Convergence, Survival, and Parameter Dependence#
A cloning step can pull positions together while adding kinetic energy. A kinetic step can dissipate that energy while moving positions apart. To understand the whole algorithm, we must keep an account of what each step transfers to the other. This chapter makes that account explicit.
There are then two different probability questions. Does a conservative swarm forget its initial state? Does a swarm that can become extinct forget its initial state conditional on survival? The second question includes a change of weights: initial states with better survival prospects contribute more to the conditional law. We carry that normalization through the proofs.
The results below assemble the cloning estimates, kinetic moment calculations, and comparison arguments already developed in this volume. They also state exactly which additional kernel or entropy estimates produce mixing, concentration, and long survival. Parameter tuning uses the constants in those estimates; measured proxies remain identified as measurements.
1. The Process and the Quantities Being Controlled#
First choose what one state means. Here it means the entire finite swarm, including its alive indicators. A distribution of such states is different from the distribution of one walker and from a deterministic density obtained as the population tends to infinity. Keeping these objects separate prevents a result about one of them from being used as a theorem about another.
Definition 567 (Swarm state and cemetery convention)
Fix a population size \(N\ge2\) and timestep \(h>0\). Let \(E_N\) be the measurable space of admissible swarm states with at least two alive walkers. Use the \(k<2\) cemetery convention of Latent Fractal Gas: Algorithm and Analytic Conditions: the process is sent to an absorbing state \(\dagger\) when this condition fails. Write
The killed one-step kernel is
Thus \(Q1\le1\). Its missing mass is the probability of absorption. Extend every nonnegative moment observable by zero at \(\dagger\). A conservative kernel, written \(P\), instead satisfies \(P1=1\) on its specified state space. A software convention that continues singleton swarms defines a different kernel and must be analyzed with its own absorbing set.
Definition 568 (Conditional evolution and quasi-stationarity)
For a probability measure \(\mu\) with \(\mu Q^n1>0\), define
A quasi-stationary distribution (QSD) is a probability measure \(\nu_N\) on \(E_N\) with
It satisfies \(\Phi_n(\nu_N)=\nu_N\) and \(\mathbb P_{\nu_N}(T_\dagger>n)=\alpha_N^n\). When absorption has positive probability under \(\nu_N\), \(\alpha_N<1\).
Throughout the chapter, \(\|\mu-\eta\|_{\rm TV}=\sup_A|\mu(A)-\eta(A)|\) for probabilities. A finite-swarm QSD is a measure on \(E_N\); its one-particle marginal is not by definition a QSD of a frozen single-particle kernel.
Definition 569 (Single-swarm moment observable)
Let \(A=\mathcal A(S)\) and \(k=|A|\). Define the alive barycentres and variances by
For a nonnegative barrier \(\phi\), put \(W_b=N^{-1}\sum_{i\in A}\phi(x_i)\), whenever finite. Set
where \(c_V,c_\mu,c_B>0\). The subscript records the observable historically used in the total variation argument; a moment inequality for \(V_{\rm TV}\) has a separate logical role from a total variation contraction.
The alive barycentre uses \(1/k\) even though the variances use \(1/N\). For \(k=N\) these conventions coincide. Bounds for a two-swarm sum, a two-swarm average, or a different alive normalization must first be converted to this convention.
Remark 167 (Analytic inputs and their domains)
The component analysis is in The Keystone Principle and the Contractive Nature of Cloning and Hypocoercivity and Convergence of the Euclidean Gas. The assembly below uses their inequalities on the states where their hypotheses hold, with constants uniform over those states. In particular, a contraction depending on occupancy of an interior reward set requires a lower bound on that occupancy. Existence of the set alone supplies no such lower bound.
The Euclidean kinetic bounds below use the specified velocity cap and Euclidean position increments. The latent algorithm additionally requires the metric, force, diffusion, and collision estimates in Latent Fractal Gas: Algorithm and Analytic Conditions. An instantaneous norm comparison between two metrics does not identify their drift inequalities.
For an unbounded position space, centred variance does not control the position of the swarm barycentre. A tightness argument therefore also needs a confining position observable with proved drift. On a domain with an absorbing boundary, \(W_b\) must have finite transition expectations. For example, a reciprocal boundary-distance singularity need not be integrable against a Gaussian position density; the barrier analysis in The Keystone Principle and the Contractive Nature of Cloning must be applied before inserting a finite \(C_b\).
2. Component Estimates That Can Be Composed#
A rate written beside a component is a promise about a particular transition. For example, friction gives an exact exponential estimate for a continuous Langevin evolution. A split numerical step needs its own estimate. We first record bounds that follow directly from capped motion, then show how sharper kinetic calculations enter when their hypotheses are verified.
Proposition 148 (Component drift inputs)
Suppose cloning and kinetics have nonnegative, state-independent comparison matrices and source vectors satisfying
The inequalities are componentwise and apply to the same observables, including the cemetery extension. If cloning has a separate intermediate state space, the observables must be defined there and the kinetic estimate must cover every intermediate state to which cloning assigns mass. This includes proposed jittered positions under the specified status-update convention. The diagonal input frequently used in the component chapters has the form
Here \(r_{C,x}=1-\kappa_x\) and \(r_{C,b}=1-\kappa_b\) when the corresponding cloning contraction estimates hold. A bounded expansion uses coefficient one; a uniform bound uses coefficient zero. Every source vector must include the jitter, collision, and discretization terms appropriate to that estimate.
The complete cloning statement Theorem 190 supplies one collection of such inputs. Its boundary component uses the favorable companion and integrable-barrier estimates Lemma 142 and Lemma 143, with revival contributions tracked in Theorem 186. The following direct kinetic bounds supply another, without an assumed equilibrium variance.
Lemma 153 (Capped displacement and removal of walkers)
Suppose a kinetic step adds no alive indices, its surviving set is \(B\subset A\), and every surviving position satisfies \(x_i'=x_i+\Delta_i\) with \(\|\Delta_i\|\le h v_{\max}\). If its output velocities satisfy \(\|v_i'\|\le v_{\max}\), then, for every \(\theta>0\),
These inequalities also hold after absorption, with the observables set to zero. They apply to two capped half-position updates whose durations sum to \(h\). For a latent metric, the displacement bound must first be established in the norm used to define \(V_x\).
Proof
For any nonempty index set \(I\),
Use \(a=\bar x_A\) for the final set \(B\). The inequality \(\|u+w\|^2\le(1+\theta)\|u\|^2+(1+\theta^{-1})\|w\|^2\) gives
Enlarge the first sum to \(A\) and use \(|B|\le N\). For velocities, the minimum over centres is at most the value at zero, so \(NV_v(S')\le\sum_{i\in B}\|v_i'\|^2\le Nv_{\max}^2\). The triangle inequality bounds the norm of the alive velocity mean by \(v_{\max}\). Empty and cemetery outputs have zero observable by convention.
Corollary 72 (Kinetic position noise and a displacement moment bound)
Suppose the kinetic step has no new alive indices and its proposed position increments satisfy
Then
For the position-noisy Euler variant EG-kin\(^+\) specified in Euclidean Gas: Canonical Transition and Operator Estimates, write its kinetic position-noise scale as \(\sigma_{\rm pos}\) to distinguish it from cloning jitter. Its update \(\Delta_i=hv_i^++\sqrt h\sigma_{\rm pos}\xi_i^x\), with capped \(v_i^+\) and independent centred \(\xi_i^x\sim N(0,I_d)\), gives
For a capped BAOAB position update without a separate position Gaussian, \(D_h^2=h^2v_{\max}^2\). These are different kinetic specifications.
Proof
Apply the minimum-over-centres calculation of the preceding lemma to the surviving subset, then enlarge the nonnegative squared-increment sum to all input alive indices before taking expectations. For EG-kin\(^+\), independence and centring remove the mixed term, while \(\mathbb E\|\xi_i^x\|^2=d\). The cap bounds the remaining deterministic velocity contribution.
Proposition 149 (Positional drift from the cloning estimate and the cap)
Assume \(P_CV_x\le r_{C,x}V_x+b_{C,x}\) with \(0\le r_{C,x}<1\), and the kinetic step satisfies Lemma 153. Then
For \(r_{C,x}>0\), choose \(0<\theta<(1-r_{C,x})/r_{C,x}\) to obtain \(r_x<1\). For \(r_{C,x}=0\), every \(\theta>0\) works. The positional decrement per full step is \(\kappa_x^{\rm step}=1-r_x\). Under Corollary 72, replace \(h^2v_{\max}^2\) in \(b_x\) by \(D_h^2\).
Proof
Apply \(P_C\) to the kinetic inequality. Since \(P_C1\le1\), its constant term increases by at most the displayed amount. Substitute the cloning bound and collect the coefficient and source. The restriction on \(\theta\) is exactly \((1+\theta)r_{C,x}<1\).
Proposition 150 (Kinetic velocity rates and their time convention)
For a fixed alive population of \(N\) walkers evolving by the continuous Euclidean Langevin equation with friction \(\gamma\), forces bounded by \(F_{\max}\), independent Brownian increments, and diffusion matrices bounded by \(\sigma_{\max}\), the Itô estimates of Hypocoercivity and Convergence of the Euclidean Gas take the form
For the exact continuous transition over time \(h\), these imply
A BAOAB or Boris-BAOAB estimate can use these coefficients only after its split step has been bounded. If the split step instead has an error estimate \(P_KV_v\le(e^{-a_vh}+\eta_v(h))V_v+b_v(h)\), its decrement is \(1-e^{-a_vh}-\eta_v(h)\) when positive. The capped output bound \(P_KV_v,P_KE_v\le v_{\max}^2\) remains available independently of this sharper friction estimate.
Proof
For the continuous equations, centring the velocity force term gives \(2N^{-1}\sum_i(v_i-\mu_v)\cdot F_i\); the part involving the mean force vanishes. Young’s inequality bounds it by \(\epsilon V_v+F_{\max}^2/\epsilon\). The centred quadratic variation is at most \(d\sigma_{\max}^2\). For the mean velocity, Young’s inequality gives \(2\mu_v\cdot\bar F\le\gamma\|\mu_v\|^2+F_{\max}^2/\gamma\), and independence bounds its quadratic variation by \(d\sigma_{\max}^2/N\). These are the two stated differential inequalities.
Multiplying an inequality \(y'\le-ay+B\) by \(e^{at}\) and integrating gives \(y(h)\le e^{-ah}y(0)+B(1-e^{-ah})/a\). In particular, \(1-e^{-ah}\le ah\); replacing \(e^{-ah}\) by the smaller number \(1-ah\) is not an upper bound. Changes of alive set, conditioning on survival, interactions in the diffusion noise, and split-step remainders must be treated in the corresponding transition estimate before using this calculation.
Proposition 151 (Boundary coefficients under sequential updates)
For a nonnegative, transition-integrable barrier observable, suppose
Then
In particular, when \(r_{C,b}=1-\kappa_b\) and \(r_{K,b}=1-\kappa_{\rm pot}h\) are valid nonnegative coefficients, the full-step decrement is
A uniform expected barrier bound can instead be entered as \(r_{K,b}=0\). The choice of a barrier observable does not modify the algorithm’s force, reward, or absorbing boundary.
Proof
Apply \(P_C\) to the kinetic inequality, use \(P_C1\le1\), and substitute its bound on \(W_b\). Expanding \(1-(1-\kappa_b)(1-\kappa_{\rm pot}h)\) yields the decrement.
Proposition 152 (Wasserstein control requires a coupling observable)
Let \(\widehat P\) be a coupling of two conservative full-step swarm kernels. Suppose a nonnegative observable \(D(S,\widetilde S)\) dominates \(W_2^2(\mu_S,\mu_{\widetilde S})\) for the empirical measures under consideration and satisfies
Then
The centred variance identities in Wasserstein-2 Control via Keystone-Based Variance Proxy provide parts of such an observable. Full phase-space contraction additionally needs control of the barycentre difference, velocity difference, and their coupling under the same transition. When \(b_D>0\), the displayed result is control to a nonzero error level. For killed swarms, the coupling must also specify survival conditioning; a conservative coupling inequality cannot be used unchanged.
Proof
Iterate the affine inequality by conditional expectation and use the domination of \(W_2^2\) by \(D\). The geometric sum is \((1-r_D^n)/(1-r_D)\).
3. A Complete Comparison-Matrix Proof#
The matrices act on observables, which reverses the apparent order of the algorithm. The swarm clones first and moves second. To estimate its final energy, first bound the energy after motion, and then average that bound over cloning. This gives the kinetic matrix followed algebraically by the cloning matrix. The same calculation determines how noise injected by cloning is subsequently damped.
Theorem 203 (Foster-Lyapunov drift for the composed kernel)
Let \(P_C,P_K\) be nonnegative sub-Markov kernels between the input, intermediate, and output spaces, with the component observables defined on all three. Let the component estimates of Proposition 148 hold with constant nonnegative matrices and finite nonnegative source vectors. For cloning followed by kinetics, the observable kernel is \(Q=P_CP_K\), and
If \(\mathbf w>0\) and \(0\le r<1\) satisfy
then \(V=\mathbf w^\top\mathbf V\) satisfies
For the historical observable \(V_{\rm TV}\), take \(\mathbf w=(c_V,c_V,c_\mu,c_B)^\top\) whenever those weights satisfy the matrix inequality. Constants are uniform in \(N\) only if the component bounds and weights, including their positive lower bounds when comparing individual components, have that uniformity.
Proof
For component \(j\), first use the kinetic estimate:
All coefficients are nonnegative. Substitute \(P_CV_\ell\le\sum_i(A_C)_{\ell i}V_i+(\mathbf b_C)_\ell\) and use \(P_C1\le1\). This proves the vector estimate with the displayed product order and source. Multiplication by \(\mathbf w^\top\) gives
Finiteness and uniformity follow exactly from the quantities used in these inequalities. State-dependent matrices require bounding their expectations through the intermediate state; their values cannot be multiplied as fixed matrices.
Theorem 204 (Positive weights and the spectral criterion)
For a finite nonnegative matrix \(M\), the following are equivalent:
There are \(\mathbf w>0\) and \(r<1\) with \(\mathbf w^\top M\le r\mathbf w^\top\).
The spectral radius \(\rho(M)\) is less than one.
For the two-component matrix
weights \((1,\omega)\) have positive effective decrements precisely when
If \(a,b>0\), such weights exist exactly when
The resulting full-step decrement is \(\min(\delta_x-\omega b,\delta_v-a/\omega)\). Zero off-diagonal coefficients are handled directly by the two strict inequalities.
Proof
For the first implication, the weighted norm \(\|z\|_{1,\mathbf w}=\sum_iw_i|z_i|\) satisfies \(\|Mz\|_{1,\mathbf w}\le r\|z\|_{1,\mathbf w}\), since \(M\ge0\). Hence \(\rho(M)\le r<1\).
Conversely, if \(\rho(M)<1\), the finite-dimensional Neumann series converges. Set
Its entries are finite and at least one, and \(\mathbf w^\top M=\mathbf w^\top-\mathbf1^\top\). Thus \(r=\max_i(1-1/w_i)<1\) gives the required inequality.
For the two-component calculation,
Divide the second coefficient by \(\omega\). Positivity of both decrements is the displayed interval condition, whose endpoints are ordered exactly when \(ab<\delta_x\delta_v\).
Theorem 205 (Full-step rate and source)
For any admissible positive weights define
If \(r(\mathbf w)<1\), these are a proved decrement and source for the weighted moment observable. In the diagonal case,
and any fixed positive weights work. Off-diagonal terms require the weight criterion of Theorem 204.
For \(h>0\) and \(0<r<1\), the corresponding physical-time exponential rate of this moment bound is \(-h^{-1}\log r\). It is not, by definition, a total variation, relative entropy, Wasserstein, or extinction rate.
Proof
Each component coefficient is bounded by \(r(\mathbf w)w_i\), so the preceding theorem applies. In the diagonal case the quotient cancels \(w_i\). Finally, \(r^n=\exp[-(-\log r/h)nh]\).
Lemma 154 (Iteration for conservative and killed kernels)
If \(QV\le rV+b\), \(Q1\le1\), and \(0\le r<1\), then
Consequently, when \(\mu Q^n1>0\),
Proof
The first formula follows by induction, applying \(Q\) to the preceding inequality. The second uses \(Q^j1\le1\). Integration against \(\mu\) and division by the positive survival probability gives the conditional bound.
The denominator in the last expression is essential. A small unnormalized moment can mean that most runs have died. To bound the size of the surviving swarm, we must also control survival. The next section supplies two ways to complete a convergence argument, according to which kernel is being studied.
Measure the finite-step moment generator
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4. From Component Drift to Mixing#
Drift brings trajectories back toward a controlled region. Mixing asks whether two trajectories can lose their memory of where they started. A common part in their transition laws provides that loss of memory. For a conservative process, we combine this common part with drift. For a killed process, we compare whole surviving blocks so that the preference for better-surviving paths is included.
4.1. Conservative drift and minorization#
Theorem 206 (A weighted coupling form of Harris’ argument)
Let \(P\) be a probability kernel on a standard Borel space. Let \(V\) be finite and nonnegative, with
Suppose \(R>2b/(1-r)\) and there are \(\epsilon\in(0,1]\) and a probability measure \(\eta\) such that
Choose \(\beta>0\) so that \(\beta(rR+2b)\le\epsilon\); if \(rR+2b=0\), any \(\beta>0\) works. Define
There is a \(\rho_H<1\) such that
for all probabilities with finite \(V\) moment. Hence \(P\) has a unique invariant probability \(\pi\) in this class, and \(\|\mu P^n-\pi\|_\beta\le\rho_H^n\|\mu-\pi\|_\beta\).
Proof
Use the cost
Its optimal transport cost between two probabilities is their displayed weighted variation norm: couple their common part on the diagonal and couple the mutually singular residual measures arbitrarily. Every coupling has at least this cost, since off-diagonal mass must account for both residual measures.
Write \(s=V(x)+V(y)\). If \(s\ge R\), any coupling gives
where
The strict inequality follows from \((1-r)R>2b\); the ratio for \(s\ge R\) is bounded by its endpoint value and its limit \(r\).
If \(s<R\), both states are in the minorized set. With probability \(\epsilon\), draw the same point from \(\eta\); couple the remaining transition mass arbitrarily. Its probability of unequal points is at most \(1-\epsilon\), and its total expected \(V\) moment is no greater than the unconditioned moment sum. Therefore
for \(x\ne y\). For equal initial points use identical transitions. Integrating these couplings gives contraction with \(\rho_H=\max(\rho_1,1-\epsilon/2)<1\).
Probabilities with finite \(V\) moment form a complete space under \(\|\cdot\|_\beta\): a Cauchy sequence converges in the Banach space of signed measures with weight \(1+\beta V\), and positivity and total mass one pass to the limit. The drift bound maps this space into itself. The contraction mapping theorem now gives the invariant measure and the asserted convergence.
Remark 168 (What must be verified for the gas)
This is the weighted coupling method of Hairer and Mattingly, Yet another look at Harris’ ergodic theorem. The comparison theorem above supplies the drift input when its component hypotheses hold. The other input is a minorization of the same probability kernel on the stated level set. On an unbounded domain, the level set needs the established confining position control as well as centred variance bounds. The theorem does not replace a killed kernel \(Q\) by a probability kernel: normalizing \(Q(x,\cdot)\) separately at every step generally differs from conditioning a complete trajectory on survival to time \(n\).
Theorem 207 (Accessibility and irreducibility)
Let \(Q\) be a sub-Markov kernel. Suppose a measurable set \(C\), an integer \(m\), and a nonzero finite measure \(\eta\) satisfy \(Q^m(x,\cdot)\ge\epsilon\eta(\cdot)\) for all \(x\in C\), with \(\epsilon>0\). Suppose also that, for every \(x\), some \(n=n(x)\ge0\) has \(Q^n(x,C)>0\). Then \(Q\) is \(\eta\)-irreducible: for every \(A\) with \(\eta(A)>0\), some iterate from every \(x\) has positive mass on \(A\).
Proof
For the indicated \(n\),
This identifies the irreducibility measure as \(\eta\). Lebesgue irreducibility needs the corresponding accessibility of every set of positive Lebesgue measure; a minorization on one interior ball alone does not identify that larger irreducibility measure.
Theorem 208 (One-step common mass excludes periodic classes)
Let \(P\) be a probability kernel with \(P(x,\cdot)\ge\epsilon\eta(\cdot)\) for all \(x\) in its state space, where \(\epsilon>0\) and \(\eta\) is a probability. Then \(P\) has no decomposition into two or more nonempty disjoint cyclic classes \(D_0,\ldots,D_{d-1}\) with \(P(x,D_{j+1})=1\) for \(x\in D_j\) (indices modulo \(d\)). It also satisfies
Proof
If such cyclic classes existed, the minorization and \(P(x,D_{j+1})=1\) would force \(\eta(D_{j+1})=1\) for every \(j\), contradicting disjointness. For the contraction, write \(P=\epsilon\eta+(1-\epsilon)R\) with \(R\) a probability kernel and use contraction of total variation under \(R\). The case \(\epsilon=1\) is immediate.
4.2. A complete killed-block criterion#
Theorem 209 (QSD convergence from a two-sided surviving-block bound)
Fix \(N\). Suppose that for an integer \(m\ge1\), a probability measure \(\eta_N\) on \(E_N\), and constants \(0<c_N\le C_N<\infty\),
Then \(Q\) has a unique QSD \(\nu_N\). With \(\rho_N=1-c_N/C_N\in[0,1)\),
for every initial probability \(\mu\). At exponent zero the right-hand side is interpreted as one. All these conditional laws are well-defined. Its one-step survival eigenvalue satisfies
The same existence and uniqueness conclusion follows from the alternative full-block conditioned contraction criterion in Proposition 123. These are sufficient criteria for the specified killed finite-swarm kernel, with constants that may depend on \(N\).
Proof
Put \(T=Q^m\) and suppress \(N\). Its bounds imply \(c\le T1\le C\) and \(Q1\ge Q^m1\ge c\), so every finite survival probability is positive. Let \(h_j=T^j1\). For a horizon \(k\) define probability kernels
Since \(h_{k-j}(x)\le C\eta(h_{k-j-1})\), these satisfy
The common probability measure on the right is independent of \(x\). Consequently each \(R_{j,k}\) contracts total variation by at most \(\rho=1-c/C\), by the decomposition used in Theorem 208.
Starting \(R_{0,k},\ldots,R_{k-1,k}\) from \(\mu_k(dx)=h_k(x)\mu(dx)/\mu h_k\) gives the final law \(\mu T^k/\mu T^k1\): the factors \(h_j\) cancel along a path. Two initial probabilities therefore give final laws at distance at most \(\rho^k\), even though their initial reweightings differ. Write \(F_k(\mu)=\mu T^k/\mu T^k1\). We have proved
Normalization cancels in composition, so \(F_kF_l=F_{k+l}\). For every \(l\ge0\), \(\|F_{k+l}(\mu)-F_k(\mu)\|_{\rm TV}\le\rho^k\). Thus \(F_k(\mu)\) converges in the complete total variation space of probabilities to a limit \(\nu\) independent of \(\mu\). The map \(F_1\) is continuous because \(T1\ge c\) and \(T1\le C\); taking limits in \(F_1F_k=F_{k+1}\) gives \(F_1(\nu)=\nu\). The diameter bound proves uniqueness of this fixed point and \(\|F_k(\mu)-\nu\|_{\rm TV}\le\rho^k\).
The one-step conditional map \(\Phi_1\) commutes with \(F_1\), so \(\Phi_1(\nu)\) is another fixed point of \(F_1\) and equals \(\nu\). Hence \(\nu Q=\alpha\nu\) with \(\alpha=\nu Q1>0\). Every QSD is a fixed point of \(F_1\), proving uniqueness. Finally, if \(n=km+j\) with \(0\le j<m\), then \(\Phi_n(\mu)=F_k(\Phi_j(\mu))\), giving the displayed bound. Integrating the two-sided bound on \(Q^m1\) against \(\nu\) gives \(c\le\alpha^m\le C\); combine this with \(\alpha\le1\) for the eigenvalue bounds.
Remark 169 (Analytic scope of the block estimate)
A proof of the displayed two-sided bound must handle the full swarm, intermediate killing, all alive-status transitions, and the actual collision and cap maps. Gaussian density bounds on a bounded interior region are usable ingredients. Uniform ellipticity and bounded positions alone do not establish the required bound for a split, interacting kernel.
In particular, positive companion probabilities do not imply positive cloning probabilities for all alive walkers: competitive cloning can have probability zero. A one-step velocity Gaussian also has only \(Nd\) noise coordinates for a \(2Nd\) phase-space update. A full density claim needs a multi-step controllability or change-of-variables argument. A hard radial projection can produce boundary atoms; a smooth cap has a different density transformation. The reference measure \(\eta_N\) must accommodate the actual transition law.
The general theory of conditioned convergence is developed by Champagnat and Villemonais. The theorem above gives an elementary sufficient condition with its complete proof; it also isolates the kernel estimate needed to use that route here. For unbounded latent spaces, the volume’s confining estimates and a compatible conditioned mixing argument are required rather than a new compactness assumption.
Survival and conditional shape in a killed gas
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5. Moments, Survival, Entropy, and Concentration#
A QSD looks stationary only after the surviving runs have been renormalized. Before renormalization its mass decreases by a factor \(\alpha_N\) each step. That factor enters the moment balance. Likewise, a large swarm survives for a long time only if many walkers can survive together without a common failure mechanism. A count of interior walkers is one part of that argument; probability estimates for their joint failures are another.
5.1. Moment bounds under a QSD#
Proposition 153 (Eigenmeasure, symmetry, and marginal statements)
For any QSD \(\nu_N\) with eigenvalue \(\alpha_N\):
\(\nu_NQ^n=\alpha_N^n\nu_N\) and its survival law is geometric in the step index. If \(\alpha_N<1\), \(\mathbb E_{\nu_N}T_\dagger=1/(1-\alpha_N)\).
If \(Q\) commutes with a measurable symmetry and its QSD is unique, \(\nu_N\) has that symmetry. Exchangeability follows when the actual swarm kernel commutes with every permutation of walker labels.
Its marginal densities and correlations are determined by the full eigenmeasure equation. A Gibbs spatial density or Gaussian velocity marginal requires a separate solution or approximation theorem for that equation, including cloning, capping, and killing.
Proof
Iteration proves the eigenmeasure identity. Summing \(\mathbb P_{\nu_N}(T_\dagger>n)=\alpha_N^n\) over \(n\ge0\) gives the mean lifetime. If \(g\) is a symmetry, the pushforward \(g_\#\nu_N\) satisfies the same eigenmeasure equation and normalization; uniqueness identifies it with \(\nu_N\). Taking coordinate projections of the equation gives the marginal relations, with interactions retained inside the projected transition.
Theorem 210 (Equilibrium moment bounds with the survival eigenvalue)
Suppose \(QV\le rV+b\), \(\nu_NQ=\alpha_N\nu_N\), and \(\nu_NV<\infty\). If \(\alpha_N>r\), then
More generally, if \(Q\mathbf V\le M\mathbf V+\mathbf b\), \(\nu_N\mathbf V\) is finite componentwise, and \(\rho(M)<\alpha_N\), then
For a conservative invariant measure, \(\alpha_N=1\). Under the block bounds of Theorem 209, the integrability premise holds for every nonnegative \(V\) with \(\eta_NV<\infty\), since \(\nu_N\le(C_N/\alpha_N^m)\eta_N\). In that case \(r<c_N^{1/m}\) is a directly checkable sufficient survival-gap condition and gives \(\nu_NV\le b/(c_N^{1/m}-r)\).
Proof
Integrating the scalar drift gives \(\alpha_N\nu_NV\le r\nu_NV+b\). Subtract the finite term \(r\nu_NV\) and divide by \(\alpha_N-r>0\).
For the vector bound, put \(u=\nu_N\mathbf V\). Then \(u\le(M/\alpha_N)u+\mathbf b/\alpha_N\). Iterating \(l\) times yields
The first term tends to zero because \(u\) is finite and \(\rho(M/\alpha_N)<1\). The nonnegative series converges to the stated inverse. Finally \(\alpha_N^m\nu_N=\nu_NQ^m\le C_N\eta_N\), proving the integrability criterion. The bounds are inequalities; stationarity does not turn an upper drift estimate into an exact moment formula.
5.2. Direct survival estimates#
Lemma 155 (Barrier control counts interior walkers)
Suppose \(W_b=N^{-1}\sum_{i\in A}\phi(x_i)\le L\) and \(\phi\ge H>0\) outside a chosen interior set \(K\). Then
In particular, after a cloning step with \(N\) alive walkers, at least \(N(1-L/H)\) lie in \(K\) when \(H>L\).
Proof
Every alive index outside \(K\) contributes at least \(H\) to \(NW_b\le NL\). Therefore the number of such indices is at most \(NL/H\). Subtract it from \(|A|\).
Proposition 154 (Joint failure bounds and the extinction scale)
Suppose every state in a class \(G_N\) has \(m_N\ge2\) identified distinct walker indices whose final alive indicators are part of the next swarm state. Assume for every subset \(I\) of these indices and every state \(S\in G_N\),
Then
If a QSD exists and \(\nu_N(G_N^c)\le a_N\), its one-step extinction hazard satisfies
Consequently \(\mathbb E_{\nu_N}T_\dagger\ge [a_N+m_Np^{m_N-1}]^{-1}\) whenever the right side is defined and \(\alpha_N<1\). If \(m_N\ge qN\) for a fixed \(q>0\) and \(a_N\le Ae^{-bN}\), this is an exponential lower bound on the mean lifetime, up to the polynomial factor \(m_N\). An exponential upper bound requires a separate lower bound on the extinction hazard.
Proof
If fewer than two identified walkers survive, some subset of \(m_N-1\) of them has entirely died. There are \(m_N\) such subsets. Apply the union bound and the joint failure estimate. Integrate \(1-Q1\) against \(\nu_N\), bounding it by one on \(G_N^c\). The eigenmeasure survival formula gives the lifetime bound. For \(m_N\ge qN\), \(p^{m_N-1}\) decays exponentially in \(N\).
The joint failure hypothesis follows, for example, from conditional independence of the identified deaths with individual probabilities at most \(p\), or from sequential conditional failure bounds at most \(p\). Shared cloning randomness and state-dependent interactions must be included in the conditioning before either property can be invoked.
Proposition 155 (Eventual absorption from a uniform block hazard)
If \(Q^m1(S)\le1-\delta\) for every \(S\in E_N\), with \(m\ge1\) and \(\delta>0\), then
Thus absorption is almost sure. Conversely, a uniform one-step upper hazard \(1-Q1\le\delta\) gives \(\mathbb P_\mu(T_\dagger\le n)\le n\delta\).
Proof
For the first statement, apply the block bound conditionally at times \(0,m,2m,\ldots\). Sum the resulting tail bound in blocks of length \(m\) to bound the expected lifetime. For the second, sum the conditional probability of first absorption at each of the \(n\) steps.
Gaussian velocity noise alone does not establish a uniform block hazard for capped motion. In particular, a walker farther than \(hv_{\max}\) from a boundary cannot cross it in a single capped position step. The exit estimate must use the positions, intermediate updates, and the stated block length.
5.3. Entropy and concentration for the specified limiting law#
Theorem 211 (Entropy dissipation implies total variation convergence)
Let \(\nu\) be a fixed probability law and let \(\mu_t\ll\nu\) be the evolution being studied, which may be a conditioned evolution when its entropy identity has been established. Suppose a nonnegative functional \(\mathcal E(t)\) obeys
Then
For a discrete bound \(\mathcal E_{n+1}\le r_E\mathcal E_n\), replace \(e^{-\lambda t/2}\) by \(r_E^{n/2}\). The entropy and hypocoercive estimates in Hypocoercive Entropy: From Kinetic Mixing to the Full Swarm and Logarithmic Sobolev inequalities and entropy convergence can be used here when their reference measure and evolution coincide with \(\nu\) and \(\mu_t\). In particular, Theorem 287 gives the entropy route under its stated modified-Fisher dissipation assumptions. For a QSD this includes the killing and normalization contribution to the entropy balance, identified in Proposition 178.
Proof
Integration gives \(\mathcal E(t)\le e^{-\lambda t}\mathcal E(0)\). For completeness, let \(A_+=\{d\mu/d\nu\ge1\}\) and set \(p=\mu(A_+)\), \(q=\nu(A_+)\). Then \(p-q=\|\mu-\nu\|_{\rm TV}\). Jensen’s inequality on \(A_+\) and its complement gives
As a function of \(p\), the right side vanishes with zero derivative at \(p=q\), and its second derivative is \(1/[p(1-p)]\ge4\). It is therefore at least \(2(p-q)^2\), with the endpoint cases obtained by continuity or infinite entropy. Combining this inequality with the dissipation estimate proves the result.
Theorem 212 (Joint-law LSI gives concentration of swarm averages)
Suppose a probability law \(\nu_N\) on the specified swarm coordinates satisfies
Let \(F\) be a bounded observable in the associated form domain with \(\|\nabla F\|^2\le L^2/N\) almost everywhere, where \(L>0\). Then
For \(F=N^{-1}\sum_{i=1}^N\varphi(z_i)\) and \(\|\nabla\varphi\|\le L\), the gradient condition follows directly in the product Euclidean metric, even when the law is interacting. An \(N\)-uniform LSI yields an \(N\)-uniform concentration coefficient in the exponent. Different metric or diffusion conventions require the corresponding gradient bound in that Dirichlet form. A law with alive-status strata also requires an LSI that controls the stated observable across those strata.
Proof
Put \(X=F-\nu_NF\) and \(\psi(\theta)=\log\nu_Ne^{\theta X}\). Apply the LSI to \(f=e^{\theta X/2}\) and divide by \(\nu_Ne^{\theta X}\). The result is
For \(\theta>0\), integrate the inequality for \((\psi(\theta)/\theta)'\), using \(\psi(0)=\psi'(0)=0\), to obtain \(\psi(\theta)\le\theta^2L^2/(2\rho_NN)\). Exponential Markov inequality gives \(\nu_N(X\ge t)\le\exp[-\theta t+\theta^2L^2/(2\rho_NN)]\). Minimize at \(\theta=\rho_NNt/L^2\) and repeat with \(-X\). For the empirical average, \(\|\nabla F\|^2=N^{-2}\sum_i\|\nabla\varphi(z_i)\|^2\le L^2/N\).
Remark 170 (Concentration inputs and finite-population limits)
The LSI in this result is an inequality for the full law \(\nu_N\), with its specified form and constant. It is not supplied by a scalar moment drift or by applying an independent-sample bounded-differences inequality to correlated walkers. The uniform-law hypotheses and the resulting LSI are stated in Corollary 94; the choice of conservative, quasi-stationary, or conditioned-process reference law is specified in Definition 608. The adaptive metric hypotheses are treated in The Geometric Gas: Regularity, Stability, and Entropy.
Finite-\(N\) conditional mixing does not by itself establish existence or uniqueness of a nonlinear mean-field stationary law. Passing to that limit also uses tightness, identification of the limiting equation, control of survival normalization, and the appropriate propagation-of-chaos estimates. Those arguments belong to The Mean-Field Law of the Euclidean Gas, The Discrete Population Limit and Propagation of Chaos, and Continuum Limits: Hypotheses and Consistency. Exchanging the limits \(N\to\infty\), \(n\to\infty\), and \(h\to0\) requires the uniform estimates stated there.
Proposition 156 (Mixing time from the proved contraction)
Under Theorem 209, for \(0<\rho_N<1\) and \(0<\varepsilon<1\), the step count
ensures \(\|\Phi_n(\mu)-\nu_N\|_{\rm TV}\le\varepsilon\) for every \(\mu\). When \(\rho_N=0\), one block suffices. Under a bound \(\|\mu P^n-\pi\|\le B_\mu\rho_H^n\), use \(\lceil\log(B_\mu/\varepsilon)/(-\log\rho_H)\rceil_+\) instead. Physical time is the step count multiplied by \(h\).
Proof
Solve the corresponding geometric inequality for the integer number of blocks or steps. Neither formula identifies its contraction factor with \(1-\kappa_{\rm drift}\); the required minorization or entropy constants enter through the theorem that established convergence.
6. Parameter Dependence and Sensitivity#
Once the inequalities are assembled, parameter dependence becomes a concrete calculation. Changing a parameter can alter a diagonal damping term, an interaction term, or the injected noise. A table of guessed powers hides these possibilities. We instead differentiate a specified bound, keeping its domain of validity in view.
Definition 570 (Algorithm parameters and derived analysis quantities)
A historical analysis vector is
Here \(\sigma_x\) is the position jitter standard deviation; \(\alpha_{\rm rest}\in[0,1]\) is the inelastic restitution coefficient; \(\lambda_{\rm alg}\ge0\) weights velocity in companion distance; \(\epsilon_c,\epsilon_d>0\) are the cloning and diversity companion scales; \(\gamma>0\) is friction; \(\sigma_v\) specifies the chosen velocity-noise convention; \(h>0\) is the numerical step; and \(N\ge2\) is an integer. The quantities \(\kappa_{\rm wall}\) and \(d_{\rm safe}\) parametrize a declared boundary potential and interior set when that model uses them.
The symbol \(\lambda\) is a derived cloning frequency or a parameter of an explicitly specified rate model. The actual algorithm uses its fitness-based cloning probabilities; introducing an independent \(\lambda\) must not change those probabilities implicitly. This vector is a selection of coordinates used for analysis, not an exhaustive list of the latent algorithm’s controls. The full definitions, including fitness and metric regularizers, are in Latent Fractal Gas: Algorithm and Analytic Conditions and Parameter Constraints and Tuning.
Logarithmic derivatives below use a chosen set of strictly positive, continuous coordinates. Zero restitution or zero velocity-distance weight requires ordinary or one-sided derivatives. Treating \(N\) as continuous is an explicit relaxation whose proposed values must later be checked at integers.
Proposition 157 (What a parameter can change in a proved bound)
Suppose a scalar moment estimate has differentiable coefficients \(r(\mathbf P)<1\) and \(b(\mathbf P)>0\). Its conservative equilibrium bound is \(B=b/(1-r)\), so for any continuous parameter \(p\),
For a QSD with a differentiable eigenvalue \(\alpha(\mathbf P)>r(\mathbf P)\), the corresponding bound is \(B_Q=b/(\alpha-r)\) and
Thus a parameter can affect damping, the additive source, and survival at the same time. Classification as affecting only one of these quantities requires those other derivatives to vanish in the particular bound being used.
Proof
Differentiate the two quotient formulas on their stated domains.
Definition 571 (Rate sensitivity matrix)
Choose positive differentiable rate functions \(\kappa_1(\mathbf P),\ldots,\kappa_q(\mathbf P)\) obtained from specified analytic inequalities, and positive continuous coordinates \(p_1,\ldots,p_s\). Put \(x_j=\log p_j\) and
Rates may describe component moments, a conditioned block contraction, or entropy dissipation, but their type and units must be recorded. The minimum of rates of different quantities has no automatic convergence interpretation. For empirically fitted rates, the same matrix is a local sensitivity estimate of the fit.
Theorem 213 (Explicit sensitivity from the comparison matrix)
Suppose \(M(\mathbf P)\) is differentiable and positive weights \(\mathbf w\) are held fixed. For every component with
its logarithmic sensitivity is
Since \(M=A_KA_C\),
If weights are themselves functions of the parameters, their derivatives must also be included. At a change of the active minimum rate, use the one-sided directional calculation below.
Proof
Differentiate the component quotient with fixed \(w_i\), then multiply by \(p_j/\kappa_i\). The remaining identities are the product rule applied to the proved composition formulas.
Definition 572 (Moment-bound sensitivity matrix)
For positive differentiable bounds \(B_i(\mathbf P)\), define \((M_B)_{ij}=\partial\log B_i/\partial\log p_j\). For a scalar conservative bound \(B=b/\kappa\),
For a QSD bound replace \(\kappa\) by \(\alpha-r\). These derivatives describe the upper bound; they equal derivatives of an actual equilibrium moment only when an additional identity establishes equality with that bound.
Theorem 214 (Singular directions of the sensitivity matrix)
For a real \(q\times s\) sensitivity matrix \(M_\kappa\) of rank \(r\), there are orthonormal right and left singular vectors with singular values \(\sigma_1\ge\cdots\ge\sigma_r>0\). A right singular vector \(v_i\) produces first-order rate change \(M_\kappa v_i=\sigma_i u_i\). Moreover,
A vector in this kernel has zero first-order rate change at the point where \(M_\kappa\) was evaluated. This is a local statement; it does not by itself produce a family of parameters with exactly equal rates.
Proof
The symmetric positive semidefinite matrix \(M_\kappa^\top M_\kappa\) has an orthonormal eigenbasis. Its positive eigenvalues are \(\sigma_i^2\). For their unit eigenvectors \(v_i\), set \(u_i=M_\kappa v_i/\sigma_i\). Then \(u_i^\top u_j=\delta_{ij}\). Complete these vectors to orthonormal bases to obtain a singular value decomposition. The zero eigenspace is precisely \(\ker M_\kappa\), proving the dimension formula. Differentiability gives \(\log\boldsymbol\kappa(x+tv)=\log\boldsymbol\kappa(x) +tM_\kappa v+o(t)\), which gives the local interpretation.
Proposition 158 (Conditioning on the identifiable subspace)
When \(r\ge1\), define the restricted condition number \(\operatorname{cond}_+(M_\kappa)=\sigma_1/\sigma_r\). For \(v\perp\ker M_\kappa\),
The upper factor controls forward sensitivity. The lower factor controls inversion on the identifiable subspace. If the matrix has a nontrivial kernel, arbitrary parameter changes cannot be recovered uniquely from rate changes.
Proof
Expand \(v\) in the right singular vectors with positive singular values and bound \(\sum_i\sigma_i^2v_i^2\) between \(\sigma_r^2\sum_iv_i^2\) and \(\sigma_1^2\sum_iv_i^2\).
Theorem 215 (Finite perturbations and the minimum rate)
Let \(g(x)=\log\boldsymbol\kappa(e^x)\) be continuously differentiable on a convex neighbourhood. If \(\|Dg(x)\|_2\le L\) there, then
If \(Dg\) is \(L_1\)-Lipschitz on that neighbourhood, then
For positive component rates with a valid common interpretation, \(\log\kappa_{\min}=\min_i\log\kappa_i\), and
Proof
Integrate \(Dg(x+tu)u\) over \(t\in[0,1]\). For the remainder, subtract \(Dg(x)u\) and use \(\|Dg(x+tu)-Dg(x)\|\le L_1t\|u\|\). Finally, for any two vectors \(a,b\), \(\min_i a_i\le\min_i b_i+\|a-b\|_\infty\); exchange \(a,b\) to obtain the absolute-value inequality.
7. Tuning a Specified Bound#
A slow component can limit a weighted moment estimate, so balancing component rates is often sensible. But the slow component may already be at its own maximum, or improving it may increase the noise source or reduce survival. We therefore optimize a stated bound over stated constraints. A numerical optimizer finds candidates; the inequalities determine what those candidates mean.
7.1. Optimization and the active rates#
Definition 573 (Parameter optimization problem)
Let \(\mathcal F\) be the set on which the chosen component, metric, integrability, and transition estimates hold. For a proved moment bound, one possible objective is
with a weight normalization such as \(\sum_iw_i=1\) and, when needed, positive lower bounds on the weights. Source constraints can impose \(b/(1-r)\le B_{\max}\) for conservative moments or \(b/(\alpha-r)\le B_{\max}\) for an integrable QSD with \(\alpha>r\).
Optimizing TV mixing instead uses its actual bound, for example \(-m^{-1}\log\rho_N\), together with the cost per block. Entropy optimization uses its proved dissipation constant. Integer population sizes are compared as discrete choices.
Theorem 216 (Directional derivatives of a minimum)
Let \(f(x)=\min_{1\le i\le q}f_i(x)\) with all \(f_i\) continuously differentiable near \(x\). Define the active set \(I(x)=\{i:f_i(x)=f(x)\}\). Then
If every \(f_i\) is affine, \(f\) is concave. In that case every convex combination of the active gradients is a supergradient: it satisfies \(f(y)\le f(x)+g\cdot(y-x)\) for all \(y\).
Proof
Inactive functions have a strictly positive gap at \(x\) and remain inactive for sufficiently small positive movement along a fixed direction. Taylor expansion for the finitely many active functions gives the directional formula. For affine branches and active \(i\), \(f(y)\le f_i(y)=f(x)+\nabla f_i(x)\cdot(y-x)\). Taking a convex combination gives the supergradient inequality. The minimum of affine functions is concave by this same inequality, or directly from its hypograph as an intersection of half-spaces.
Theorem 217 (A conditional balancing principle)
At an interior local maximum \(x_*\) of \(\min_i f_i(x)\), if exactly one branch \(i_*\) is active, then \(\nabla f_{i_*}(x_*)=0\). Consequently, if all possible uniquely active branches have nonzero gradients, every interior local maximizer has at least two active branches. Boundary constraints can instead prevent further improvement of a single active branch.
Proof
If a unique branch is active, continuity and the finite positive gap to the other branches make the same branch active in a neighbourhood. The minimum therefore equals the differentiable function \(f_{i_*}\) locally. The ordinary first-order condition for an interior local maximum gives zero gradient.
Theorem 218 (Exact balancing in a two-rate allocation model)
Consider the auxiliary optimization problem
Its unique optimizer is
This solves the displayed linear resource model. Applying it to algorithmic parameters requires independently establishing those rate formulas and that resource constraint.
Proof
Substitute \(\gamma=B-\lambda\). The first branch increases strictly and the second decreases strictly. Their minimum increases up to their unique intersection and decreases afterwards. Solve \(a\lambda=b(B-\lambda)\) to obtain the formulas.
7.2. Parameter couplings with explicit assumptions#
Proposition 159 (Restitution and friction enter different energy terms)
For one collision group with conserved mean velocity \(\bar v\) and collision rule \(v_i'=\bar v+\alpha_{\rm rest}(v_i-\bar v)\),
Thus restitution multiplies the relative energy of that group by \(\alpha_{\rm rest}^2\). If an independently proved complete-step velocity bound is \(QV_v\le r_v(\alpha_{\rm rest},\gamma)V_v+ b_v(\alpha_{\rm rest},\gamma)\), its conservative moment bound is \(b_v/(1-r_v)\), and its QSD moment bound, under the integrability and survival gap conditions, is \(b_v/(\alpha_N-r_v)\).
Proof
Expand the square in the collision rule. The cross term vanishes because \(\sum_i(v_i-\bar v)=0\). Apply Theorem 210 to the complete-step estimate. Overlapping collision groups must follow the algorithm’s actual update convention; conservation for each isolated group cannot be summed as a conservation law for overlapping overwrites.
Proposition 160 (Jitter and contraction in a declared scalar model)
Suppose a conservative positional estimate has, on its admissible parameter range, the specific coefficients
Then its invariant moment bound is \((B_0+B_1\sigma_x^2)/(c\lambda)\) when the invariant moment is finite. A sufficient condition for this bound to be at most \(V_*\) is
For the actual competitive cloning kernel, \(c\), \(\lambda\), and the source must be derived from its probabilities and the kinetic composition. The simple ratio above applies only to the displayed scalar model.
Proof
Integrate the given inequality against the invariant law and rearrange. The target-bound condition is the resulting scalar inequality solved for \(\lambda\).
Proposition 161 (Companion geometry at a fixed swarm state)
For the Euclidean softmax companion law with fixed candidate set,
define \(\Delta_x=\|x_i-x_j\|^2-\|x_i-x_l\|^2\) and \(\Delta_v=\|v_i-v_j\|^2-\|v_i-v_l\|^2\). Then
These identities quantify companion reweighting at a fixed state. They do not specify how fitness or the stationary law changes after the swarm evolves.
Proof
The common softmax denominator cancels in the ratio. Differentiate its logarithm while holding the state and candidate set fixed.
7.3. A reviewable tuning procedure#
Algorithm 5 (Selecting parameters from analytic bounds)
Input: a specified Euclidean or latent transition, a target observable, and admissible parameter ranges.
Choose the object of study: conservative law, killed finite-swarm QSD, nonlinear mean-field law, or a stated continuous-time limit.
Collect the component inequalities for that transition, retaining their alive-normalization, metric, integrability, and occupancy hypotheses.
Form \(M=A_KA_C\) and \(\mathbf b=A_K\mathbf b_C+\mathbf b_K\). Find positive weights with \(\mathbf w^\top M<\mathbf w^\top\).
For a mixing target, add the required minorization, killed-block, or entropy-dissipation estimate. For a QSD moment target also check integrability and \(\alpha_N>r\).
Evaluate the resulting rate, source, and computational cost. Compare feasible parameter choices and verify the proposed choice in the original inequalities.
Output: parameters together with the bound they satisfy, or a list of analytic inputs still requiring verification for that proposed choice.
Algorithm 6 (Projected optimization of a declared objective)
Fix a continuously parametrized feasible region where the selected bound is valid, and keep integer parameters in an outer search. At each iterate:
Evaluate the objective and all active component inequalities.
Compute their derivatives, including parameter-dependent weights and source terms. For a minimum objective use the active directional derivative from Theorem 216.
Propose a feasible direction and a step size. A projection requires a specified projection rule and a region on which it is defined.
Accept the proposed point only after checking feasibility and the intended improvement of the actual objective; otherwise shorten the step or stop.
Report the final objective, active constraints, and verification residuals.
This procedure defines a numerical search. Global optimality requires the convexity or other structural hypotheses of a separate optimization theorem.
Definition 574 (Pareto comparison)
A feasible parameter choice is Pareto optimal for specified objectives, such as a proved mixing rate, moment bound, and cost, if no other feasible choice improves at least one objective without worsening another. All objectives must have declared directions of improvement and be evaluated for the same model and parameter convention.
Algorithm 7 (Empirical tuning with analytic checks)
For a fixed parameter choice, record empirical cloning frequency, component moment changes, alive counts, and the costs relevant to the chosen objective. Estimate sensitivities with repeated runs and quantified sampling error. Use those estimates to propose a new parameter choice, then re-evaluate the analytic constraints and run the updated configuration.
Diagnostics in src/fragile/fractalai/convergence_bounds.py provide computed
bounds or proxies according to their documented inputs. A fitted decay rate,
a positive proxy, or a finite trajectory cannot establish a uniform kernel
inequality. Online adaptation also changes the process into a time-dependent
kernel unless the controller is included in the state; convergence claims
then require estimates uniform along that adaptation or a separate adaptive
process analysis.
The calculations now have distinct jobs. The comparison matrix bounds moments of the actual composed step. Kernel overlap or an entropy estimate controls loss of memory. The QSD eigenvalue controls survival. A joint-law LSI controls fluctuations of observables. Keeping those jobs explicit lets us reuse each proved estimate without asking it to answer a different probability question.