Logarithmic Sobolev inequalities and entropy convergence#
Overview#
An entropy estimate answers a distributional question: how much information about the initial law survives after the swarm evolves? A variance estimate answers a different question: how widely spread are the walkers? A cloud can have the right variance and the wrong distribution. To pass from one question to the other, we need a functional inequality for a specified probability law.
This chapter develops that argument in three stages. First we prove logarithmic Sobolev inequalities, including bounds independent of the number of particles. Next we show how velocity diffusion and position transport combine to dissipate a modified entropy. Finally we incorporate the cloning operator and the normalization required when conditioning on survival. Each stage keeps its reference measure explicit.
The proofs apply on unbounded phase space with a confining potential. Nonconvex potentials are included through quadratic radial confinement and a lower Hessian bound, or through a bounded perturbation of a convex confining potential. The constants retain the dependence on the potential and the interaction strength; increasing the population does not remove that dependence.
Reading guide
The main static estimate is Corollary 94. The kinetic argument is proved in Lemma 237 and Theorem 285. The exact entropy balance for a killed process is Proposition 178; the full evolution is assembled in Theorem 287.
The algorithm and its finite-step kernel are specified in Euclidean Gas: Canonical Transition and Operator Estimates and Latent Fractal Gas: Algorithm and Analytic Conditions. The complementary drift, mean-field, and regularity estimates are developed in Convergence, Survival, and Parameter Dependence, The Mean-Field Law of the Euclidean Gas, The Discrete Population Limit and Propagation of Chaos, and The Geometric Gas: Regularity, Stability, and Entropy.
1. Probability laws and entropy#
Definition 608 (Finite-particle laws and survival conditioning)
Fix a population size \(N\). Let \(E_N\) be the nonabsorbed swarm state space. A conservative Markov semigroup \(P_t\) has an invariant probability law \(\pi_N\) when
For a killed process, write
A quasi-stationary distribution \(\nu_N\) satisfies
The law conditioned on survival is
In discrete time, replace \(Q_t\) by a sub-Markov kernel \(Q^n\) and \(e^{-\lambda_Nt}\) by \(\vartheta_N^n\), where \(\nu_NQ=\vartheta_N\nu_N\) and \(0<\vartheta_N\leq1\).
A one-particle marginal of \(\nu_N\), a limiting mean-field stationary law, and an invariant law of a process that revives killed walkers are separate probability measures. Their identification requires an equation or a limiting theorem.
Definition 609 (Relative entropy and full Fisher information)
For probability measures \(\mu\ll\pi\) on a continuous phase space, put \(h=d\mu/d\pi\) and
For an unnormalized nonnegative function \(g\), define
For \(S=(x_1,v_1,\ldots,x_N,v_N)\), the full Fisher information is
The integral is understood by lower semicontinuous extension when necessary. Relative entropy is \(+\infty\) if absolute continuity fails.
Definition 610 (Full-gradient logarithmic Sobolev inequality)
A probability law \(\pi\) satisfies an LSI with constant \(C_{\mathrm{LSI}}\) if
for every function in the associated Sobolev space. Equivalently, for probability densities \(h\),
For \(N\) particles, the gradient in (15.1) is the sum of the position and velocity gradients over all particles. An \(N\)-uniform LSI means that the same upper bound for \(C_{\mathrm{LSI}}\) works for all \(N\) in the stated family of laws.
Remark 212 (Velocity diffusion and the gradient in an LSI)
For the kinetic generator
the actual carré du champ is
An LSI with only this gradient cannot hold for every phase-space function under a law with a nontrivial spatial marginal: take \(f=f(x)\) nonconstant. Its entropy can be positive while \(\Gamma_L(f,f)=0\).
The full-gradient inequality (15.1) is a property of the measure. Nonreversibility does not prevent that inequality. It changes how the dynamics use it, which is the purpose of the hypocoercive calculation below.
Imagine perturbing only the spatial density while leaving the conditional velocities unchanged. At the first instant, velocity noise sees no error. Transport then turns spatial variation into position-velocity correlations, and diffusion can act. This delay explains why the useful quantity contains gradients and a cross term, rather than entropy alone.
Proposition 176 (Entropy decomposition across status strata)
Suppose a joint law has the disintegration \(\pi(ds,dz)=p_s\pi_s(dz)\) over alive/dead status patterns \(s\). For \(g_s=\bigl(\pi_s(f^2)\bigr)^{1/2}\),
Consequently, full-gradient LSIs on the continuous strata control the first term. Controlling the entropy of the entire status-bearing law also requires a functional inequality for the second term, with a discrete form connecting the status patterns.
Proof
Insert \(\log(f^2/\pi(f^2))=\log(f^2/\pi_s(f^2))+\log(\pi_s(f^2)/\pi(f^2))\) into the entropy integral and sum over \(s\). A function constant on each stratum has zero continuous gradient, so the second term cannot be omitted when more than one status has positive probability.
2. Complete proofs of the static LSI#
2.1. Curvature#
Theorem 279 (Bakry-Émery criterion)
Let \(\pi(dz)=Z^{-1}e^{-V(z)}dz\) on \(\mathbb R^m\), with \(V\in C^2\) and
Then (15.1) holds with \(C_{\mathrm{LSI}}\leq\rho^{-1}\).
Proof
Use the auxiliary elliptic generator \(\mathscr L=\Delta-\nabla V\cdot\nabla\), which preserves \(\pi\). For a smooth positive probability density \(h\), let \(h_t=e^{t\mathscr L}h\) and \(u_t=\log h_t\). Integration by parts gives
Differentiating the Fisher information, using \(\partial_tu_t=\mathscr Lu_t+|\nabla u_t|^2\), gives the exact identity
For completeness, the local identity underlying (15.5) is
Apply the product rule to \(\int h_t|\nabla u_t|^2d\pi\); the terms containing \(\nabla|\nabla u_t|^2\) cancel after integration by parts, leaving (15.5). Thus \(I_\pi(h_t)\leq e^{-2\rho t}I_\pi(h)\).
For bounded \(h\) bounded away from zero, ergodicity of this uniformly confining elliptic diffusion and bounded convergence give \(H_\pi(h_t)\to0\). One can obtain this ergodicity directly from synchronous coupling: strong convexity implies contraction of two trajectories by \(e^{-\rho t}\). Integrating (15.4) therefore yields
Apply this to \(h=f^2/\pi(f^2)\). Truncation, positive regularization, and lower semicontinuity extend the result to the Sobolev domain. Smooth approximation of \(V\) gives the stated \(C^2\) formulation.
2.2. Tensorization#
Theorem 280 (Tensorization of the logarithmic Sobolev inequality)
If \(\pi_i\) satisfies (15.1) with constant \(C_i\), then \(\pi=\bigotimes_{i=1}^N\pi_i\) satisfies it with
The test function \(f\) may depend jointly on every coordinate; it need not factorize.
Proof
For two factors \(\pi=\mu\otimes\nu\), set \(g(x)=\bigl(\int f(x,y)^2\nu(dy)\bigr)^{1/2}\). The entropy chain rule gives
Cauchy-Schwarz implies
with the zero-denominator case handled by regularization. Apply the two LSIs and iterate. The resulting bound is
which proves (15.6). The product structure is required of the reference measure, not of the evolving density.
2.3. Bounded perturbations and nonconvex potentials#
Theorem 281 (Bounded-density perturbation of an LSI)
Suppose \(\mu\) satisfies (15.1) with constant \(C_0\) and \(\pi=w\mu\), where
Then
In particular, if \(d\pi=Z_B^{-1}e^{-B}d\mu\) and \(\operatorname{osc}(B)=\sup B-\inf B<\infty\), then
Proof
For \(g\geq0\) and \(c>0\), set \(\psi_c(g)=g\log(g/c)-g+c\geq0\). Minimizing over \(c\) gives
Choose \(c=\mu(g)\). Then
For the exponential tilt, the ratio of the upper and lower density bounds is \(e^{\operatorname{osc}(B)}\); its normalization cancels.
Empirical density in a nonconvex reward landscape
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2.4. Entropy, transport, and quadratic confinement#
Theorem 282 (Entropy-transport-Fisher inequality)
Let \(\pi(dx)=Z^{-1}e^{-V(x)}dx\) have finite second moment and \(\nabla^2V\succeq-KI\) for \(K\geq0\). For a probability law \(\mu\) with finite relative Fisher information and finite second moment,
For \(\nabla^2V\succeq\kappa I\) with \(\kappa\geq0\), the final term is instead \(-\kappa W_2^2/2\).
Proof
First consider smooth densities and a regular optimal gradient transport \(T\) from \(\mu\) to \(\pi\). Put \(T_s=(1-s)\operatorname{id}+sT\) and \(\mu_s=(T_s)_\#\mu\). The change-of-variables formula expresses the entropy as the sum of an internal term containing \(-\log\det((1-s)I+sDT)\) and a potential term \(\int V(T_s(x))\mu(dx)\). The internal term is convex in \(s\), because \(DT\) is positive semidefinite and \(-\log\det\) is convex. The second derivative of the potential term is at least \(-K\int|T-x|^2d\mu\).
Thus, with \(H(s)=D_{\mathrm{KL}}(\mu_s\Vert\pi)\),
The continuity equation for the interpolation gives
Cauchy-Schwarz proves the claim. A positive lower curvature bound gives the stated negative term by the same calculation. The usual smooth approximation of the transport interpolation extends the inequality to the finite-information domain; equivalently, the argument is the integrated above-tangent inequality for entropy.
Lemma 235 (Weighted estimate from an elliptic Lyapunov function)
Let \(\mathscr L=\Delta-\nabla V\cdot\nabla\) preserve \(\pi\), and suppose a smooth positive function \(W\) satisfies
Then
In particular, \(\pi(|x|^2)\leq b/c\).
Proof
For compactly supported smooth \(f\), integration by parts and completion of the square give
Insert the Lyapunov inequality. Cutoff approximation extends the estimate to its form domain; taking cutoffs increasing to one gives the moment bound by Fatou’s lemma.
Theorem 283 (LSI from a quadratic Lyapunov bound)
Let \(\pi(dx)=Z^{-1}e^{-V(x)}dx\) on \(\mathbb R^d\), with \(V\in C^2\) and \(\nabla^2V\succeq-KI\) for some finite \(K\geq0\). Suppose the Lyapunov bound of Lemma 235 holds. Then \(\pi\) satisfies a full-gradient LSI, without a convexity assumption on \(V\).
One finite constant can be constructed as follows. Choose \(R\) with \(\ell=cR^2-b>0\), and let \(C_{\mathrm{loc}}\) be a Poincaré constant for \(\pi\) conditioned on the ball \(B_R\). Define
For any \(\varepsilon>0\), put
Then
Proof
Poincaré inequality. For a smooth \(f\), subtract its mean on \(B_R\) and call the resulting function \(g\). Since \(\operatorname{Var}_\pi(f)\leq\pi(g^2)\), it suffices to bound \(\pi(g^2)\). The weighted estimate gives
The local Poincaré inequality controls the last integral. Adding the inside and outside contributions yields \(\operatorname{Var}_\pi(f)\leq C_P\int|\nabla f|^2d\pi\). The constant \(C_{\mathrm{loc}}\) is finite: the density has positive upper and lower bounds on the ball, so the usual Poincaré inequality on a ball transfers by comparison of measures.
Defective entropy inequality. For \(\mu=h\pi\), apply the weighted estimate to \(\sqrt h\):
The independent coupling of \(\mu\) and \(\pi\) gives
Substitute into Theorem 282 and use \(\sqrt{B_0I}\leq\varepsilon I+B_0/(4\varepsilon)\). It follows that
By homogeneity this is \(\operatorname{Ent}_\pi(f^2)\leq4A_1\int|\nabla f|^2d\pi+B_1\pi(f^2)\).
Removal of the defect. The classical Rothaus centering inequality states
Apply the defective inequality to \(f-\pi(f)\), then use the Poincaré bound already proved. This gives
which is the claimed LSI. The transport-and-centering argument and its classical ingredients are also given in Cattiaux, Guillin, and Wu, Section 3.3.
Corollary 92 (The confining potential supplies the Lyapunov function)
Suppose \(U\in C^2(\mathbb R^d)\) satisfies
Then the spatial Gibbs law proportional to \(e^{-U/\theta}\) has an LSI constant \(C_x<\infty\). In Theorem 283, one may take
A radial coercivity bound holding outside a ball gives the displayed global bound after increasing \(b_U\) to control the compact interior.
Proof
Radial integration of the coercivity inequality gives quadratic lower growth of \(U\), up to a logarithmic term and a bounded interior contribution. Hence the Gibbs law is normalizable. For \(\mathscr L=\Delta-\theta^{-1}\nabla U\cdot\nabla\),
The stated \(s\) gives the stated positive \(c\). Apply Theorem 283. All constants concern the one-particle spatial law; the local Poincaré constant retains dependence on the interior landscape.
The Lyapunov estimate prevents probability from escaping to infinity. The local Poincaré estimate controls the remaining bounded region, including any barriers between wells. Both enter the constant. This is why confinement can replace global convexity, while the depth and shape of the interior wells still affect the rate.
2.5. The kinetic reference and its products#
Definition 611 (Kinetic Gibbs reference measure)
Write
and define
This is the invariant law of the conservative kinetic Langevin dynamics with potential \(U\). It is a reference law for the entropy estimates, and is not by definition the invariant or quasi-stationary law after cloning is added.
Theorem 284 (LSI for the kinetic reference measure)
Suppose the spatial Gibbs law has LSI constant \(C_x\), as proved by either the bounded-perturbation argument or Corollary 92. Then \(m_U\) satisfies a full-gradient LSI with
For \(U=W+B\) with \(\nabla^2W\succeq\kappa I_d\) and bounded \(\operatorname{osc}(B)\), one may take \(C_x=\theta e^{\operatorname{osc}(B)/\theta}/\kappa\). In particular, the conclusion allows nonconvex \(U\) and applies on \(\mathbb R^d\times\mathbb R^d\).
Proof
For the bounded-perturbation case, apply Theorem 279 to \(W/\theta\), then Theorem 281 to its tilt by \(e^{-B/\theta}\). For the confinement case, use the proof of Corollary 92. The Gaussian velocity law \(\mathcal N(0,\theta I_d)\) has constant \(\theta\). Tensorization proves (15.9).
Corollary 93 (Product kinetic LSI independent of population size)
For every \(N\geq1\), the reference law \(m_U^{\otimes N}\) satisfies (15.1) with the same \(C_0\) from (15.9).
Proof
Apply Theorem 280 to the \(N\) copies of \(m_U\).
The distinction between a one-particle perturbation and a whole-swarm perturbation matters. A bounded nonconvex feature in each particle’s potential is handled before tensorization, so its cost is paid once in the constant. If instead we compare the entire swarm density to a product density in one step, the oscillation of the whole logarithmic density ratio is the quantity we must bound.
Corollary 94 (N-uniform LSI for specified joint laws)
Let \(\pi_N\) be a family of continuous joint probability laws on \(\mathbb R^{2dN}\). Each of the following gives an \(N\)-uniform full-gradient LSI:
Product reference: \(\pi_N=m_U^{\otimes N}\) under Theorem 284; take \(C_*=C_0\).
Bounded joint tilt: \(d\pi_N=Z_N^{-1}e^{-B_N}dm_U^{\otimes N}\) with \(\sup_N\operatorname{osc}(B_N)\leq B_*<\infty\); take \(C_*=e^{B_*}C_0\).
Joint curvature: \(d\pi_N=Z_N^{-1}e^{-V_N}dS\) with \(\nabla^2V_N\succeq\rho_* I_{2dN}\) uniformly in \(N\), for \(\rho_*>0\); take \(C_*=1/\rho_*\).
Contractive additive-noise flow: \(\pi_N\) is the invariant law in Theorem 296, with \(\lambda_{\min}(Q_N)\geq q_*>0\), \(r_N\geq r_*>0\), and \(c_{Q_N}\leq c_*<\infty\); take \(C_*=2c_*/(r_*q_*)\).
Thus, in the respective cases,
This conclusion applies to a swarm invariant law or a QSD when that particular law satisfies one of the stated structural hypotheses. For a law carrying discrete status variables, (15.3) specifies the additional entropy term.
Proof
The first case is Corollary 93. Apply (15.7) to obtain the second. Apply Theorem 279 in dimension \(2dN\) to obtain the third. The fourth follows from the flow and entropy-interpolation proof of Theorem 296. None of these proofs introduces a factor of \(N\) in the constant.
Proposition 177 (A normalized interaction with an explicit curvature bound)
Let
where \(\nabla^2V_0\succeq\rho_0I\), \(\varepsilon\geq0\), and the Hessian of \(W\) in both variables satisfies \(\|\nabla^2 W\|_{\mathrm{op}}\leq L_W\). If \(\varepsilon L_W<\rho_0\), then
uniformly in \(N\).
Proof
For \(\xi=(\xi_1,\ldots,\xi_N)\),
Add the lower bound for \(\sum_iV_0\) and apply the curvature case of (15.10).
Remark 213 (Identifying the law in the static estimate)
Stationarity of the kinetic Gibbs law does not imply stationarity under cloning. For a conservative full generator \(L=L_{\mathrm{kin}}+J\), a proposed density \(\pi\) must satisfy \(L^*\pi=0\). For a killed generator \(A\), a proposed QSD must satisfy \(A^*\nu=-\lambda\nu\), together with the relevant boundary conditions.
Exchangeability or propagation of chaos does not supply either density equation or an \(N\)-uniform bound on the joint density ratio. In particular, \(\operatorname{osc}(B_N)\leq Nb\) in the second case of (15.10) yields the bound \(e^{Nb}C_0\), not an \(N\)-uniform constant.
3. The kinetic entropy calculation#
3.1. The modified functional#
Definition 612 (Hypocoercive quadratic form)
For constants \(a,c>0\) and \(ac>b^2\), let
Define
If \(g_-I\preceq G\preceq g_+I\), then
An LSI with constant \(C_*\) consequently gives
The cross term records whether the position error and velocity error point in related directions. It may have either sign. The condition \(ac>b^2\) ensures that the total gradient contribution remains positive. We will choose all three coefficients explicitly, so this positivity is part of the proof.
3.2. Exact commutator identities#
Lemma 236 (Kinetic Fisher-information identities)
Let \(\pi=m_{U_N}\) be a kinetic Gibbs law on \(\mathbb R^{2dN}\) with potential \(U_N(x)\). Suppose \(\|\nabla^2U_N\|_{\mathrm{op}}\leq M\) globally. For the relative density \(h=f/\pi\), the forward kinetic equation becomes
For smooth positive solutions with justified integration by parts,
and, writing \(u=\log h\) and \(H_U=\nabla^2U_N\),
Proof
The fluctuation-dissipation relation \(D=\gamma\theta\) and density (15.8) give (15.14) by expanding \(\pi^{-1}L_{\mathrm{kin}}^*(\pi h)\). The velocity part is symmetric in \(L^2(\pi)\), and the Hamiltonian part is antisymmetric. This proves (15.15).
The derivative commutators are
More explicitly, put
Differentiating \(I_G\) and integrating by parts gives
To see the cancellation in this identity, use \(\partial_tu=\mathcal Ku+D|\nabla_vu|^2\) and \(\nabla\mathcal Ku=\mathcal Kq+Bq\) in the derivative of \(\int h q^{\mathsf T}Gq\,d\pi\). The product rule and invariance replace the terms containing \(\mathcal K\) by \(-2D\sum_j(\partial_{v_j}q)^{\mathsf T}G(\partial_{v_j}q)\); the derivative of \(D|\nabla_vu|^2\) cancels the corresponding first-gradient cross terms. The remaining drift terms are precisely \(GB+B^{\mathsf T}G\).
Taking separately the velocity, cross, and position entries of (15.17) proves (15.16). The three second-derivative terms must be retained together: their combination is nonpositive because \(G\) is positive definite.
3.3. An explicit choice of coefficients#
Lemma 237 (Dissipation of the modified entropy)
Under Lemma 236, set
Then \(G=\eta\begin{pmatrix}2I&I\\I&2I\end{pmatrix}\) has eigenvalues \(\eta\) and \(3\eta\), and
Proof
The Hessian terms in (15.16) satisfy
Also \(|I_{xv}|\leq\sqrt{I_xI_v}\). Combine (15.15)–(15.17), keeping the nonpositive second-derivative quadratic form, to obtain
Young’s inequality in the form
therefore yields
The choice (15.18) gives \(\eta(2M+L_M^2)\leq D/2\) and \(\eta\leq D/2\). This proves (15.19). In particular, neither the positive definiteness of \(G\) nor the dissipation estimate uses an asymptotic approximation.
Theorem 285 (Kinetic hypocoercive entropy convergence)
Suppose the kinetic Gibbs law \(\pi=m_{U_N}\) satisfies a full-gradient LSI with constant \(C_*\) and \(\|\nabla^2U_N\|_{\mathrm{op}}\leq M\). Let \(G\) and \(\eta\) be given by (15.18). Then
The estimate holds for initial densities with finite \(\Phi_G\), and after a positive time \(t_0\) for any solution for which \(\Phi_G(h_{t_0})<\infty\). If \(C_*\) and \(M\) are independent of \(N\), then \(r\) is independent of \(N\).
Proof
Combine (15.13) and (15.19) to obtain \(\dot\Phi_G\leq-r\Phi_G\). Grönwall’s inequality proves (15.20). Approximation extends the smooth calculation to finite-\(\Phi_G\) data. Applying the same argument with initial time \(t_0\) gives the positive-time statement.
Corollary 95 (Nonconvex product potentials)
Let \(U_N(x)=\sum_iU(x_i)\), where \(U\) satisfies Theorem 284 and \(\sup_x\|\nabla^2U(x)\|_{\mathrm{op}}\leq M\). Then (15.20) holds with \(C_*=C_0\) from (15.9), uniformly in \(N\). Global convexity of \(U\) is not required.
Proof
The full Hessian of \(U_N\) is block diagonal with operator norm at most \(M\). The reference law is \(m_U^{\otimes N}\), whose LSI is Corollary 93. Apply (15.20).
This proof retains both pieces of the classical argument: an inequality for the equilibrium law and a dynamical calculation that transfers velocity dissipation to position. The first gives \(C_*\); the second gives \(\eta\). A small coefficient in front of Fisher information does not make that information bounded by entropy. That is why the initial factor in (15.20) is the modified entropy \(\Phi_G(h_0)\).
The method is the entropic hypocoercivity construction described in Villani’s monograph. The calculation above supplies the coefficients and proof needed here.
4. Cloning, killing, and the full evolution#
4.1. The generator and its target law#
Definition 613 (Full diffusion-jump generator)
On a continuous swarm state space, a conservative diffusion-jump realization has backward generator
For the Euclidean kinetic part,
The jump kernel \(r_N\) includes the entire selected cloning update: companion sampling, acceptance, position jitter, and the coupled velocity update. It need not factorize over offspring. Additional geometric forces and diffusion enter \(b_N\) and \(a_N\) with the corresponding Itô correction when the model is specified in Stratonovich form.
With an interior killing rate \(\kappa_N\geq0\), the killed generator is
For \(\partial_t\widetilde f=A^*\widetilde f\) and \(f=\widetilde f/\int\widetilde f\), on a domain without absorbing boundary flux,
A QSD density \(\nu\) therefore solves
For a discrete algorithm, the object to analyze is its actual sub-Markov one-step kernel \(Q_\tau\). A diffusion-jump generator represents a separately specified continuous-time realization or a justified scaling limit; it is not an exact replacement of a finite BAOAB-and-cloning step.
Conditioning changes the equation because probability is removed and the survivors are renormalized. The last term in (15.22) puts the total mass back to one. It is present even when the killed dynamics are linear. Calling the QSD an ordinary invariant measure would erase precisely this term.
4.2. Exact entropy production, including normalization#
Proposition 178 (Entropy identity for a conditioned diffusion-jump process)
Let \(A=L-\kappa\) be as in (15.21), with bounded nonnegative \(\kappa\), no absorbing boundary flux, and a positive QSD density \(\nu\) satisfying (15.23). Set \(h=f/\nu\) for the normalized solution (15.22), and define
Both functions are nonnegative on their domains. Put
Then, whenever the quantities are finite and the generator identities are justified,
Equivalently, the last two terms are
In particular,
For a conservative invariant law, \(\kappa=0\), this reduces to \(\dot H=-\mathcal D_\nu(h)\), without requiring reversibility.
Proof
Mass preservation of the normalized equation gives
The diffusion chain rule and the jump identity give, pointwise,
Integrating and using (15.23),
Thus
Since \(\int\kappa\psi(h)d\nu=\int\kappa h\log h\,d\nu-\bar\kappa_f+\lambda_N\), this is (15.25). Finally, \(H_\nu(h)=\int\psi(h)d\nu\), \(\psi\geq0\), and \(\bar\kappa_f\leq\sup\kappa\) imply (15.26).
Proposition 179 (Absorbing kinetic boundary)
For kinetic motion in a smooth spatial domain \(\Omega\), let
denote the outgoing boundary, with the analogous sum over particle boundary faces for a swarm. Impose the absorbing incoming trace on \(f\) and \(\nu\). Suppose the outgoing traces and the integrations below exist, and interior jumps do not cross the boundary except through an included killing term. Define
Then the normalized forward equation has source \(\ell_f f\), and
The outgoing trace of \(h\) is \(f/\nu\) wherever \(\nu>0\); the expression is interpreted by its entropy extension otherwise.
Proof
The mass equation gives \(d\int\widetilde f/dt=-\ell_{\widetilde f}\). In the entropy calculation, the transport integration by parts contributes \(-\int_{\partial_+E_N}(v\cdot n)f\log h\). The corresponding eigenmeasure identity is
Use \(\lambda_N=\nu(\kappa)+\int_{\partial_+E_N}(v\cdot n)\nu\), and combine the three boundary terms into \(-\int_{\partial_+E_N}(v\cdot n)\nu\psi(h)\). The interior terms are those of (15.25).
Remark 214 (Boundary and normalization estimates)
The boundary entropy term in (15.27) is nonnegative before its minus sign. The normalization term \(\ell_fH_\nu(h)\) is also present. A moment bound on \(f\) does not remove it or supply a uniform bound on the boundary flux. Soft killing, reflecting boundaries, and absorption at the validity boundary therefore require their respective equations.
4.3. Cloning as a Markov update#
Lemma 238 (Entropy dissipation for a common invariant cloning kernel)
Let \(P\) be a conservative Markov kernel with \(\pi P=\pi\). Then
If the stronger channel estimate \(D_{\mathrm{KL}}(\mu P\Vert\pi)\leq q_JD_{\mathrm{KL}}(\mu\Vert\pi)\) holds with \(q_J<1\), a Poisson jump generator \(J=\omega(P-I)\) dissipates entropy at least at rate \(\omega(1-q_J)\).
Proof
Construct the joint laws \(\mu(dS)P(S,dS')\) and \(\pi(dS)P(S,dS')\). Their relative entropy is \(D_{\mathrm{KL}}(\mu\Vert\pi)\). Marginalizing to \(S'\) cannot increase relative entropy, and the second marginal is \(\pi\). This proves (15.28).
For the last assertion, convexity of relative entropy gives
where \(P^\dagger\) is the density operator relative to \(\pi\). Multiply by \(\omega\).
Lemma 239 (Cloning contribution to the modified functional)
Under the common-invariant-law hypothesis of (15.28), suppose the cloning density operator satisfies
for every density in the form domain. Then the generator \(J=\omega(P-I)\) satisfies
Here (15.29) is a gradient estimate for the full kernel on its stated domain, rather than a consequence of a status-distance contraction.
Proof
The map \((s,p)\mapsto p^{\mathsf T}Gp/s\) is convex for \(s>0\), so \(I_G(h)=\int(\nabla h)^{\mathsf T}G\nabla h/h\,d\pi\) is convex. Consequently,
Add the nonpositive entropy contribution from (15.28), and replace \(A_J-1\) by its positive part.
Theorem 286 (Kinetic and cloning convergence with a common target)
Suppose the conservative kinetic dynamics satisfy Theorem 285 with target \(\pi\) and coefficient \(\eta\). Suppose also that \(\pi P=\pi\) and (15.29) holds for the cloning kernel. If
then the combined generator \(L_{\mathrm{kin}}+\omega(P-I)\) satisfies
The rate is \(N\)-uniform when all the constants in these hypotheses are \(N\)-uniform.
Proof
Add (15.19) and (15.30), and use \(I_G\leq3\eta I_\pi\). This gives \(\dot\Phi_G\leq-\delta_*I_\pi\). Close with (15.13) and apply Grönwall.
The common-target condition can be checked by starting the system at the proposed target and applying each operator. If cloning changes that law, the separate contraction argument does not apply. We must then compute the derivative for the full stationary equation. This is particularly relevant for fitness selection: the kinetic and selection contributions can balance only after they are added.
4.4. The law-dependent full-generator estimate#
Lemma 240 (First variation of modified entropy)
Let \(\nu\) be a fixed positive reference law and \(G\) a constant positive definite matrix. For a differentiable density curve \(h_t\), put \(r_t=\partial_t h_t\) and \(q_t=\nabla\log h_t\). Then
For the normalized killed evolution, the quantity to substitute is exactly
with \(\ell_f=\bar\kappa_f\) in (15.22), and with the total killing flux in (15.27).
Proof
Differentiate \(h\log h\) and \((\nabla h)^{\mathsf T}G\nabla h/h\) under the integral. The quotient rule gives the second line of (15.32). Equation (15.33) follows by differentiating the normalized killed density. No invariance of \(\nu\) under the conservative part is used.
Theorem 287 (Entropy convergence for the full normalized swarm evolution)
Let \(\nu_N\) be the QSD of the specified finite-particle killed dynamics, or let it be the invariant law of a specified conservative dynamics. This statement concerns a continuous state space. An entire law carrying discrete status variables requires the additional status entropy and form from (15.3).
Suppose that, on the solution class under consideration:
The actual law \(\nu_N\) satisfies the full-gradient LSI (15.2) with constant \(C_N\).
A constant positive definite matrix \(G_N\) satisfies \(G_N\preceq g_{+,N}I\).
The full derivative (15.32), with the full source (15.33), satisfies
\[ \frac{d}{dt}\Phi_{G_N}(h_t) \leq-\delta_N I_{\nu_N}(h_t), \qquad \delta_N>0. \tag{15.34} \]
Then
An \(N\)-uniform conclusion follows when \(C_N\leq C_*\), \(g_{+,N}\leq g_*\), and \(\delta_N\geq\delta_*>0\) uniformly in \(N\). The first bound is supplied by Corollary 94 whenever the QSD itself satisfies one of its density hypotheses. The conservative kinetic and common-target cloning cases have the explicit dissipation proofs (15.19) and (15.31).
Proof
The LSI and the upper matrix bound give
Substitute this into (15.34), apply Grönwall, and use \(H_{\nu_N}\leq\Phi_{G_N}\). The stated uniform bounds give \(r_N\geq\delta_* /(C_*/2+g_*)\).
Remark 215 (What must be estimated for the full swarm law)
For an application beyond the explicitly proved cases, (15.34) requires the derivative of the actual cloning kernel and the killing normalization in the same reference law. Bounded fitness values do not by themselves bound derivatives of the kernel. A bound of the form \(c\sqrt{I_x}+bI_x\) cannot be replaced by \(CI_x\) near \(I_x=0\); its square-root term must cancel in the full stationary calculation or remain as a forcing term.
The regularity estimates Lemma 198, Lemma 199, Theorem 258, and Theorem 259 in C³ Regularity of the Fractal Gas Fitness give derivative bounds for the normalized measurements and fitness field under their stated weighted-kernel hypotheses. They are inputs to the full kernel derivative calculation. They do not identify its stationary density or supply (15.34) without that calculation.
5. A second route through the conditioned process#
There is another exact way to handle survival. Weight each state by its future survival amplitude, evolve a conservative transformed process, and undo the weight afterward. This replaces a nonlinear normalization in time by a change of measure. The cost of that change must be retained, especially when the number of particles grows.
Proposition 180 (Doob transform and its invariant law)
Suppose the killed semigroup has a positive right eigenfunction \(\eta_N\) and left eigenmeasure \(\nu_N\) satisfying
Define
Then \(P_t^\eta\) is conservative and preserves \(\widehat\pi_N\). Its generator is
For (15.21), this adds the diffusion drift \(2a_N\nabla\log\eta_N\) and replaces the jump rate by
For the reweighting map \(\mathcal R_w\mu=w\mu/\mu(w)\),
Proof
The right eigenfunction identity gives \(P_t^\eta1=1\). The left eigenmeasure identity gives
Expanding \(A(\eta_NF)\) proves the generator formula and (15.37); the zero-order terms cancel by the eigenfunction equation. Finally,
Reweighting by \(1/\eta_N\) proves (15.38).
Lemma 241 (Relative entropy under bounded reweighting)
If \(0<m\leq w\leq M<\infty\), then
Proof
Apply to both laws the same experiment: accept a point \(S\) with probability \(w(S)/M\). The relative entropy of the joint point-and-acceptance laws is \(D_{\mathrm{KL}}(\alpha\Vert\beta)\). Its conditional chain rule is the sum of a nonnegative Bernoulli entropy and the two conditional entropies, weighted by the acceptance and rejection probabilities under \(\alpha\). The accepted conditional laws are \(\mathcal R_w\alpha\) and \(\mathcal R_w\beta\). Their weight is \(\alpha(w)/M\geq m/M\). Dropping the other nonnegative terms proves (15.39).
Theorem 288 (Transfer of entropy convergence to the QSD)
Under (15.36), suppose \(0<m_N\leq\eta_N\leq M_N<\infty\), and put \(R_N=M_N/m_N\). If the transformed process satisfies
then the original survival-conditioned process satisfies
If a stronger estimate \(H_{\widehat\pi_N}(\widehat h_t)\leq C e^{-rt}H_{\widehat\pi_N}(\widehat h_0)\) is available, then
Proof
Apply (15.39) with \(w=1/\eta_N\) to (15.38), then use the transformed entropy estimate. For (15.41), apply (15.39) a second time with \(w=\eta_N\) to the initial laws.
Remark 216 (Uniformity of the survival change of measure)
The invariant law of the transformed process is \(\widehat\pi_N=\eta_N\nu_N\), not \(\nu_N\). Its LSI must be established for that law. The estimates (15.40)–(15.41) preserve population-uniform constants only when the reweighting costs are also uniform. On an absorbing domain the right eigenfunction may approach zero at the boundary, so the bounded-reweighting argument is not automatic; the direct balance (15.27) remains available.
6. Quadratic-form estimates and parameter conditions#
Definition 614 (Auxiliary gradient form and iterated form)
For \(\lambda>\mu^2\), define
Let \(\mathcal Q(f,g)\) be its polarization. For a linear generator \(L\), define
This is an auxiliary first-derivative form; it is distinct from the actual diffusion-jump carré du champ of \(L\).
Lemma 242 (Additivity in the generator)
For \(L=L_1+L_2\) and a fixed form \(\mathcal Q\),
Proof
Expand (15.42), using linearity of \(L\) and bilinearity of \(\mathcal Q(f,g)\).
Theorem 289 (Exact absorption of a spatial-gradient penalty)
Suppose estimates for the specified generators have established
where \(\beta\geq0\) and \(\varepsilon\geq0\). Then
Proof
Complete the square:
Hence \(|\nabla_xf|^2\leq\mathcal Q(f)/(\lambda-\mu^2)\). Add the two assumed generator estimates and substitute this inequality. Because the spatial penalty has a minus sign, replacing \(1/(\lambda-\mu^2)\) by the smaller \(1/\lambda\) would not be a valid lower bound.
Corollary 96 (Both constraints in a minimum must hold)
If \(\alpha=c_1\min\{\gamma,\alpha_U/\sigma_v^2\}\), define
Then \(\rho_{\mathcal Q}>0\) is equivalent to the two simultaneous conditions
Proof
For real numbers \(a,b,T\), the inequality \(\min\{a,b\}>T\) holds exactly when both \(a>T\) and \(b>T\).
Corollary 97 (Uniformity of the absorbed coefficient)
If the constants in (15.43) satisfy uniform bounds with
then the resulting auxiliary-form bound is uniform in \(N\). Here \(\lambda_N^{\mathrm{form}}\) is the form parameter, distinct from the QSD killing eigenvalue.
Proof
Take the infimum of the coefficient in (15.43).
Remark 217 (Scope of the auxiliary curvature calculation)
The conclusion (15.43) is an algebraic consequence of its two generator inequalities. It does not establish those inequalities. For a jump generator, (15.24) contains a nonlocal entropy remainder, and an estimate for \(\mathcal Q_{2,J}(f)\) must be connected to the entropy/Fisher calculation for that kernel. The complete kinetic entropy calculation is (15.15)–(15.19); the common-invariant cloning estimate is (15.29)–(15.30).
A contraction in a distance counting alive/dead differences cannot provide a bound on spatial derivatives: that distance vanishes between different configurations with the same status pattern. Similarly, a global Lyapunov integral estimate is not a pointwise curvature inequality.
7. What selection and Gaussian smoothing provide#
7.1. Fisher information after Gaussian noise#
Lemma 243 (Gaussian smoothing and absolute Fisher information)
Let \(X\) have any probability law on \(\mathbb R^m\), let \(Z\sim\mathcal N(0,\delta^2I_m)\) be independent, and let \(p\) be the density of \(Y=X+Z\). Then
For a smooth positive reference density \(\pi\),
provided the last integral is finite.
Proof
Differentiating the Gaussian convolution gives the score identity
Conditional Jensen yields
Apply \(|a-b|^2\leq2|a|^2+2|b|^2\) to \(\nabla\log(p/\pi)=\nabla\log p-\nabla\log\pi\) to obtain (15.46).
Remark 218 (Which coordinates have actually been smoothed)
The bounds (15.45)–(15.46) concern a full Gaussian convolution in the stated \(m\) coordinates. They apply to a noisy offspring law in those coordinates. If some walkers are unchanged, or the velocity noise is coupled to conserve group momentum, the entire swarm update is not automatically such a convolution. A joint Fisher estimate must use the covariance and degeneracies of that actual update.
For \(m=2dN\), the absolute Fisher bound is proportional to \(N\). Dividing entropy and Fisher information by \(N\) makes an intensive estimate; it does not by itself prove the joint LSI (15.10).
7.2. Heat flow and its reference density#
Imagine running two swarms and pairing their walkers after the complete cloning stage. Each pair carries a position and velocity error. The kinetic step turns that error into a separation between two Gaussian means; the noise covariance tells us how distinguishable those means are. An error along a direction with little noise costs more relative entropy than the same error along a direction with substantial noise. The matrix below measures precisely this cost for the declared BAOAB step and position noise.
The geometric clusters organize that cost. Within each block, split the paired errors into their average and their deviations from that average. The average measures displacement of the whole block; the deviations measure its internal deformation. Their cross term is exactly zero. Intersecting the two transported cluster partitions lets every paired row belong to one block without changing either swarm’s collision groups. Shared rotations and donor motion are already present in those rows.
This identity preserves the cluster strategy: it connects its geometric errors to entropy without counting all possible interacting pairs. A bound uniform in population size still requires controlling the resulting cluster cost. The identity itself introduces no factor from the number of pairs.
Theorem 290 (Full kinetic entropy bound from geometric cluster errors)
Use the actual quadratic-force configuration of Theorem 197, with \(h\ne2\) and positive OU and final position-noise amplitudes. Let \(P,R\) be laws of nonextinct marked input swarms. Apply the complete actual cloning stage to each, including measurement, retained fitness, donor selection, frozen gates, revival, component rotations, and position jitter. Denote the resulting physical laws by \(\mathcal C_N(P)\) and \(\mathcal C_N(R)\).
For any coupling \(\Gamma\) of these laws, write paired post-cloning rows as \(Z_i,\widetilde Z_i\in\mathbb R^{2d}\) and \(\Delta_i=Z_i-\widetilde Z_i\). Set \(c=h/2\), \(k=1-c^2\), \(a=e^{-\gamma h}\), \(q=B\sqrt{(1-a^2)/(2\gamma)}\), and \(s=\sigma_x\sqrt h\). Define the actual BAOAB mean matrix and noise covariance by
Let \(K_N\) denote the complete update retaining every physical terminal output, including extinct outputs, and \(Q_N\) its restriction to survival. For \(a_P=PQ_N1>0\) and \(a_R=RQ_N1>0\),
To express this cost in the geometric error clusters of The Keystone Principle and the Contractive Nature of Cloning, transport their labels through the chosen row pairing and take their common refinement, keeping any residual labels as separate blocks. Call the resulting partition \(\mathcal G\). It is a proof partition of the paired rows, not a change to the algorithm’s clusters or collision components. For \(G\in\mathcal G\), put \(\bar\Delta_G=|G|^{-1}\sum_{i\in G}\Delta_i\) and \(e_i=\Delta_i-\bar\Delta_G\). With \(M=A^{\mathsf T}\Sigma_1^{-1}A\), the entropy cost has the exact decomposition
Both terms vanish for identical paired configurations. Taking \(R=\nu_N\) uses the actual QSD output \(RQ_N/a_R=\nu_N\) without imposing a common invariant law on the individual algorithm stages.
Proof
Every dead input slot is revived at the scheduled cloning stage because the input is nonextinct. Thus all post-cloning rows undergo the same declared kinetic schedule. Conditional on all post-cloning coordinates, the pre-cap outputs are independent Gaussians with means \(AZ_i\) and common covariance \(\Sigma_1\). The final position noise is added after the last force kick; this gives the displayed covariance, whose scalar-block determinant is \(s^2k^2q^2>0\).
For coupled deterministic inputs the relative entropy of these Gaussian product laws equals \(\frac12\sum_i|A\Delta_i|_{\Sigma_1^{-1}}^2\). Integrate both Gaussian laws against the same coupling \(\Gamma\) and use joint convexity of relative entropy. The first and second mixtures have the correct actual input marginals; no independence of post-cloning rows is asserted. Applying the same deterministic velocity cap and terminal marking to both mixtures cannot increase relative entropy, proving the first inequality.
Disintegrate \(PK_N\) and \(RK_N\) according to terminal survival. The relative entropy chain rule includes the nonnegative Bernoulli relative entropy, \(a_P D(PQ_N/a_P\Vert RQ_N/a_R)\), and the nonnegative extinct-branch relative entropy. Discarding the other terms proves the second inequality. If the total entropy is infinite, that upper bound remains valid in the extended sense.
For each block \(G\), expand \(\Delta_i=\bar\Delta_G+e_i\). The cross terms sum to zero because \(\sum_{i\in G}e_i=0\). This proves the exact cluster decomposition, also when the partition depends measurably on the coupled configuration. The complete collision noise and changed donor velocities enter \(\mathcal C_N(P)\) and \(\mathcal C_N(R)\); none are omitted from the cost.
Theorem 291 (Relative entropy along heat flow)
Let \(\partial_tp_t=\tfrac12\Delta p_t\). For a fixed smooth positive reference density \(q\), integrations by parts give
If the reference also solves \(\partial_tq_t=\tfrac12\Delta q_t\), then
For a fixed reference \(q\), the equation preserving it is instead
and this equation satisfies \(\dot H=-I(p\Vert q)/2\). If \(q\) has LSI constant \(C_q\), its entropy therefore decays at least as \(e^{-t/C_q}\).
Proof
For fixed \(q\),
which is (15.47). For a moving heat-flow reference, there is the additional term \(-\frac12\int p\Delta q/q\). Use \(\Delta q/q=\Delta\log q+|\nabla\log q|^2\) and \(\int p\Delta\log q=-\int p\nabla\log p\cdot\nabla\log q\) to obtain the square in (15.48).
For the reference-preserving equation, integration by parts directly gives \(\dot H=-I/2\). Combining this with \(H\leq C_q I/2\) proves the final assertion.
Example 6 (Smoothing changes a Gaussian target)
Take \(p_0=q=\mathcal N(0,\theta I_m)\). Heat flow gives \(p_t=\mathcal N(0,(\theta+t)I_m)\), so
Thus Gaussian convolution does not contract relative entropy to an arbitrary fixed target, even when that target satisfies an LSI.
7.3. A valid selection-energy calculation#
Proposition 181 (Symmetrization for capped selection rates)
Let \(\mu\) be a probability law and let an energy \(E\) have oscillation at most \(L<\infty\). Consider the specified pairwise selection mechanism with rate
which replaces a donor state \(z\) by \(z'\), with pairs distributed as \(\mu\otimes\mu\). Before adding offspring noise, its contribution to mean energy is
where \(c_L=(1-e^{-aL})/L\) for \(L>0\), with \(c_0=a\).
Proof
Swap \(z\) and \(z'\) in the full product integral and average. For \(\Delta=E(z')-E(z)>0\), \(p(z,z')-p(z',z)=\omega(e^{-a\Delta}-1)\); for \(\Delta<0\), use the opposite sign. This gives the equality in (15.49).
The function \(s\mapsto(1-e^{-as})/s\) decreases on \((0,\infty)\) because \(e^{as}\geq1+as\). Thus \(1-e^{-as}\geq c_Ls\) for \(0\leq s\leq L\). Finally, \(\iint(E(z')-E(z))^2\mu(dz)\mu(dz')=2\operatorname{Var}_\mu(E)\).
Remark 219 (Energy descent and relative entropy)
The pair rate in (15.49) is stated explicitly. Other acceptance rules, sampled fitness values, or offspring noise require their own generator contribution. The symmetrization uses the full product integral; swapping variables also swaps a domain restricted by \(E(z')>E(z)\).
The conclusion is an energy-variance estimate. It does not imply \(\operatorname{Var}_\mu(E)\geq cD_{\mathrm{KL}}(\mu\Vert\pi)\) for arbitrary \(\mu\). For example, increasingly narrow smooth densities can have vanishing energy variance while their entropy relative to a fixed smooth \(\pi\) diverges. Entropy convergence instead uses the full balances (15.25), (15.32), or a common-target Markov contraction.
These calculations keep the useful pieces of the selection argument. Selection can lower a specified energy; Gaussian noise can bound the roughness of an offspring density. To prove convergence, those pieces must refer to the same evolution and target. Neither changing the target during smoothing nor replacing entropy by an energy variance supplies that connection.
7.4. Stability of a normalized companion distribution#
Lemma 244 (Companion-set perturbation with shared weights)
Let \(U_1,U_2\) be two candidate sets with identical positive weights \(w_j\) on their intersection. Define \(P_s(j)=w_j/Z_s\) on \(U_s\), where \(Z_s=\sum_{j\in U_s}w_j\). Put
Then
If \(|U_s|\geq k\), \(|U_1\triangle U_2|\leq n_c\), and \(0<w_{\min}\leq w_j\leq w_{\max}\), then for any common bounded observable \(F\),
Proof
Assume \(a\geq b\), so \(Z_1\geq Z_2\). Sum the absolute probability differences over the common set and the two disjoint parts:
The other case is symmetric. Use \(Z_s\geq kw_{\min}\) and \(a,b\leq n_cw_{\max}\) for the final bound.
Remark 220 (Candidate sets during sequential pairing)
This estimate is conditional on the candidate sets and on shared weights for common candidates. During sequential pairing, the current number of candidates may be much smaller than the original alive population. Moreover, moving common candidates changes their weights. Those effects require their own estimates before the lemma can be applied to a whole sequential matching law. For an unbounded Gaussian-distance kernel, a global positive \(w_{\min}\) is not available; the exact formula in terms of \(a,b,c\) remains valid.
8. Discrete time and the numerical kernel#
Keep a record of both the input swarm and the output swarm for one actual update. Compare two possible input laws, while using exactly the same transition rule for both. Once we condition on survival, the input law is tilted: inputs more likely to survive occur more often in this record. That is why the entropy entering the calculation belongs to the survivor-tilted input law.
Now erase the input column and retain only the output. The backward conditional term measures the distinguishing information that this erasure loses. It is nonnegative, so it enters the output entropy balance with a minus sign. Survival contributes two other terms: a covariance accounting for preferential retention of inputs, and the logarithm of the ratio of survival probabilities. Neither contribution has a fixed favorable sign on its own.
For the QSD reference, the complete surviving output has the same law as the reference input. Its reference-change term therefore vanishes exactly. This uses the full update, including the correlations created by shared component rotations. The balance tells us which terms must be controlled together to prove contraction; it does not require each algorithm stage to preserve the QSD separately.
Theorem 292 (Exact entropy balance for the complete marked update)
Let \(Q_N\) be the actual Euclidean Gas sub-Markov kernel on nonextinct, terminally consistent marked swarms. Its transition integrates the weighted measurement and donor draws, retained sampled fitness and frozen gates, connected-component rotations, revival and jitter, BAOAB, position noise, velocity cap, and terminal boundary classification. Set \(k_N=Q_N1\). For input probability laws \(P\ll R\), put
and define the survivor-tilted input laws \(P^{\rm s}=k_NP/a\), \(R^{\rm s}=k_NR/b\). Let \(\mathsf B_P(y,dx)\) and \(\mathsf B_R(y,dx)\) be the backward conditional input laws under \(P(dx)Q_N(x,dy)/a\) and \(R(dx)Q_N(x,dy)/b\), respectively. Then the following identity holds in \([0,\infty]\):
Write the last nonnegative term as \(\mathcal B_N(P,R)\). If all terms below are finite and \(P^+\ll T\), with \(\log(dR^+/dT)\) defined \(P^+\)-almost everywhere, then
In particular, for \(T=R\) and \(f=dP/dR\),
For the actual QSD reference \(R=\nu_N\), \(b=\alpha_N\) and \(R^+=\nu_N\), so the last term vanishes identically. No separate invariance of cloning, kinetics, or a Gibbs law is required.
Proof
The marked state space is a standard Borel space, so the two joint survivor laws admit the stated disintegrations. On \(\{k_N>0\}\) they have the same forward conditional kernel \(Q_N(x,dy)/k_N(x)\). The chain rule in the input coordinate therefore identifies their joint relative entropy with \(D(P^{\rm s}\Vert R^{\rm s})\). The chain rule in the output coordinate gives its decomposition into \(D(P^+\Vert R^+)\) and the expected backward relative entropy. This establishes the first identity without subtraction of infinite quantities.
The survivor input likelihood ratio is \(dP^{\rm s}/dR^{\rm s}=(b/a)f\); integrating its logarithm yields the first two terms of the second identity. The reference change follows by integrating \(\log(dP^+/dT)=\log(dP^+/dR^+)+\log(dR^+/dT)\). Finally \(P(k_N\log f)/a=P(\log f)+\operatorname{Cov}_P(k_N,\log f)/a\). The QSD substitution uses the full discrete eigenmeasure identity.
Every random choice defining \(Q_N\) has already been integrated in these joint laws. In particular, this argument does not factor the outputs of walkers that share a component rotation, and it does not replace fitness marks by their conditional means.
Proposition 182 (Conditional interaction information)
For any input swarm \(S\), let \(K_S\) be the complete conservative output law before restricting extinction, with marginals \(K_{S,i}\). For any product comparison law \(T_1\otimes\cdots\otimes T_N\) for which the entropies are finite,
The first term is the conditional total correlation. Given a frozen accepted component \(C\), put \(u_i=v_i-\bar v_C\). Its shared Haar rotation satisfies
Consequently the component output law need not equal the product of its marginals, even conditionally on the input and accepted graph.
Proof
Insert the product of the actual marginals between the two densities in the likelihood ratio. Each marginal term integrates against \(K_{S,i}\), which proves the entropy decomposition. For Haar \(R\in O(d)\), \(\mathbb ER=0\) and \(\mathbb E[R_{ap}R_{bq}]=\delta_{ab}\delta_{pq}/d\). Apply these identities to \(v_i^+=\bar v_C+\alpha Ru_i\). For a two-member component with \(u_1=-u_2\ne0\) and \(\alpha>0\), the cross-covariance is \(-\alpha^2|u_1|^2I_d/d\), which is nonzero. The entropy decomposition applies to the complete output law after the actual kinetic noises and terminal marks; its total-correlation term must be evaluated for that law rather than omitted on the basis of independent per-slot innovations.
For a two-walker collision, the velocities relative to the component mean point in opposite directions. Rotate them together and they remain opposite. Looking at either walker alone conceals this relation; looking at both reveals it immediately. The negative cross-covariance in the proposition is the numerical expression of that picture.
Independent kinetic noise is added afterward, but the resulting joint law must still carry whatever dependence survives that noise. The entropy of each individual output therefore supplies only part of the entropy relative to a product comparison law. The total-correlation term supplies the rest. Geometric error clusters and accepted collision components have different jobs here: the former organize the error of a paired-swarm comparison, while the latter determine which walkers receive a shared rotation in an actual run. Keeping both structures in the calculation lets the cluster estimate address the joint algorithm, with its correlations intact.
Definition 615 (One-step entropy dissipation)
For a conservative Markov kernel \(P\) preserving \(\pi\), define
A one-step entropy inequality with coefficient \(\varepsilon\in(0,1]\) is
This is an entropy-contraction inequality for the kernel. It is distinct from the static full-gradient LSI (15.1).
Definition 616 (Markov-kernel quadratic form)
For \(\pi P=\pi\), the quadratic form associated with \(I-P\) is
In general this is not \(\int(f-Pf)^2d\pi\), and neither expression is the entropy dissipation \(\mathcal D_P\).
Theorem 293 (Iteration of one-step entropy contraction)
Under (15.50),
For a sub-Markov kernel \(Q\), the same iteration applies if the inequality has instead been established directly for the normalized map \(\mu\mapsto\mu Q/\mu Q1\) with target QSD \(\nu\).
Proof
Rearrange (15.50) to obtain the one-step bound and iterate. Use \(1-\varepsilon\leq e^{-\varepsilon}\). The normalized map also iterates, since \(\mathcal T_Q^n\mu=\mu Q^n/\mu Q^n1\) whenever the denominators are positive.
Theorem 294 (Composition of bounds with a common functional)
Let two updates \(T_1,T_2\) satisfy, for the same nonnegative functional \(\Phi\) and the same target law,
where \(q_1,q_2,b_1,b_2\geq0\). Then
If \(q=q_1q_2<1\), iteration yields
Proof
Substitute the first bound into the second and sum the geometric series. A nonzero additive term remains in (15.53); it cannot be removed to infer convergence to the target law.
Definition 617 (Entropy-transport functional)
For laws with finite second moments and a specified target \(\pi\), put
Theorem 295 (Algebraic entropy-transport contraction)
Suppose one full update satisfies, with \(H=D_{\mathrm{KL}}(\mu\Vert\pi)\) and \(W=W_2(\mu,\pi)\),
where \(a,b,K,d_H,d_W\geq0\). Then
If \(q<1\), the geometric-series conclusion (15.53) holds with additive term \(d_H+cd_W\).
Proof
Add \(c\) times the second input inequality to the first: \(\mathcal V'\leq aH+(cK-b)W^2+d_H+cd_W\). The definition of \(q\) bounds the first two terms by \(q(H+cW^2)\).
Remark 221 (Inputs to an entropy-transport estimate)
The algebra in (15.54) retains a useful way of combining estimates. Its entropy input must be proved for the actual update. A contraction of \(W_2\) alone does not imply the negative entropy term \(-bW_2^2\). A linear mixture \((1-s)\mu+s\nu\) is also not generally a Wasserstein geodesic, so displacement convexity cannot be applied to that mixture as if it were the transport interpolation.
Lemma 245 (Numerical entropy defect)
Let an exact evolution \(T_\tau\) satisfy \(\Phi(T_\tau\mu)\leq e^{-r\tau}\Phi(\mu)\). Suppose the numerical update \(\widetilde T_\tau\) has a functional error estimate
on an invariant class of input laws, with \(p>0\), \(K,B\geq0\). Set \(q_\tau=e^{-r\tau}+K\tau^{p+1}\). For \(q_\tau<1\),
For fixed \(r>0\) and sufficiently small \(\tau\), the second term is \(O(\tau^p)\). If \(B=0\), the estimate gives exact contraction of the functional to zero.
Proof
Combine the exact bound and (15.55), then iterate the resulting affine recursion. Since \(p>0\), \(1-q_\tau=r\tau+o(\tau)\) as \(\tau\downarrow0\), which gives the stated order of the residual term.
Remark 222 (What a BAOAB weak-error estimate does not supply)
Relative entropy and Fisher information are nonlinear functionals of the evolving density. An error estimate for expectations of smooth observables is not, by itself, (15.55). That estimate needs density and derivative control in a class where the functional can be compared. The numerical invariant law may also differ from the continuous invariant law; the static LSI must then be transferred by an actual measure comparison such as (15.7), or proved directly for the numerical law.
Compare the numerical BAOAB velocity law
Rust engine experiment · Seed 7 · Poster after 7 experiment steps. Interactive view starts from the same seed.
9. Consequences and limiting laws#
9.1. Marginals and empirical observables#
Corollary 98 (Poincaré inequality and empirical observables)
If the continuous joint law \(\pi_N\) satisfies (15.10), then
In particular, for \(F_N=N^{-1}\sum_i\varphi(z_i)\) with \(\|\nabla\varphi\|_\infty\leq L\),
No independence of the coordinates is needed once the joint LSI has been established.
Proof
For a bounded smooth mean-zero \(F\), substitute \(f=1+\varepsilon F\) into (15.10). The second-order entropy expansion is \(\operatorname{Ent}_{\pi_N}((1+\varepsilon F)^2)=2\varepsilon^2\pi_N(F^2)+o(\varepsilon^2)\). Divide by \(2\varepsilon^2\) and let \(\varepsilon\to0\); approximation gives (15.57). For \(F_N\), each gradient is \(N^{-1}\nabla\varphi(z_i)\), so their squared sum is at most \(L^2/N\).
Corollary 99 (LSI for fixed marginals and their limits)
Suppose \(\pi_N\) satisfies (15.10) uniformly. Its marginal on any fixed \(k\) particles satisfies the same LSI constant \(C_*\). If these marginals converge weakly to a law \(\pi^{(k)}\), the limiting law satisfies the same inequality for smooth bounded test functions with bounded continuous squared gradient, and hence on the associated Sobolev closure when that class is a core.
Proof
Apply (15.10) to a function depending only on the first \(k\) particles; all other derivatives vanish. For the limit, apply the inequality to a fixed test function. Its entropy integrand and its squared gradient are bounded continuous functions, so both sides pass to the weak limit. Closure extends the inequality to the stated domain.
Remark 223 (Marginal convergence, empirical measures, and KL)
A finite empirical measure is atomic. Its relative entropy to a continuous positive density is therefore infinite. Entropy convergence in (15.35) concerns the law of the swarm, or a marginal obtained from that law. Smoothed empirical measures are different objects and can be studied using (15.45)–(15.47).
Relative entropy decreases under taking a marginal and implies total-variation control through Pinsker’s inequality. Neither a Wasserstein convergence bound nor propagation of chaos supplies an upper bound on KL without an additional density estimate. KL also has no general triangle identity through a third probability measure.
9.2. LSI for a non-Gibbs invariant law from flow contraction#
Theorem 296 (Additive-noise diffusion with a contractive flow)
Consider \(dZ_t=b(Z_t)dt+\Sigma dW_t\) on \(\mathbb R^m\), with constant \(\Sigma\) and a globally Lipschitz \(C^1\) drift. Suppose its synchronous flow satisfies
for a constant positive definite matrix \(Q\), where \(|z|_Q^2=z^{\mathsf T}Qz\). Let
The diffusion has a unique invariant probability law \(\pi\) with finite second moment, and this law satisfies
Consequently its Euclidean full-gradient LSI constant is at most
Proof
Invariant law. Differentiating the contraction estimate at \(t=0\) gives
Set \(z'=0\) and absorb the fixed vector \(b(0)\) by Young’s inequality. The generator then satisfies \(L|z|_Q^2\leq-(r/2)|z|_Q^2+C\), including the constant diffusion trace. Time-averaged laws are tight, and the Feller property gives an invariant law. Synchronous contraction gives its uniqueness and convergence to it in the quadratic transport distance induced by \(Q\).
Gradient estimate. Let \(J_t\) be the derivative of the flow in its initial condition. The contraction implies \(J_t^{\mathsf T}QJ_t\preceq e^{-rt}Q\). For smooth positive \(h\), differentiate \(P_th\) through the flow and apply weighted Cauchy-Schwarz:
Integrate against the invariant law to obtain \(I_{Q^{-1},\pi}(P_th)\leq e^{-rt}I_{Q^{-1},\pi}(h)\).
Entropy interpolation. The actual carré du champ is \(\Gamma_L(g,g)=|\Sigma^{\mathsf T}\nabla g|^2/2\) and satisfies \(\Gamma_L(g,g)\leq c_Q(\nabla g)^{\mathsf T}Q^{-1}\nabla g\). For bounded positive \(h\) bounded away from zero, invariance and the diffusion chain rule give
The entropy at infinite time vanishes by ergodicity. Substitute \(h=f^2/\pi(f^2)\), then extend by approximation. The Euclidean bound follows from \(Q^{-1}\preceq\lambda_{\min}(Q)^{-1}I\).
Corollary 100 (Stationary LSI for a frozen alignment field)
Consider the conservative kinetic diffusion
where \(m_UI\preceq\nabla^2U\preceq L_UI\), \(m_U>0\), \(\gamma>0\), \(\nu\geq0\), and \(a\) is a fixed \(C^1\) Lipschitz field with \(\|\nabla a\|_\infty\leq R\). Suppose
Set
Its invariant law \(\pi^a\) satisfies a full-gradient LSI with
In particular, the result applies to a stationary mean-field law whose frozen alignment field satisfies these bounds and this strict parameter inequality. It does not require that \(\pi^a\) have a Gibbs density.
Proof
For two synchronously driven solutions, write \(\xi=\Delta X\), \(\zeta=\Delta V\), \(\beta=g/2\), and \(\alpha=g^2/2\). The noise cancels, and differentiation gives
Young’s inequality bounds the last term by \((g/2)|\zeta|^2+2(L_U+\nu R)^2|\xi|^2/g\). Thus \(d|\Delta Z|_Q^2/dt\leq-r|\Delta Z|_Q^2\). The matrix is positive definite since \(\alpha-\beta^2=g^2/4>0\). With \(\Sigma=(0,\sigma I)^{\mathsf T}\), one has \(c_Q=\sigma^2/2\), because the velocity block of \(Q\) is \(I\). Apply Theorem 296.
Here the invariant density need not be known explicitly. The replacement for a density formula is a checked contraction of the stochastic flow. Noise and that contraction give an inequality for the law through entropy interpolation. The inequality on \(A_x\) retains both the friction requirement and the alignment penalty; small alignment alone does not make that coefficient positive for every friction value.
9.3. Geometric forms and frozen velocity covariance#
Proposition 183 (Frozen row-normalized alignment has a uniform Gaussian LSI)
Fix positions \(X\) and symmetric weights \(K_{ij}=K_{ji}\), with \(K_{ii}=0\) and \(0<k_*\leq K_{ij}\leq k^*\) for \(i\ne j\). For \(N\geq2\), put
where \(\gamma>0\) and \(\nu\geq0\). The frozen velocity equation
has stationary law \(\mathcal N(-A_X^{-1}b_X,\Sigma_X)\), with
Its Gaussian LSI and Poincaré constants are at most this last bound, independently of \(N\) and \(\nu\).
Proof
The matrix
is symmetric positive semidefinite: for any \(z\), \(z^{\mathsf T}(D_X-K)z=\frac12\sum_{i\ne j}K_{ij}(z_i-z_j)^2\geq0\). The inequality \((z_i-z_j)^2\leq2z_i^2+2z_j^2\) also gives \(\widetilde L_X\preceq2I\).
Set \(H_X=D_X^{1/2}\otimes I_d\) and \(\widetilde A_X=\gamma I+\nu\widetilde L_X\otimes I_d\). Then \(A_X=H_X^{-1}\widetilde A_XH_X\), with \(\widetilde A_X\succeq\gamma I\). Consequently
Solving the linear equation gives the stationary mean and
The semigroup bound yields the covariance estimate; differentiating the integrand gives the Lyapunov equation. A Gaussian of covariance \(\Sigma_X\) is a linear image of a standard Gaussian, so its LSI constant is at most \(\|\Sigma_X\|_{\mathrm{op}}\); the Poincaré estimate follows by linearizing the LSI. The same argument works with a uniformly bounded degree ratio in place of the pointwise kernel bounds.
Corollary 101 (Comparison with an adaptive geometric form)
Suppose a specified law \(\pi_N\) satisfies (15.10), and let a full phase-space matrix field \(G_N(S)\) satisfy
uniformly in \(S,N\). Then
If instead the law changes to \(\widetilde\pi_N\) with uniformly bounded density ratio \(a\leq d\widetilde\pi_N/d\pi_N\leq b\), the corresponding bound is \(2(b/a)C_*/g_*\).
Proof
The matrix inequality gives \(|\nabla f|^2\leq g_*^{-1}(\nabla f)^{\mathsf T}G_N\nabla f\). Substitute this into the LSI. For the changed measure, first apply (15.7).
Remark 224 (Frozen velocities and the full geometric law)
For positions held fixed, a linear velocity equation \(dV=-A(x)V\,dt+B(x)dW\) has Gaussian stationary covariance
If the symmetric part of \(A(x)\) is at least \(\gamma I\) and \(\|B(x)\|_{\mathrm{op}}\leq b_*\), then \(\|\Sigma_x\|_{\mathrm{op}}\leq b_*^2/(2\gamma)\), by the bound \(\|e^{-tA(x)}\|_{\mathrm{op}}\leq e^{-\gamma t}\). This gives a valid conditional Gaussian LSI for that frozen process.
It does not identify the conditional velocity law of the moving, cloning swarm. Furthermore, for any disintegration \(\pi(dx,dv)=\pi_x(dx)\pi(dv\mid x)\),
The second term requires spatial control. Uniform inequalities for the frozen Gaussian laws alone do not remove it. Likewise, row normalization of a graph Laplacian does not make it symmetric in the Euclidean inner product. The weighted norm in Proposition 183 provides the required comparison when its degree ratios are controlled.
The population-independent constant becomes useful at (15.58): the gradient of an empirical average carries a factor \(1/N\), while the sum over particles contributes \(N\). The resulting variance is proportional to \(1/N\). This conclusion is available for correlated walkers because the required control is already contained in the joint inequality.
For a limiting mean-field law, the order of the argument is equally concrete. Establish the inequality for the finite-particle laws, take a fixed marginal, then pass to its limit. Identifying that limit as the stationary solution of the mean-field equation uses the existence, uniqueness, and convergence results for that equation. A static LSI and a mean-field identification perform different jobs.
10. How the estimates fit together#
Constants and dependencies
Estimate |
Inputs |
Result |
|---|---|---|
Kinetic reference LSI |
Quadratic radial confinement and a lower Hessian bound, or bounded perturbation of a convex potential |
\(C_0\) in (15.9) |
Joint LSI |
Product law, bounded joint tilt, uniform joint curvature, or uniform flow contraction |
\(C_*\) in (15.10) |
Kinetic dissipation |
Global Hessian bound \(M\), \(D=\sigma_v^2/2\), \(\gamma>0\) |
\(\eta\) in (15.18) |
Kinetic entropy convergence |
Joint LSI and kinetic dissipation for the same Gibbs law |
Rate (15.20) |
Full conditioned evolution |
Actual QSD LSI and the complete derivative estimate (15.34) |
Rate (15.35) |
Doob-transform route |
Transformed-law estimate and bounded survival reweighting |
(15.40)–(15.41) |
Numerical iteration |
A functional defect bound for the actual numerical kernel |
(15.56) |
Empirical variance |
Full-gradient joint LSI |
\(C_*L^2/N\) in (15.58) |
The proofs of these implications are contained in this chapter. The operator-specific inputs come from the algorithm and convergence chapters, with the reference law, state space, and normalization fixed before they are combined.
The essential calculation is now visible: an LSI controls entropy by full Fisher information, and a hypocoercive derivative controls the modified entropy by that same information. Killing adds a normalization term; cloning adds a kernel contribution. Once those contributions are estimated for the actual law, the final step is Grönwall’s inequality. The resulting statement says exactly which distribution converges, which functional decays, and which constants remain independent of the population.