Self-similarity and diffusive limits for linear kinetic equations: a Wild sum approach

Abstract: We prove that linear collisional kinetic equations in the whole space without confinement mechanism display a long-time self-similar behaviour. This drastically improves the recently known results (decay estimates) about the solutions in such a context, providing the first result regarding this self-similar behaviour. As a consequence, we also establish a uniform-in-time convergence of the suitably rescaled solutions to their diffusion limit, which is also new. The class of equations considered includes some BGK type equations, some kinetic nonlocal Fokker--Planck-type equations and some kinetic (possibly fractional) Fokker--Planck equations, for which we are able to write explicitly solutions through a Wild sum (or Dyson series) or we can manage some accurate computations on the Fourier side.

• The paper develops a Wild sum and probabilistic concentration framework that proves convergence toward explicit self-similar profiles for BGK and Fokker–Planck kinetic equations.
• The results improve the leading decay rate by an additional factor of (1+t)^−γ, with γ ranging from 1/3 to the sharp 1/2 depending on the operator, fractional order, and Lebesgue space.
• The paper establishes the first uniform-in-time quantitative diffusive limit without well-prepared initial data, while identifying open problems involving optimal BGK rates, fractional cases, and more general collision operators.

Setting and main result

The paper by Cañizo, Mischler, and Tassi (2605.24685) studies the long-time behaviour of linear collisional kinetic equations on the whole space $\mathbb{R}^d \times \mathbb{R}^d$,

$\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$

where $\mathcal{L}$ acts only on velocity, preserves mass and positivity, and admits a unique probability equilibrium $F$. Three classes of operators are treated: linear BGK operators $\mathcal{L}f = F(v)\rho[f] - f$ (with $F$ a Maxwellian, a $2s$-stable law, or an equilibrium in the basin of attraction of either); nonlocal Fokker–Planck operators $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$ with Gaussian kernel $G$; and (possibly fractional) Fokker–Planck operators $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$, $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$0.

The central claim is that, in the absence of any confinement mechanism in $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$1, solutions do not merely decay algebraically but converge to an explicit self-similar profile: for probability initial data with a finite moment $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$2 in $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$3 ($\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$4 if $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$5, $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$6 otherwise),

$\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$7

where $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$8 solves the (fractional) heat equation with diffusivity $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$9 and initial data $\mathcal{L}$0, and

$\mathcal{L}$1

The exponent $\mathcal{L}$2 matches the previously known decay rate of Bouin et al. for weighted $\mathcal{L}$3 norms; the additional factor $\mathcal{L}$4 is what makes the result new. It shows that $\mathcal{L}$5 is a first-order approximation to $\mathcal{L}$6 at large time, and it proves that the earlier rate estimates are optimal. Combining with the known self-similar asymptotics of the fractional heat equation yields directly

$\mathcal{L}$7

i.e., genuine convergence to the self-similar profile.

A corollary gives the diffusive limit: for the rescaled equation $\mathcal{L}$8,

$\mathcal{L}$9

uniformly in time, without any well-preparedness assumption on the initial data. This is the first uniform-in-time quantitative diffusion limit for this class; previous results (Bouin–Mouhot) gave rates only on compact time intervals with constants whose time-dependence precluded extension to $F$0.

Summary of exponents

The second-order exponent $F$1 depends on the operator, on $F$2, and on the Lebesgue range:

Equation	Order $F$3	$F$4 range	Exponent $F$5
Standard BGK	$F$6	$F$7	$F$8
Fractional BGK	$F$9	$\mathcal{L}f = F(v)\rho[f] - f$0	$\mathcal{L}f = F(v)\rho[f] - f$1
Low-order fractional BGK	$\mathcal{L}f = F(v)\rho[f] - f$2	$\mathcal{L}f = F(v)\rho[f] - f$3	$\mathcal{L}f = F(v)\rho[f] - f$4
General BGK	$\mathcal{L}f = F(v)\rho[f] - f$5	$\mathcal{L}f = F(v)\rho[f] - f$6	$\mathcal{L}f = F(v)\rho[f] - f$7
Nonlocal KFP	$\mathcal{L}f = F(v)\rho[f] - f$8	$\mathcal{L}f = F(v)\rho[f] - f$9	$F$0
KFP	$F$1	$F$2	$F$3
Fractional KFP	$F$4	$F$5	$F$6

Here $F$7, $F$8, and $F$9 is the Berry–Esseen convergence rate of the equilibrium towards its stable attractor. The authors state that they believe the optimal value should be $2s$0 in all cases when $2s$1, but their method does not reach this everywhere — notably the BGK case saturates at $2s$2. For the classical Fokker–Planck operator the sharp rate $2s$3 is obtained in the full range $2s$4 via explicit computation of the covariance matrix eigenvalues, which scale as $2s$5.

Method: Wild sums and concentration

The core technique for the BGK and nonlocal Fokker–Planck cases is a Wild sum (Dyson series) representation obtained by iterating Duhamel's formula against the transport semigroup $2s$6. The solution is written as an expectation over a probability space $2s$7 of jump histories, where the number of jumps $2s$8 is Poisson-distributed and, conditionally on $2s$9, the scaled inter-jump times follow a flat Dirichlet distribution. This probabilistic reading converts the asymptotic analysis into a study of typical versus atypical histories.

Two structural facts drive the argument:

• Exchange lemmas: for factorised data $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$0, one has $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$1, exhibiting how the interplay of free transport and velocity-local scattering generates diffusion in $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$2. In the nonlocal case, an analogous "star convolution" identity $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$3 pushes all transport to the left and produces multivariate Gaussians (or stable laws) with explicit covariance matrices built from the sums $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$4.
• Concentration inequalities: Chebyshev bounds on Dirichlet statistics of order $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$5 control the deviation of $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$6 from its mean, while Bernstein and Chernoff bounds handle Poisson tails and exponential functionals such as $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$7.

The proof splits the history space into a probable region (where $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$8, $$\mathcal{L}f = G*_v f - f + \operatorname{div}_v(vf)$$9 are near the boundaries' expected values and $G$0 concentrates) and an improbable region. On the probable region, a four-term telescoping decomposition isolates the error between the Wild sum term and the target profile $G$1; each term decays as a power of $G$2 or $G$3, and summing against the Poisson measure with optimised parameters $G$4 yields the $G$5 rate. The improbable region contributes $G$6 (for $G$7), and balancing $G$8 gives $G$9.

For the general-equilibrium BGK case, the authors add a Berry–Esseen-type assumption quantifying the $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$0 convergence of normalised convolutions of $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$1 towards the stable law, with speed $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$2; the resulting exponent is $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$3. They concede that a unified presentation of Berry–Esseen rates across all $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$4 spaces is not available in the literature, so the assumption is stated abstractly rather than verified under minimal hypotheses.

For the nonlocal Fokker–Planck case, the analysis must be carried out in mixed Lebesgue spaces $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$5, because the equilibrium $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$6 does not belong to $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$7 in general. The transport-drift semigroup is an isometry on $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$8 but not on $$\mathcal{L}f = -(-\Delta_v)^s f + \operatorname{div}_v(vf)$$9; Minkowski's integral inequality bridges the two. A further subtlety, explicitly flagged by the authors, is that the random covariance matrix does not concentrate around its mean in the velocity block: mass concentrates along typical trajectories in $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$00 while fluctuations in $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$01 remain non-negligible, and the target covariance matrix is chosen to share exactly these heavy tails. The comparison of random versus limiting Gaussians uses a Schur-complement decomposition and a Frobenius-norm perturbation bound for covariance matrices. The final term requires controlling negative moments of $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$02; the expectation $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$03 is finite precisely for $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$04, which explains the $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$05 exponent in that case.

The Kinetic Fokker–Planck section proceeds differently: the Fourier-space solution is fully explicit, being a convolution of $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$06 with a multivariate Gaussian ($\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$07) or a multivariate stable law ($\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$08) with characteristic exponent $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$09. The proof splits into a covariance-matrix discrepancy term (eigenvalues of the relative difference scale as $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$10, giving the sharp $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$11 for $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$12) and an initial-data mixing term handled by moment interpolation. In the fractional case, Plancherel and $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$13 endpoint estimates plus interpolation give $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$14 for $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$15; for $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$16 only a suboptimal interpolated rate is available, degenerating as $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$17.

Limitations and open questions

The authors are candid about the scope of the method. Because the proof relies on explicit representations of solutions, the results are restricted to simple collision operators; extending the self-similar description to general operators under purely structural assumptions on $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$18 remains open, as does the natural candidate of the linear Boltzmann equation with Maxwell molecules. Several specific gaps are identified:

• The conjectured optimal $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$19 for $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$20 is not attained in the BGK case ($\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$21); whether the obstruction is structural to the Poisson weighting or merely technical (a sharper concentration inequality than Chebyshev would improve rates) is unresolved.
• The dependence of $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$22 on the fractional index $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$23 is not settled; the regime transition at $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$24 reflects the convexity change of $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$25, and the rate saturates as $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$26 without reaching $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$27.
• The critical tail regime $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$28 is excluded.
• For fractional KFP, the full range $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$29 is open, possibly requiring spectral-measure analysis of multivariate stable laws.
• The nonlocal-to-local limit yielding the KFP results from the NLFP theorem via a uniform-in-$\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$30 statement is described as expected rather than proved here.
• The extension of the nonlocal Fokker–Planck result to general kernels $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$31 with finite $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$32 moment, including heavy-tailed ones, is asserted to be possible but not carried out.

Conclusion

This paper establishes, for the first time, long-time self-similar behaviour and uniform-in-time diffusive limits for unconfined linear kinetic equations, upgrading the known algebraic decay rates $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$33 to two-term asymptotics $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$34 with explicit profiles. The Wild-sum/probabilistic framework — combining exchange lemmas, Dirichlet and Poisson concentration inequalities, and careful covariance-matrix perturbation analysis — is the enabling tool, and the sharp $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$35 obtained for the classical kinetic Fokker–Planck operator serves as evidence for the conjectured optimality across the other models. The main outstanding questions concern the optimality of $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$36 for BGK-type equations, the full $\partial_t f + v\cdot\nabla_x f = \mathcal{L}f,$37 range in the fractional case, and the extension beyond explicitly solvable operators.