\|T_t\|\leq K e^{-at}\) and a mass-preserving bounded perturbation \(B\) has norm at most \(b\). Then the perturbed semigroup obeys \(\|S_t\|\leq K e^{-(a-Kb)t}\). It decays when \(a>Kb\).
Proof
The variation-of-constants formula gives
\(\|S_th\|\leq Ke^{-at}\|h\|+Kb\int_0^te^{-a(t-s)}\|S_sh\|ds\).
Multiply by \(e^{at}\) and apply Grönwall. For the full nonlinear mean-field
flow the corresponding global Lipschitz estimate and attraction theorem
are proved in thm-uniqueness-uniqueness-stationary-solution.
A finite-particle QSD mixing estimate is not itself this linearized
mean-field operator estimate.
Lemma 194 (Boundedness of nonlocal integral terms)
For \(Bh(y)=\int k(x,y)h(x)dx-a(y)h(y)\), if \(\sup_x\int|k(x,y)|dy\leq K_1\) and \(\|a\|_\infty\leq K_2\), then \(\|Bh\|_1\leq(K_1+K_2)\|h\|_1\). The complete derivative of a nonlinear kernel includes its derivative with respect to the law, which requires its own analogous bound.
Proof
Apply Tonelli and the triangle inequality to the integral term, then the multiplication-operator bound. The law-dependent derivative is obtained by the product rule; its bound is not included in \(K_1\) unless explicitly verified.
Lemma 195 (Local regularity under smooth lower-order sources)
Suppose a kinetic operator \(P\) has the local regularity implication \(Pu\in H^s_{\mathrm{loc}}\Rightarrow u\in H^{s+\delta}_{\mathrm{loc}}\) for some \(\delta>0\), and a lower-order operator \(B\) preserves every local Sobolev class under consideration. A distributional solution of \(Pu=Bu\) that starts in one such class is locally smooth.
Proof
If \(u\in H^s_{\mathrm{loc}}\), then \(Bu\) belongs to that class; the specified estimate gains \(\delta\) derivatives. Iterate to obtain every finite Sobolev order and use local Sobolev embedding. The kernel and coefficient regularity needed for \(B\) must hold at every iteration; an arbitrary bounded operator on \(L^1\) is not enough for this bootstrap.
The last conversion has a simple time interpretation. Let the dynamics first reduce the integral error, then reserve a fixed interval \(\delta\) to smooth what remains. The smoothing constant depends on that interval. To turn the result into a density-ratio estimate on a set, one also needs a positive lower bound for the reference density on that set.
Lemma 196 (Fixed-time smoothing after convergence in an integral norm)
Let an evolution \(\Phi_t\) fix \(q\), satisfy \(\|\Phi_tf-q\|_1\leq Ke^{-at}\|f-q\|_1\), and have a fixed-time estimate \(\|\Phi_\delta f-\Phi_\delta g\|_\infty\leq C_\delta\|f-g\|_1\) on the invariant class under study. Then for \(t\geq\delta\),
Proof
Apply the smoothing estimate to \(\Phi_{t-\delta}f\) and \(q\), then the integral-norm estimate. A corresponding density-ratio estimate follows on each set where \(q\) has a known positive lower bound. Its constants retain that set dependence.
Consequences for the Volume#
The main bound combines the mass balance with entropy convergence for the same measure and reference. Its uniformity in particle number is inherited from those inputs. The empirical measure of a simulation uses the separate transport and observable estimates, retaining the sampling error. This is how the mass, entropy, and finite-particle analyses fit together.
Theorem 257 (Convergence of mass, shape, and transport)
The mass estimate Lemma 183 and the entropy estimates in Logarithmic Sobolev inequalities and entropy convergence, whenever they apply to the same alive law and reference, imply the quantitative bound of Theorem 252. Its constants are independent of \(N\) when the input mass lower bound, LSI constant, entropy rates, and defect bounds are independent of \(N\).
For normalized QSD laws, Theorem 253 gives Hellinger and canonical HK convergence directly. For finite empirical measures, the transport and observable bounds in Quantitative Error Bounds retain their sampling error.
Proof
Substitute the proved input inequalities into the mass-shape identity and the entropy-to-transport bound. No step in that substitution introduces a new population factor. The empirical statement follows from Proposition 190 and Lemma 266.