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The integral equation of the drift

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

Give every edge of a graph a continuous path \(w_e\). This lesson solves the equations \[ Z_e(t)=w_e(t)+\int_0^tb_e(s,Z(s))\,ds \] for all edges at once, where \(b\) is the drift of the free uniform spanning forest. There are infinitely many unknown paths, each equation involves all of them, and the paths \(w_e\) are arbitrary: they need not be bounded in \(e\). The bound \(\tfrac12\) of the drift, which compares two families of numbers through the largest difference, is exactly what the classical method of successive approximations needs. The lesson is formulated for a general drift with such a bound.

The lesson follows [OpenAI-F, Section 4, the lemma on deterministic inversion]. It continues Tilted laws and the drift. It uses the Lebesgue integral on intervals, measurable functions, monotone convergence, and the monotone class theorem for families of sets (core course Measure and Integration).

1. Drifts and families of paths

Let \(I\) be a countable set. On \([0,\infty)\times\mathbb R^I\) we use the \(\sigma\)-algebra generated by \(t\) and the coordinates \(y_i\), and call its measurable functions Borel functions.

Definition 1.1. A drift with the constant \(L\), where \(0\le L<\infty\), is a family \(b=(b_i)_{i\in I}\) of Borel functions \(b_i\colon[0,\infty)\times\mathbb R^I\to[0,1]\) such that \[ \sup_{i\in I}|b_i(t,y)-b_i(t,y')|\le L\,\sup_{j\in I}|y_j-y'_j| \] for every \(t\ge0\) and all \(y,y'\in\mathbb R^I\) for which the supremum on the right is finite.

By Proposition 4.3 of the lesson on tilted laws and the drift, the drift of the free uniform spanning forest of a graph is a drift with the constant \(\tfrac12\), with \(I\) the set of edges.

Let \(\mathcal C\) be the set of the continuous real functions on \([0,\infty)\), and \(\mathcal C_I\) the set of the families \(z=(z_i)_{i\in I}\) of elements of \(\mathcal C\). For \(z\in\mathcal C_I\) and \(t\ge0\) let \(z(t)=(z_i(t))_{i\in I}\in\mathbb R^I\). No bound in \(i\) is required.

Lemma 1.2. Let \(b\) be a family of Borel functions with values in \([0,1]\), \(z\in\mathcal C_I\) and \(i\in I\). Then \(s\mapsto b_i(s,z(s))\) is a Borel function on \([0,\infty)\) with values in \([0,1]\), and \[ t\mapsto\int_0^tb_i(s,z(s))\,ds \] is continuous and increasing, and it changes by at most \(|t-t'|\) between \(t'\) and \(t\).

Proof. The map \(s\mapsto(s,z(s))\) from \([0,\infty)\) to \([0,\infty)\times\mathbb R^I\) is measurable, because each of its coordinates is continuous. Its composition with the Borel function \(b_i\) is measurable. The integrand lies between \(0\) and \(1\). \(\blacksquare\)

2. Existence and uniqueness

Theorem 2.1. Let \(b\) be a drift with the constant \(L\). For every \(w\in\mathcal C_I\) there is exactly one \(Z=Z(w)\in\mathcal C_I\) with \[ Z_i(t)=w_i(t)+\int_0^tb_i(s,Z(s))\,ds \] for all \(i\in I\) and \(t\ge0\). For every \(i\), the function \(Z_i-w_i\) is increasing and changes by at most \(|t-t'|\) between \(t'\) and \(t\).

Proof. Successive approximations. Let \(Z^0=w\) and, for \(k\ge0\), \[ Z^{k+1}_i(t)=w_i(t)+\int_0^tb_i(s,Z^k(s))\,ds . \] By Lemma 1.2 these are elements of \(\mathcal C_I\). We show by induction that for all \(k\ge0\) and \(t\ge0\) \[ \sup_{i\in I}|Z^{k+1}_i(t)-Z^k_i(t)|\le\frac{L^kt^{k+1}}{(k+1)!}, \] with \(L^0=1\). For \(k=0\) the difference is the integral of \(b_i(s,w(s))\) over \([0,t]\), which is at most \(t\). Let \(k\ge1\). By the bound for \(k-1\), the two families of numbers \(Z^k(s)\) and \(Z^{k-1}(s)\) differ by at most \(L^{k-1}s^k/k!\) in every coordinate, so the inequality of Definition 1.1 applies to them, and \[ \begin{aligned} |Z^{k+1}_i(t)-Z^k_i(t)|&\le\int_0^tL\cdot\frac{L^{k-1}s^k}{k!}\,ds\\ &=\frac{L^kt^{k+1}}{(k+1)!}. \end{aligned} \]

Convergence. Fix \(T\ge0\), and let \(R_k=\sum_{m\ge k}L^mT^{m+1}/(m+1)!\). This is the remainder of a convergent series, so \(R_k\to0\). For \(t\le T\) and \(k\le l\), every coordinate of \(Z^l(t)-Z^k(t)\) has absolute value at most \(R_k\). So for every \(i\) the functions \(Z^k_i\) converge uniformly on \([0,T]\), for every \(T\); the limit \(Z_i\) is continuous, and \[ \sup_{i\in I}\ \sup_{0\le t\le T}|Z_i(t)-Z^k_i(t)|\le R_k . \] By Definition 1.1, \(|b_i(s,Z(s))-b_i(s,Z^k(s))|\le LR_k\) for \(s\le T\). So the integral in the definition of \(Z^{k+1}_i(t)\) converges to \(\int_0^tb_i(s,Z(s))\,ds\) as \(k\to\infty\), for \(t\le T\), and \(Z\) satisfies the equation.

Uniqueness. Let \(z\in\mathcal C_I\) be a solution. Then \(|z_i(t)-Z^0_i(t)|\le t\) for all \(i\). If every coordinate of \(z(s)-Z^k(s)\) is at most \(L^ks^{k+1}/(k+1)!\) in absolute value, then \[ \begin{aligned} |z_i(t)-Z^{k+1}_i(t)|&\le\int_0^tL\cdot\frac{L^ks^{k+1}}{(k+1)!}\,ds\\ &=\frac{L^{k+1}t^{k+2}}{(k+2)!}. \end{aligned} \] So this holds for all \(k\), the right side tends to \(0\), and \(z=Z\).

The last statement follows from the equation and Lemma 1.2. \(\blacksquare\)

All estimates in the proof concern differences of two families with the same \(w\). The family \(w\) itself may be unbounded in \(i\), and its coordinates need not be equicontinuous.

Example 2.2. Let \(I\) have one element, and \[ b(t,y)=\frac{e^{y-t/2}}{1+e^{y-t/2}} . \] This is the function of Exercise 5.1 of the lesson on tilted laws and the drift for \(p=\tfrac12\): the conditional probability that a hidden variable with the values \(0\) and \(1\), each with probability \(\tfrac12\), is \(1\), given an observation \(y\) at the time \(t\). Its derivative in \(y\) is \(b(1-b)\le\tfrac14\), so it is a drift with the constant \(\tfrac14\). Let \(w\) be the constant path with the value \(c\). The solution is \[ Z(t)=\frac t2+2\operatorname{arsinh}\bigl(\sinh\bigl(\tfrac c2\bigr)e^{t/4}\bigr). \] Indeed, let \(u(t)=Z(t)-\tfrac t2\). Then \(\sinh(u/2)=\sinh(c/2)e^{t/4}\), so \(u(0)=c\), and differentiating gives \(\tfrac12\cosh(u/2)\,u'=\tfrac14\sinh(u/2)\), that is \(u'=\tfrac12\tanh(u/2)\). On the other hand \(b(t,Z(t))=e^u/(1+e^u)\), which is \(\tfrac12+\tfrac12\tanh(u/2)\). So \(Z'(t)=b(t,Z(t))\) and \(Z(0)=c\).

For \(c=0\) the solution is \(Z(t)=\tfrac t2\). For \(c>0\), \(Z(t)-t\) converges to \(2\log(2\sinh(c/2))\), and for \(c<0\), \(Z(t)\) converges to \(-2\log(2\sinh(|c|/2))\). So the slope \(Z(t)/t\) tends to \(1\), \(\tfrac12\) or \(0\) according to the sign of \(c\): the solution decides between the two values of the hidden variable, and an arbitrarily small \(c\ne0\) decides it.

3. Causality and relabelling

Proposition 3.1. Let \(b\) be a drift with the constant \(L\).

  1. If \(w,w'\in\mathcal C_I\) agree on \([0,T]\) in every coordinate, then \(Z(w)\) and \(Z(w')\) agree on \([0,T]\) in every coordinate.
  2. Let \(\sigma\colon I\to I'\) be a bijection of countable sets and \(b'\) a drift on \(I'\). For \(y\in\mathbb R^I\) let \(\sigma y\in\mathbb R^{I'}\) be given by \((\sigma y)_{\sigma(i)}=y_i\), and define \(\sigma w\) for \(w\in\mathcal C_I\) in the same way. If \(b'_{\sigma(i)}(t,\sigma y)=b_i(t,y)\) for all \(i,t,y\), then \(Z'(\sigma w)=\sigma Z(w)\), where \(Z'\) is the solution map of \(b'\).

Proof. (1) By induction on \(k\), the approximations \(Z^k(w)\) and \(Z^k(w')\) agree on \([0,T]\), because \(Z^{k+1}(t)\) is formed from \(w(t)\) and the values of \(Z^k\) on \([0,t]\).

(2) Let \(z'=\sigma Z(w)\). Then \(z'(s)=\sigma(Z(w)(s))\), and for every \(i\) \[ \begin{aligned} &z'_{\sigma(i)}(t)\\ &\ =w_i(t)+\int_0^tb_i(s,Z(w)(s))\,ds\\ &\ =(\sigma w)_{\sigma(i)}(t)+\int_0^tb'_{\sigma(i)}(s,z'(s))\,ds . \end{aligned} \] So \(z'\) solves the equation of \(b'\) for \(\sigma w\), and by the uniqueness in Theorem 2.1 it is \(Z'(\sigma w)\). \(\blacksquare\)

4. Measurability

On \(\mathcal C_I\) we use the \(\sigma\)-algebra generated by the evaluations \(z\mapsto z_i(t)\), \(i\in I\), \(t\ge0\). A map \(F\) from a measurable space \(\Omega\) to \(\mathcal C_I\) is measurable exactly if \(\omega\mapsto F_i(\omega)(t)\) is measurable for all \(i\) and \(t\). For one coordinate this is the Borel \(\sigma\)-algebra of the uniform convergence on bounded intervals (Exercise 6.4).

Lemma 4.1. Let \(\Omega\) be a measurable space, \(t\ge0\), and \(g\colon[0,t]\times\Omega\to[0,1]\) measurable for the product of the Borel \(\sigma\)-algebra of \([0,t]\) and the \(\sigma\)-algebra of \(\Omega\). Then \(\omega\mapsto\int_0^tg(s,\omega)\,ds\) is measurable.

Proof. Let \(\lambda\) be the Lebesgue measure on \([0,t]\). For a measurable set \(D\subseteq[0,t]\times\Omega\) let \(D_\omega=\{s:(s,\omega)\in D\}\), and let \(\mathcal D\) be the family of the measurable \(D\) for which \(\omega\mapsto\lambda(D_\omega)\) is measurable. It contains the products \(B\times A\) of measurable sets, for which the function is \(\lambda(B)\) on \(A\) and \(0\) elsewhere; these products are closed under intersections and generate the product \(\sigma\)-algebra. The family \(\mathcal D\) contains the whole space, with \(D\subseteq D'\) in \(\mathcal D\) it contains \(D'\setminus D\), because \(\lambda((D'\setminus D)_\omega)=\lambda(D'_\omega)-\lambda(D_\omega)\), and it contains the union of an increasing sequence of its elements. A family of sets with these three properties that contains a family closed under intersections contains the \(\sigma\)-algebra generated by it (the monotone class theorem). So \(\mathcal D\) consists of all measurable sets. This is the assertion for indicator functions. It follows for finite linear combinations of indicator functions, and for \(g\) as the limit of the increasing sequence \(g_n=2^{-n}\lfloor2^ng\rfloor\), by monotone convergence. \(\blacksquare\)

Lemma 4.2. Let \(\Omega\) be a measurable space, \(F\colon\Omega\to\mathcal C_I\) measurable, and \(b\) a family of Borel functions as in Lemma 1.2. Then \((s,\omega)\mapsto b_i(s,F(\omega)(s))\) is measurable on \([0,\infty)\times\Omega\), for every \(i\).

Proof. Let \(j\in I\). For \(n\ge1\) the function \((s,\omega)\mapsto F_j(\omega)(\lfloor ns\rfloor/n)\) is measurable, because on each of the countably many sets \([m/n,(m+1)/n)\times\Omega\) it is the measurable function \(\omega\mapsto F_j(\omega)(m/n)\). As \(n\to\infty\) these functions converge to \(F_j(\omega)(s)\), by the continuity of the paths. So \((s,\omega)\mapsto F_j(\omega)(s)\) is measurable. Hence \((s,\omega)\mapsto(s,F(\omega)(s))\) is measurable as a map to \([0,\infty)\times\mathbb R^I\), and so is its composition with \(b_i\). \(\blacksquare\)

Theorem 4.3. Let \(b\) be a drift with the constant \(L\). The solution map \(Z\colon\mathcal C_I\to\mathcal C_I\) is measurable.

Proof. We show by induction that the approximations \(Z^k\) of the proof of Theorem 2.1 are measurable maps from \(\mathcal C_I\) to \(\mathcal C_I\). \(Z^0\) is the identity. If \(Z^k\) is measurable, then by Lemma 4.2 with \(\Omega=\mathcal C_I\) and \(F=Z^k\), and by Lemma 4.1, the function \(w\mapsto Z^{k+1}_i(w)(t)\), which is \[ w_i(t)+\int_0^tb_i(s,Z^k(w)(s))\,ds, \] is measurable for all \(i\) and \(t\). Finally \(Z_i(w)(t)\) is the limit of the \(Z^k_i(w)(t)\). \(\blacksquare\)

5. The forest

Let \(G\) be an infinite connected graph in which every vertex lies on finitely many edges, with the edge set \(E\), and let \(b=(b_e)_{e\in E}\) be the drift of its free uniform spanning forest. It is a drift with the constant \(\tfrac12\). So every family \(w=(w_e)_{e\in E}\) of continuous paths has a unique solution \(Z(w)\) of the equations at the head of this lesson, and \(Z\) is measurable and has the properties of Proposition 3.1. By Proposition 4.3 of the lesson on tilted laws and the drift, the relabelling by an isomorphism of graphs satisfies the condition of Proposition 3.1; so the solution map commutes with isomorphisms of graphs.

For every edge, \(Z_e(w)(t)-w_e(t)\) is increasing in \(t\) with a slope between \(0\) and \(1\). Define a set of edges \(\Psi(w)\): an edge \(e\) belongs to \(\Psi(w)\) if and only if \[ \limsup_{n\to\infty}\frac{Z_e(w)(n)}{n}>\frac12, \] with \(n\) running through the positive integers. The map \(\Psi\colon\mathcal C_E\to\{0,1\}^E\) is measurable, by Theorem 4.3, and it commutes with isomorphisms of graphs. It is defined for every \(w\), without any exceptional set.

The theorem of [OpenAI-F, Section 4] is that \(\Psi\) turns noise into the forest: if the \(w_e\) are independent Brownian motions, then \(\Psi(w)\) has the law \(\mathsf{FUSF}_G\). Its proof uses Brownian motion and is outside this course. In outline: if a forest \(X\) with the law \(\mathsf{FUSF}_G\) is observed as \(Y_e(t)=tX_e+B_e(t)\) with independent Brownian motions \(B_e\), then the paths \(Y_e(t)-\int_0^tb_e(s,Y(s))\,ds\) are again independent Brownian motions; by the uniqueness in Theorem 2.1, \(Y\) is the solution \(Z\) for these paths; and \(Y_e(n)/n\to X_e\). Example 2.2 shows, for one coordinate and a constant path, how the slope of the solution takes the values \(0\) and \(1\).

6. Exercises

6.1. Let \(b\) be a drift with the constant \(L\), \(T\ge0\) and \(\delta\ge0\), and let \(w,w'\in\mathcal C_I\) with \(|w_i(t)-w'_i(t)|\le\delta\) for all \(i\) and all \(t\le T\). Show that \(|Z_i(w)(t)-Z_i(w')(t)|\le\delta e^{Lt}\) for all \(i\) and \(t\le T\).

6.2. Let \(b\) be a drift with the constant \(0\). Describe the solution map.

6.3. In Example 2.2, prove the statements about the behaviour of \(Z(t)\) as \(t\to\infty\).

6.4. A set \(O\subseteq\mathcal C\) is open for the uniform convergence on bounded intervals if for every \(z\in O\) there are a positive integer \(n\) and \(r>0\) such that every \(z'\in\mathcal C\) with \(|z'(t)-z(t)|<r\) for all \(t\le n\) lies in \(O\). Show that the \(\sigma\)-algebra on \(\mathcal C\) generated by the evaluations \(z\mapsto z(t)\) is the \(\sigma\)-algebra generated by the open sets.

7. Solutions

6.1. Let \(Z^k\) and \(Z'^k\) be the approximations for \(w\) and \(w'\). We show by induction that every coordinate of \(Z^k(t)-Z'^k(t)\) is at most \(\delta\sum_{m\le k}(Lt)^m/m!\) in absolute value, for \(t\le T\). For \(k=0\) this is the assumption. If it holds for \(k\), then by Definition 1.1 \[ \begin{aligned} &|Z^{k+1}_i(t)-Z'^{k+1}_i(t)|\\ &\quad\le\delta+\int_0^tL\,\delta\sum_{m\le k}\frac{(Ls)^m}{m!}\,ds\\ &\quad=\delta\sum_{m\le k+1}\frac{(Lt)^m}{m!}. \end{aligned} \] The sums are at most \(e^{Lt}\), and the approximations converge to the solutions.

6.2. The inequality of Definition 1.1 with \(L=0\) says that \(b_i(t,y)=b_i(t,y')\) whenever \(y-y'\) is bounded. In the equation, \(Z(s)-w(s)\) has coordinates between \(0\) and \(s\), so \(b_i(s,Z(s))=b_i(s,w(s))\), and \[ Z_i(w)(t)=w_i(t)+\int_0^tb_i(s,w(s))\,ds . \] The first approximation is the solution.

6.3. For \(c=0\), \(\sinh(u/2)=0\) and \(Z(t)=\tfrac t2\). Let \(c>0\) and \(a=\sinh(c/2)>0\). Since \(\operatorname{arsinh}(x)=\log\bigl(x+\sqrt{x^2+1}\bigr)\) and \(x+\sqrt{x^2+1}=2x\,(1+\varepsilon(x))\) with \(\varepsilon(x)\to0\) as \(x\to\infty\), \[ 2\operatorname{arsinh}(ae^{t/4})-\frac t2\ \longrightarrow\ 2\log(2a), \] so \(Z(t)-t\to2\log(2a)\). For \(c<0\), \(\operatorname{arsinh}\) is odd, so \(Z(t)=\tfrac t2-2\operatorname{arsinh}(|a|e^{t/4})\), and by the first case \(Z(t)\to-2\log(2|a|)\).

6.4. Let \(\mathcal A\) be the \(\sigma\)-algebra generated by the evaluations. For \(t\ge0\) and real \(\alpha\) the set \(\{z:z(t)<\alpha\}\) is open: for \(z\) in it take \(n\ge t\) and \(r=\alpha-z(t)\). So every evaluation is measurable for the \(\sigma\)-algebra generated by the open sets, and \(\mathcal A\) is contained in it. For the other inclusion we show that every open set belongs to \(\mathcal A\). Let \(P\) be the set of the continuous functions on \([0,\infty)\) that have rational values at the points \(k/N\), \(k\ge0\), for some positive integer \(N\), are linear between these points, and are constant from some point on. The set \(P\) is countable. For \(p\in P\), a positive integer \(n\) and a rational \(r>0\) let \[ U(p,n,r)=\Bigl\{z:\sup_{t\le n}|z(t)-p(t)|<r\Bigr\}. \] By the continuity of \(z\) and \(p\), the supremum over \(t\le n\) equals the supremum over the rational \(t\le n\). So \(U(p,n,r)\) is the union, over the rational \(r'<r\), of the intersections over the rational \(t\le n\) of the sets \(\{z:|z(t)-p(t)|\le r'\}\), and it belongs to \(\mathcal A\). Let \(O\) be open and \(z\in O\), with \(n\) and \(r\) as in the statement; we may take \(r\) rational. Since \(z\) is uniformly continuous on \([0,n]\), there is \(p\in P\) with \(|p(t)-z(t)|<r/2\) for \(t\le n\): take \(N\) so large that \(z\) varies by less than \(r/8\) on intervals of length \(1/N\) in \([0,n]\), and rational values of \(p\) at the points \(k/N\) within \(r/8\) of those of \(z\); between two such points \(p(t)\) lies between its values at the two points, which are within \(r/4\) of \(z(t)\). Then \(z\in U(p,n,r/2)\), and \(U(p,n,r/2)\subseteq O\) because every element of it is within \(r\) of \(z\) on \([0,n]\). So \(O\) is the union of the countably many sets \(U(p,n,\rho)\), with \(\rho\) rational, that it contains, and \(O\in\mathcal A\).

References