Lanford’s theorem

Notes taken by Pantelis Tassopoulos

Lent 2026

1. Introduction

Large-scale classical systems, albeit being governed by a set of deterministic laws, are usually computationally intractable and one is naturally inclined to investigate the emergence of a simplified limiting dynamics. In physics this line of inquiry led to the growth of statistical mechanics which has seen a lot of success. However, the connection between classical mechanics and the ‘emergent’ statistical mechanics has has not been rigorously established, until very recently where substantial progress has been made.

We will be interested in the behaviour of a model of an idealised, dilute gas, see Figure 1. The ‘gas’ is comprised of $N\gg 1$ identical particles of diameter $\varepsilon > 0$ in $d(\ge 2)$-dimensional Euclidean space that move freely in a rectilinear fashion until they collide elastically with each other, from time to time.

It is natural to ask what may happen as one takes $N\to \infty$, in particular whether there is an emergent dynamics. We will investigate the so-called ‘Boltzmann-Grad’ limit where we tune $N\to \infty$ and $\varepsilon \to 0$ in such a way as to keep $N\varepsilon^{d-1}=1$. This is to ensure that the times between collisions are of unit order.

Now, for $N\ge 1$ particles at time $t\ge 0$, let

\[\overline{f}^N_t \stackrel{\mathrm{def}}{=} \frac{1}{N}\sum_{i=1}^N \mathbf{1}_{\{(x_i(t), v_i(t)\}}\,,\]

denote the empirical measure of the particles, which formally can be viewed as a measure on $\mathbb{R}^{2d}$.

The fact that the particle dynamics converge is the content of Lanford’s theorem. It states that if the empirical measures at the start, $\overline{f}^N_0\to f_0$ where $f_0$ is some function satisfying appropriate conditions (under a suitable notion of convergence), then there exists a time horizon $T_L (f_0)>0$ such that for $t\in [0, T_L (f_0)]$, $\overline{f}^N_t\to f_t$ in the same sense, where $(f_t: t\in [0, T_L (f_0)])$ is a solution to Boltzmann’s equation.

This result was a culmination of ¹ and ², where a ‘short time’ convergence result was established. More recently, there has been another breakthrough in ³ establishing long-time convergence, replacing $T_L (f_0)$ with the maximal existence interval of the solution to Boltzmann’s equation with initial data $f_0$.

In this short note, we will explore a means of constructing solutions to the Boltzmann equation by means of an `expansion on trees’, studying the collision histories of particles. A lot of the remaining work in the proof of Lanford’s theorem is to prove analogous results for the particle (or hard sphere) dynamics, which we do not pursue here.

2. Geometry of collisions

We now start to describe the model by explaining the mechanism and geometry of collisions between two particles. Consider two particles of diameter $\varepsilon > 0$ with positions $x_1, x_2\in \mathbb{R}^d$ and velocities $v_1, v_2\in \mathbb{R}^d$ respectively at some time $t\ge 0$.

Upon collision, the particles exchange the component of their momentum (hence velocities) orthogonal to their plane of contact (with normal vector $\omega = (x_1 - x_2)/\varepsilon$). Mathematically, the particles have exit velocities

\[\begin{cases} v'_1 & = v_1 - \langle v_1 - v_2, \omega \rangle\\ v'_2 & = v_2 + \langle v_1 - v_2, \omega \rangle\,. \end{cases}\]

Observe the for the particles to approach a collision, one must necessarily have

\[\frac{1}{2\varepsilon}\frac{\mathrm{d}}{\mathrm{d}t} |x_1-x_2|^2 = \langle v_1 - v_2, \omega\rangle \le 0 \qquad \mathrm{(pre - collisional)}\,.\]

One can readily verify that in an elastic collision (as above):

• the total momentum is conserved, i.e. $v_1 + v_2 = v’_1 + v’_2$;
• the total kinetic energy is conserved, i.e. $|v_1|^2 + |v_2|^2 = |v'_1|^2 + |v'_2|^2$;
• the outgoing velocities satisfy

\[\langle v'_1 - v'_2, \omega\rangle = -\langle v_1 - v_2, \omega\rangle\ge 0 \qquad \mathrm{(post - collisional)}\,.\]

3. Boltzmann’s equation

Boltzmann’s equation with unknown $f = f_t(x, v)$ on $[0, \infty)\times \mathbb{R}^{2d}$ is given by

\[\dot{f} + \overbrace{\langle v, \nabla_x f\rangle}^{\mathrm{ballistic\, transport\, term}} = \overbrace{Q(f, f)}^{\mathrm{collision\, term}}\,,\]

where $Q \equiv Q^+ - Q^-$ and for $f, g$ functions on $[0, \infty)\times \mathbb{R}^{2d}$,

\[\begin{cases} Q^+ (f,g)(x, v) &= \displaystyle\int_{\mathbb{R}^d \times S^{d-1}} f(x, v_1)g(x, v_2) \langle v'_1 - v'_2, \omega\rangle^- \mathrm{d} \omega \mathrm{d} u\qquad x, v\in \mathbb{R}^{d}\\ Q^- (f,g)(x, v) &= \displaystyle\int_{\mathbb{R}^d \times S^{d-1}} f(x, v)g(x, u) \langle v - u, \omega\rangle^- \mathrm{d} \omega \mathrm{d} u\qquad x, v\in \mathbb{R}^{d}\,, \end{cases}\]

with $v_1 = v - \langle v - u, \omega \rangle$ and $v_2 = u + \langle v - u, \omega \rangle$. Informally, the $Q^-$ term represents the rate of pre-collisional particles to a particle at position $x$ with velocity $v$; hence more particles are lost the higher $Q^-$ is, hence the minus sign in the definition of $Q$. Similarly, the $Q^+$ term represents the total ‘rate’ of pre-collisional particles such that were they to become incident at location $x$, the first one would have velocity $v$; thus the positive sign.

Formally, integrating out both sides in the differential equation, one obtains the integral equation

\[f_t = P_t f_0 + \displaystyle \int^t_0 P_{t-s}Q(f_s, f_s)\mathrm{d} s\,,\tag{BE}\]

where $P_t f(x, v) = f(x - vt, v)$ for $t\ge 0$, $x, v\in \mathbb{R}^d$, is the density propagator for ballistic transport.

3.1 Sub-Gaussian solutions

We now impose some asymptotic growth conditions on the functions that we will ask to solve Boltzmann’s equation. We make the following definition.

Definition 1 (Sub-Gaussian) > For $T>0$, and $f$ a measurable function on $[0, T]\times \mathbb{R}^{2d}$, we say that it is sub-gaussian if there exist constants $A, \beta \in (0, \infty)$ such that

\[|f_t(x, v)| \le A\mathrm{e}^{-\beta |v|^2}\qquad t\in [0, T]\,, x, v\in \mathbb{R}^d\,.\]

Remark: One can easily check for $\beta \in (0, \infty)$ that $f = \mathrm{e}^{-\beta |v|^2}$ (for $v\in \mathbb{R}^d$) is a (stationary) solution to the Boltzmann equation.

This definition is nicely compatible with the Boltzmann equation as all terms that appear in the integral equation are well-defined and finite. We now come to a notion of solution to the Boltzmann equation incorporating this kind of growth condition.

Definition 2 (Sub-Gaussian solution to Boltzmann’s equation) > For $T>0$, and $f$ a measurable function on $[0, T]\times \mathbb{R}^{2d}$, we say that it is a sub-gaussian solution to the Boltzmann integral equation, if there exist constants $A, \beta \in (0, \infty)$ such that

\[|f_t(x, v)| \le A\mathrm{e}^{-\beta |v|^2}\qquad t\in [0, T]\,, x, v\in \mathbb{R}^d\,.\]

Moreover, one has uniqueness of sub-Gaussian solutions, as expressed in the following proposition.

Proposition 1 (Uniqueness of sub-Gaussian solutions) > Let $f, g$ be sub-Gaussian solutions to the Boltzmann integral equation on $[0, T]$, for some $T > 0$ and suppose that $f_0 = g_0$. Then, $f_t = g_t$ for all $t\in [0, T]$.

Proof

Suppose that $f$ and $g$ are sub-Gaussian solutions with constants $A$ and $\beta$. We will show that there exists a time $$ t_0 = t_0(A,\beta,d) > 0 $$ such that $f_t = g_t$ for all $t \le t_0$. The claim then follows by repeated application of this property. Write $$ |v|^2 + |u|^2 = r^2 $$ and note that, for $s < t$, $$ |v-u|^2 \mathrm{e}^{-2(t-s)\beta r^2} \le 2r^2 \mathrm{e}^{-2(t-s)\beta r^2} \le \frac{1}{(t-s)\beta}, $$ so that $$ |v-u| \mathrm{e}^{-(t-s)\beta r^2} \le \frac{1}{\sqrt{(t-s)\beta}} . $$ Set $m_t = \lvert f_t - g_t \rvert$. Subtracting the integral equations for $f$ and $g$, we obtain for $x,v \in \mathbb{R}^d$, with $x(s) = x - (t-s)v$: $$ \begin{aligned} m_t(x,v) &\le \int_0^t \int_{\mathbb{R}^d \times S^{d-1}} \Bigl( |g_s(x(s),v_1)|\,m_s(x(s),v_2) + m_s(x(s),v_1)\,|f_s(x(s),v_2)| \Bigr) \langle v_1 - v_2, \omega \rangle^{-} \,\mathrm{d}\omega\,\mathrm{d}u\,\mathrm{d}s \\ &\quad + \int_0^t \int_{\mathbb{R}^d \times S^{d-1}} \Bigl( |g_s(x(s),v)|\,m_s(x(s),u) + m_s(x(s),v)\,|f_s(x(s),u)| \Bigr) \langle v-u, \omega \rangle^{-} \,\mathrm{d}\omega\,\mathrm{d}u\,\mathrm{d}s . \end{aligned} $$ We use that $$ \int_{S^{d-1}} \langle v,\omega\rangle^{-}\,\mathrm{d}\omega = V_{d-1}|v|, $$ where $V_{d-1}$ denotes the volume of the $(d-1)$-dimensional unit ball. Define $$ y_t = \sup_{x,v\in\mathbb{R}^d} m_t(x,v)\,\mathrm{e}^{(1-t)\beta|v|^2}. $$ Then $$ m_t(x,v) \le y_t \mathrm{e}^{-(1-t)\beta|v|^2}, $$ and since $ \lvert v_1 \rvert^2 + \lvert v_2 \rvert^2 = r^2 $, we obtain $$ m_t(x,v) \le 4A V_{d-1} \int_0^t \int_{\mathbb{R}^d} y_s \mathrm{e}^{-(1-s)\beta r^2} |v-u|\,\mathrm{d}u\,\mathrm{d}s . $$ Hence, for $t \le \tfrac{1}{2}$, $$ \begin{aligned} m_t(x,v)\mathrm{e}^{(1-t)\beta|v|^2} &\le 4A V_{d-1} \int_0^t \int_{\mathbb{R}^d} y_s \mathrm{e}^{-(t-s)\beta r^2 - (1-t)\beta|u|^2} |v-u| \,\mathrm{d}u\,\mathrm{d}s \\ &\le C \int_0^t \frac{y_s}{\sqrt{t-s}}\,\mathrm{d}s, \end{aligned} $$ where $$ C = C(A,\beta,d) = 4A V_{d-1}(2\pi)^{d/2}\beta^{-(d+1)/2}, $$ and we used $$ \int_{\mathbb{R}^d} \mathrm{e}^{-\beta|u|^2/2}\,\mathrm{d}u = \left(\frac{2\pi}{\beta}\right)^{d/2}. $$ Set $$ t_0 = (4C)^{-2} \wedge \frac{1}{2} $$ and note that $$ C \int_0^{t_0} \frac{1}{\sqrt{t_0-s}}\,\mathrm{d}s = 2C\sqrt{t_0} \le \frac{1}{2} . $$ Let $$ S = \sup_{t \le t_0} y_t . $$ Then $S \le 2A$ and $S \le S/2$, hence $S=0$. Therefore, $$ f_t = g_t \qquad \text{for all } t \le t_0 . \qquad \blacksquare $$

However, existence of sub-Gaussian solutions is a lot more subtle and it is an open problem as to whether for sub-Gaussian initial data $f_0$, there exists a sub-Gaussian solution for all $[0, T]$, $T> 0$.

For sub-Gaussian solutions to the Boltzmann integral equation, one can prove the following conservation law:

\[\displaystyle \int_{\mathbb{R}^{2d}}f_t(x, v)\mathrm{d} x \mathrm{d} v = \mathrm{constant}\,,\]

and that if the initial data $f_0 \ge 0$ everywhere, then $f_t \ge 0$ for all $t$.

3.2 Collision histories

It is useful to keep track of collision histories of particles in this model. For a given particle at a given time $t> 0$, we will want to trace back time and uncover its collision history. We will consider actual collisions between the particle of interest and another particle, denoted pictorially with a filled node or more algebraically $[1, 1]$, and potential collisions that for some intervening reason, did not happen, denoted pictorially with an open node or algebraically $(1,1)$.

Using these notions of collision, we establish the following ‘tree’ spaces using an induction principle.

Definition 3 (Tree spaces) Let $\mathbb{I}$ denote the set generated recursively by $1\in \mathbb{I}$, and the inductive step $\tau_1, \tau_2 \in \mathbb{I}$ implies $[\tau_1, \tau_2]\in \mathbb{I}$. Similarly, let $\mathbb{I}^{(2)}$ denote the set generated recursively by $1\in \mathbb{I}^{(2)}$, and the inductive step $\tau_1, \tau_2 \in \mathbb{I}^{(2)}$ implies $[\tau_1, \tau_2], (\tau_1, \tau_2)\in \mathbb{I}^{(2)}$.

3.3 Small time expansion for sub-Gaussian initial data

Having established the above notation for our two types of trees based on two types of collision, we will ‘build’ solutions to the Boltzmann integral equation, by ‘summing over all collision histories’.

Now, given measurable $f_0\ge 0 $ on $\mathbb{R}^{2d}$, define recursively for $\tau \in\mathbb{T}^{(2)}$, $(g^\tau_t: \tau \in \mathbb{T}^{(2)}\,, t\ge 0 )$ as follows

\[\begin{cases} g^{\mathbf{1}}_t &\equiv P_t f_0\\ g^{[\tau_1, \tau_2]}_t &\equiv \int^t_0 P_{t-s} Q^+(g^{\tau_1}_s, g^{\tau_2}_s)\mathrm{d} s\,,\qquad\hfill \tau_1, \tau_2\in \mathbb{T}^{(2)}\\ g^{(\tau_1, \tau_2)}_t &\equiv \int^t_0 P_{t-s} Q^-(g^{\tau_1}_s, g^{\tau_2}_s)\mathrm{d} s\,,\qquad \hfill\tau_1, \tau_2\in \mathbb{T}^{(2)} \end{cases}\tag{pseudo-particle equations}\]

One can write the pseudo-particle equations more explicitly for fixed $x, v\in \mathbb{R}^d$ as,

\[\begin{cases} g^{\mathbf{1}}_t (x,v) &= f_0(x-tv, v)\\ g^{[\tau_1, \tau_2]}_t (x, v) &=\displaystyle\int^t_0\int_{S^{d-1}\times \mathbb{R}^d} g^{\tau_1}_s (x-(t-s)v, v_1) g^{\tau_2}_s(x-(t-s)v, v_2)\cdot \langle v_1-v_2 , \omega\rangle^-\mathrm{d} \omega\mathrm{d} u\mathrm{d} s\,,\qquad \tau_1, \tau_2\in \mathbb{T}^{(2)}\\ g^{(\tau_1, \tau_2)}_t (x, v) &= \displaystyle\int^t_0\int_{S^{d-1}\times \mathbb{R}^d} g^{\tau_1}_s (x-(t-s)v, v) g^{\tau_2}_s(x-(t-s)v, u)\cdot \langle v-u , \omega\rangle^-\mathrm{d} \omega\mathrm{d} u\mathrm{d} s\,.\qquad \tau_1, \tau_2\in \mathbb{T}^{(2)} \end{cases}\]

Note the first term corresponds to the history with no collisions (——, or $\mathbf{1}$), that is the pure transport term. Then for $\tau_1, \tau_2\in \mathbb{T}^{(2)}$, $g^{[\tau_1, \tau_2]}_t (x, v)$, $x, v\in \mathbb{R}^d$ corresponds to a term counting the rate of collisions of particles in $g^{\tau_1}$ against particles from $g^{\tau_2}_s$ (that is with collision histories $\tau_1$ and $\tau_2$ respectively, see diagram on the left in Figure 3) that set them on a fixed course with velocity $v$ ending up in location $x\in \mathbb{R}^d$ at time $t\ge 0$. Similarly, $g^{(\tau_1, \tau_2)}_t (x, v)$, $x, v\in \mathbb{R}^d$ corresponds to a term counting the rate of possible collisions of particles in $g^{\tau_1}$ against particles from $g^{\tau_2}_s$ (see diagram on the right in Figure 3), where the former move along a fixed rectilinear course with velocity $v$ ending up in location $x\in \mathbb{R}^d$ at time $t\ge 0$. Particles involved in collisions of the second form are also referred to as ghost particles.

How will every such term contribute to the total density? Clearly, for $\mathbf{1}\in \mathbb{T}^{(2)}$, we want the $g^{\mathbf{1}}_t$ to count positively, as this is just the transport of particles. However, we may lose some due to potential collisions, hence we discount the $g^{(\mathbf{1}, \mathbf{1})}_t$ term. Similarly, the term $g^{[\mathbf{1}, \mathbf{1}]}_t$ must reinforce the total density as it presents with an additional way to obtain particles with a fixed velocity at a given space-time location.

For more general collision histories $\tau \in \mathbb{T}^{(2)}$, the way to determine the sign, $\mathrm{sign}(\tau)$ ($= +1$ for a reinforcing history and $=-1$ for a discounting history) is by induction. That is fixing $\tau \in \mathbb{T}^{(2)}$, the node of the tree with an additional filled node or open node inserted in exactly one edge connecting a root node to an interior node, the sign will be $\pm \mathrm{sign}(\tau)$ respectively. The reasons for this are analogous to the foregoing discussion.

An equivalent definition is to start by declaring $\mathrm{sign}(\mathbf{1}) = 1$ and for $\tau_1, \tau_2 \in \mathbb{T}^{(2)}$, declare $\mathrm{sign}([\tau_1, \tau_2]) = \mathrm{sign}(\tau_1)\mathrm{sign}(\tau_2)$ and $\mathrm{sign}((\tau_1, \tau_2)) = -\mathrm{sign}(\tau_1)\mathrm{sign}(\tau_2)$. Below is an illustration of some trees in $\mathbb{T}^{(2)}$, and their signs. The sign essentially depends on the parity of the number of circular brackets in the algebraic representation of $\tau \in \mathbb{T}^{(2)}$.

The following proposition now gives control over the contributions $g^\tau_t$, $t\ge 0$ from histories with many collisions.

Proposition 2 (Small time control, many collisions) Fix any $\delta\in (0, 1/2]$ and suppose the initial density $f_0$ is sub-Gaussian with \(f_0(x, v) \le A_0 \mathrm{e}^{-\beta |v|^2}\,,\qquad x, v\in \mathbb{R}^d\) for some $A_0, \beta>0 $. Then, there exists $C \equiv C(\beta, d, \delta)<\infty$ such that for all $\tau \in \mathbb{T}^{(2)}$, $t\ge 0$, $x, v\in \mathbb{R}^d$, \(g^{\tau}_t (x, v) \le (A_0)^m (Ct)^{m-1} \mathrm{e}^{-(1-\delta)\beta |v|^2}\,.\)

Proof

Set $$ C = C(\beta_0,\delta,d) = c(\beta_0,d)\,\delta^{-1/2}, \qquad c(\beta_0,d) = V_{d-1}(2\pi)^{d/2}\beta_0^{-(d+1)/2}. $$ Fix $t_0 \ge 0$ and set $$ \lambda = \frac{\delta}{t_0}, \qquad B = C\sqrt{t_0}, \qquad \beta(t) = (1-\lambda t)\beta_0 . $$ Note that $\beta(t_0) = (1-\delta)\beta_0$. We will show that, for all $m \ge 1$, all $\tau \in \mathbb{T}^{(2)}$ with $m$ leaves, all $t \in [0,t_0]$, and all $x,v \in \mathbb{R}^d$, $$ g_t^{\tau}(x,v) \le A_0^{m}(B\sqrt{t})^{m-1} \mathrm{e}^{-\beta(t)|v|^2}. \tag{8} $$ The main bound follows by taking $t=t_0$. The inequality $(8)$ holds for $m=1$ since $f_0$ is sub-Gaussian. Suppose inductively that it holds for $m$ and that $\tau=[\tau_1,\tau_2]$ has $m+1$ leaves, where $\tau_1$ and $\tau_2$ have $m_1$ and $m_2$ leaves respectively. In the following calculation we use the fact that $$ |v_1|^2 + |v_2|^2 = r^2 = |v|^2 + |u|^2 . $$ Since $m_1,m_2 \le m$, for $t \in [0,t_0]$ we have $$ \begin{aligned} g_t^{\tau}(x,v) &= \int_0^t \int_{\mathbb{R}^d \times S^{d-1}} g_s^{\tau_1}\bigl(x-(t-s)v, v_1\bigr)\, g_s^{\tau_2}\bigl(x-(t-s)v, v_2\bigr) \langle v_1 - v_2, \omega \rangle^{-}\, \mathrm{d}\omega\,\mathrm{d}u\,\mathrm{d}s \\ &\le A_0^{m_1+m_2}(B\sqrt{t})^{m_1+m_2-2} \int_0^t \int_{\mathbb{R}^d \times S^{d-1}} \mathrm{e}^{-\beta(s)r^2} \langle v-u,\omega\rangle_{+}\, \mathrm{d}\omega\,\mathrm{d}u\,\mathrm{d}s \\ &= A_0^{m+1}(B\sqrt{t})^{m-1} V_{d-1} \int_0^t \int_{\mathbb{R}^d} |v-u|\,\mathrm{e}^{-\beta(s)r^2} \,\mathrm{d}u\,\mathrm{d}s . \end{aligned} $$ Note that $|v-u| \le \sqrt{2}\,r$ and $$ \beta(s) = \beta(t) + \lambda (t-s)\beta_0 . $$ We use the inequality $$ \int_0^t r\,\mathrm{e}^{-\lambda(t-s)\beta_0 r^2}\,\mathrm{d}s \le \frac{r t}{2\lambda\beta_0} $$ to obtain $$ \begin{aligned} V_{d-1} \int_0^t \int_{\mathbb{R}^d} |v-u|\,\mathrm{e}^{-\beta(s)r^2} \,\mathrm{d}u\,\mathrm{d}s &\le \sqrt{2}V_{d-1} \int_{\mathbb{R}^d} \mathrm{e}^{-\beta(t)r^2} \int_0^t r\,\mathrm{e}^{-\lambda(t-s)\beta_0 r^2}\,\mathrm{d}s \,\mathrm{d}u \\ &\le V_{d-1} \mathrm{e}^{-\beta(t)|v|^2} \int_{\mathbb{R}^d} \mathrm{e}^{-\beta(t)|u|^2}\,\mathrm{d}u\, \frac{t}{\lambda\beta_0} \\ &\le B\sqrt{t}\,\mathrm{e}^{-\beta(t)|v|^2}, \end{aligned} $$ where for the final inequality we used $$ \int_{\mathbb{R}^d} \mathrm{e}^{-\beta(t)|u|^2}\,\mathrm{d}u = \left(\frac{\pi}{\beta(t)}\right)^{d/2} \le \left(\frac{2\pi}{\beta_0}\right)^{d/2}. $$ Hence, $$ g_t^{\tau}(x,v) \le A_0^{m+1}(B\sqrt{t})^{m} \mathrm{e}^{-\beta(t)|v|^2}. $$ In the case $\tau=(\tau_1,\tau_2)$, the initial equation is replaced by $$ g_t^{\tau}(x,v) = \int_0^t \int_{\mathbb{R}^d \times S^{d-1}} g_s^{\tau_1}\bigl(x-(t-s)v, v\bigr)\, g_s^{\tau_2}\bigl(x-(t-s)v, u\bigr) \langle v-u,\omega\rangle^{-}\, \mathrm{d}\omega\,\mathrm{d}u\,\mathrm{d}s, $$ and a similar argument leads to the same bound for $g_t^{\tau}(x,v)$, so the induction proceeds. $\blacksquare$

Now, we are in a position to construct the sub-Gaussian solution to the Boltzmann integral equation, by adding all of the contributions from the various collision histories.

Proposition 3 (Construction of sub-Gaussian solution) Under the same conditions as in Proposition 2, set $T_0 = (8A_0C)^{-1}$. Then for all $0\le t \le T_0$, the following series converges absolutely and uniformly: \(f_t \equiv \sum_{\tau \in \mathbb{T}^{(2)}}\mathrm{sign}(\tau) \cdot g^{\tau}_t\,,\) and defines a sub-Gaussian solution of the Boltzmann integral equation.

Proof

We start with some combinatorial observations. For $m\ge 1$, let $\mathbb{T}_m, \mathbb{T}^{(2)}_m$ denote the trees in $\mathbb{T}, \mathbb{T}^{(2)}$ respectively, which have $m$ root nodes (and so $m-1$ interior nodes). Then, one observes that $$ |\mathbb{T}_m| = \sum_{k=1}^{m-1} |\mathbb{T}_{k}|\cdot |\mathbb{T}_{m-k}|\,, $$ which is the same recursion as that of the Catalan numbers, $C_m = \frac{1}{m+1} {2m \choose m}$, $m\ge 0$. Since $|\mathbb{T}_1| = C_0 = 1$, we conclude $ |\mathbb{T}_m| = C_{m-1}\le 4^{m-1}$, $m\ge 1$. Moreover, since the number of trees in $\mathbb{T}^{(2)}_m$ can be estimated by $|\mathbb{T}_m|\cdot 2^{m-1}$, due to the possible choices of 'hollow' or 'filled in' interior nodes, one obtains by Stirling's formula, the bound $|\mathbb{T}^{(2)}_m| \le 2^{m-1}\cdot 4^{m-1}\le 8^{m-1}$. Now, using Proposition 2, $$ g_t\equiv \sum_{\tau \in \mathbb{T}^{(2)}}g^\tau _t (x, v) \le A_0 \underbrace{\sum_{m=1}^\infty 8^{m-1}A^{m-1}_0 (Ct)^{m-1}}_{\le 2 \text{ (for } t\le T_0)}\cdot \mathrm{e}^{-(1-\delta)|v|^2}\le 2\mathrm{e}^{-(1-\delta)|v|^2}\,,\qquad x, v\in \mathbb{R}^d\,. $$ Now, by the absolute convergence of $g$, we have for $f_t$ as in the statement, $$ \begin{aligned} f_t \equiv \sum_{\tau \in \mathbb{T}^{(2)}}f^\tau _t &= f^{\mathbf{1}}_t + \sum_{\tau_1, \tau_2 \in \mathbb{T}^{(2)}}\mathrm{sign}([\tau_1, \tau_2])f^{[\tau_1, \tau_2]}_t+ \sum_{\tau_1, \tau_2 \in \mathbb{T}^{(2)}}\mathrm{sign}((\tau_1, \tau_2))f^{(\tau_1, \tau_2)}_t\\ &= P_t f_0 + \sum_{\tau_1, \tau_2 \in \mathbb{T}^{(2)}}\displaystyle \mathrm{sign}(\tau_1)\cdot \mathrm{sign}(\tau_2)\int^t_0 P_{t-s}Q^+ (f^{\tau_1}_s, f^{\tau_2}_s)\mathrm{d} s\\ &\quad - \mathrm{sign}(\tau_1)\cdot \mathrm{sign}(\tau_2)\displaystyle \int^t_0 P_{t-s}Q^- (f^{\tau_1}_s, f^{\tau_2}_s)\mathrm{d} s\\ &= P_t f_0 + \displaystyle \int^t_0 P_{t-s}Q^+ (g_s, g_s)\mathrm{d} s - \displaystyle \int^t_0 P_{t-s}Q^- (g_s, g_s)\mathrm{d} s\qquad \text{(bi-linearity of } Q^\pm \text{)}\,. \end{aligned} $$ $\blacksquare$

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