John Bradstreet Walsh, PhD 1966, University of Illinois Urbana-Champaign, under the supervision of Joseph Leo Doob. An early explorer of stochastic partial differential equations (SPDE). His approach to make sense of SPDE is via random fields, seeing them as multidimensional stochastic differential equations (SDE) rather than as an infinite-dimensional system of SDE. He should not be confused with Joseph Leonard Walsh (1895-1973) or with Bertram John Walsh (1938-) famous for harmonic analysis.
The Gaussian free field (GFF) is the natural equilibrium measure of an Ornstein-Uhlenbeck dynamics, for which we can study the long time behavior. This short post explores this idea, on the discrete one-dimensional GFF. The SDE of the process is a discrete version of the stochastic heat equation (SHE).
A discrete pinned stochastic heat equation (SHE). Let $\Delta\in\mathcal{M}_n$ be the discrete Laplacian on $\Lambda=\{1,\ldots,n\}$, with zero Dirichlet conditions, namely \[ (\Delta u)_x=u_{x-1}+u_{x+1}-2u_x,\quad x\in\Lambda, \] for all $u\in\mathbb{R}^\Lambda\equiv\mathbb{R}^{|\Lambda|}=\mathbb{R}^n$, with the convention $u_0=u_{n+1}=0$. Here $u$ is seen as a function from $\Lambda$ to $\mathbb{R}$, extended by zero at the (outer) boundary $\partial\Lambda=\{0,n+1\}$. It turns out that the $n\times n$ matrix $\Delta$ is tridiagonal and Toeplitz symmetric \[ \Delta=J-2I \quad\text{where}\quad J_{xy}=\mathbf{1}_{|x-y|=1},\quad x,y\in\Lambda. \]
We consider now the Markov diffusion process ${(U_t)}_{t\in\mathbb{R}_+}$, $U_t=(U_{t,x}:x\in\Lambda)$, with state space $\mathbb{R}^n$, solution of the stochastic differential equation \[ \mathrm{d}U_t=\Delta U_t\mathrm{d}t+\sqrt{2}\mathrm{d}B_t,\quad t\in\mathbb{R}_+,\quad U_0=u_{\mathrm{in}}\in\mathbb{R}^\Lambda, \] where ${(B_t)}_{t\in\mathbb{R}_+}$ is a standard Brownian motion on $\mathbb{R}^n$. In other words ${(U_t)}_{t\in\mathbb{R}_+}$ is the solution of the system of stochastic differential equations \[ \mathrm{d}U_{t,x}=(U_{t,x-1}+U_{t,x+1}-2U_{t,x})\mathrm{d}t+\sqrt{2}\mathrm{d}B_{t,x},\quad t\in\mathbb{R}_+, x\in\Lambda, \quad U_{t,0}=U_{t,n+1}=0. \] This is a discrete in space version of the stochastic heat equation (SHE) \[ \partial_tu(t,x)=\partial_{xx}u(t,x)+\sqrt{2}\xi(t,x),\quad t\in\mathbb{R}_+, x\in [0,1], \] where $\xi$ is a space-time white noise, and more generally \[ \partial_tu(t,x)=\Delta u(t,x)+\sqrt{2}\xi(t,x),\quad t\in\mathbb{R}_+, x\in [0,1]^d \] see for instance [W]. The stochastic process ${(U_t)}_{t\in\mathbb{R}_+}$ is nothing but an overdamped Langevin process with potential $V(u)=\frac{1}{2}\langle(-\Delta)u,u\rangle$. It is actually Gaussian, known as the Ornstein-Uhlenbeck process. The Mehler formula for ${(U_t)}_{t\in\mathbb{R}_+}$ reads \[ U_t=\mathrm{e}^{t\Delta}u_{\mathrm{in}}+\sqrt{2}\int_0^t\mathrm{e}^{(t-s)\Delta}\mathrm{d}B_s \sim\mathcal{N}\Bigl(\mathrm{e}^{t\Delta}u_{\mathrm{in}},(-\Delta)^{-1}(I-\mathrm{e}^{2t\Delta})\Bigr). \] In particular, using the fact that $-\Delta$ is positive-definite, the long time behavior writes \[ U_t \xrightarrow[t\to\infty]{\mathrm{law}} \mu_\Delta \quad\text{where}\quad\mu_\Delta=\mathcal{N}(0,(-\Delta)^{-1}). \] There is an explicit formula for $(-\Delta)^{-1}$, but we do not use it in the sequel: \[ (-\Delta)^{-1}_{xy} =\min(x,y)-\frac{xy}{n+1}. \] We recognize the 1D Green function plus a vanishing boundary term as $n\to\infty$. Note that we have $(-\Delta)^{-1}_{xy}\geq0$ on $\Lambda\cup\partial\Lambda=\{0,1,\ldots,n+1\}$, with equality when $x$ or $y$ is in $\partial\Lambda$.
Gaussian free field (GFF). The law $\mu_\Delta=\mathcal{N}(0,(-\Delta)^{-1})$ is known as the Gaussian free field on $\Lambda$. This normal on $\mathbb{R}^\Lambda$ has density proportional to \[ u\in\mathbb{R}^\Lambda\mapsto \exp\Bigl(-\frac{1}{2}\langle(-\Delta)u,u\rangle\Bigr) =\exp\Bigl(-\frac{1}{4}\sum_{\substack{x,y\in\Lambda\cup\partial\Lambda\\|x-y|=1}}(u_x-u_y)^2\Bigr). \] The equality between the two sums is algebra but can be seen as a discrete integration by parts. The eigenvalues/eigenvectors of $-\Delta$, namely $-\Delta\varphi_{n,k}=\lambda_{n,k}\varphi_{n,k}$, are given by \begin{align*} \varphi_{n,k}(x) &=\sqrt{\frac{2}{n+1}}\sin\left(\frac{k\pi x}{n+1}\right)\\ \lambda_{n,k} &= 2-2\cos\left(\frac{k\pi}{n+1}\right) =4\sin^2\left(\frac{k\pi}{2(n+1)}\right), \qquad k\in\{1,\dots,n\}. \end{align*} See for instance [S]. In particular $\lambda_{n,1}\leq\cdots\leq\lambda_{n,n}$ and the curvature is \[ \lambda_{n,1} = 2-2\cos\left(\frac{\pi}{n+1}\right) \underset{n\to\infty}{\sim}\frac{\pi^2}{(n+1)^2}\xrightarrow[n\to\infty]{}0. \] The process ${(U_t)}_{t\in\mathbb{R}_+}$ is rigid: curvature and spectral gap are equal to $\lambda_{n,1}$. In order to make them of order $1$ as $n\to\infty$, we could consider the stochastic differential equation \[ \mathrm{d}U_t'=(n+1)^2\Delta U_t'\mathrm{d}t+\sqrt{2(n+1)^2}\mathrm{d}B_t, \quad U_0'=u_{\mathrm{in}}. \] The invariant law of ${(U'_t)}_{t\in\mathbb{R}_+}$ is still $\mu_\Delta=\mathcal{N}(0,(-\Delta)^{-1})$, but the spectral gap is now $\sim\pi^2$. Actually the process ${(U'_t)}_{t\in\mathbb{R}_+}$ has the law of the time change ${(U_{(n+1)^2t})}_{t\in\mathbb{R}_+}$. But regarding the long time behavior and the cutoff phenomenon, the overdamped Langevin processes ${(U_t)}_{t\in\mathbb{R}_+}$ and ${(U_t')}_{t\in\mathbb{R}_+}$ are outside the scope of [CF], because the spectral gap goes to zero or because the diffusion coefficient goes to infinity.
Long time behavior and diffusive behavior instead of cutoff phenomenon. Following [BCL] we can take advantage of the Gaussian nature of the distributions and use the available explicit formulas for various distances and divergences. Let us take a look for instance at the Kullback-Leibler divergence or relative entropy. We set \[ D_t:=\mathrm{H}(\mathrm{Law}(U_t)\mid\mu_\Delta) \quad\text{where}\quad \mathrm{H}(\nu\mid\mu):=\int\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\log\frac{\mathrm{d}\nu}{\mathrm{d}\mu}\mathrm{d}\mu. \] We have the explicit formula \begin{align*} \mathrm{H}(\mathcal{N}(m_1,K_1)\mid\mathcal{N}(m_2,K_2)) &=\frac{1}{2}K_2^{-1}(m_1-m_2)\cdot(m_1-m_2)\\ &\quad +\frac{1}{2}\mathrm{Tr}\bigl(K_2^{-1}K_1-I\bigr)-\frac{1}{2}\log\det(K_2^{-1}K_1). \end{align*} and in view of the Mehler formula, we would like to use it with \begin{align*} \mathcal{N}(m_1,K_1) &=\mathcal{N}(\mathrm{e}^{t\Delta}u_{\mathrm{in}},(-\Delta)^{-1}(I-\mathrm{e}^{2t\Delta}))\sim U_t\\ \mathcal{N}(m_2,K_2) &=\mathcal{N}(0,(-\Delta)^{-1}) \end{align*} which gives \begin{align*} K_2^{-1}(m_1-m_2)\cdot(m_1-m_2) &=u_{\mathrm{in}}\cdot(-\Delta)\mathrm{e}^{2t\Delta}u_{\mathrm{in}}\\ \mathrm{Tr}(K_2^{-1}K_1-I) &=\mathrm{Tr}((-\Delta)(-\Delta)^{-1}(I-\mathrm{e}^{2t\Delta})-I) =-\mathrm{Tr}(\mathrm{e}^{2t\Delta})\\ \log\det(K_2^{-1}K_1) &=\log\det(I-\mathrm{e}^{2t\Delta}). \end{align*} Therefore, using the spectral decomposition $-\Delta\varphi_{n,k}=\lambda_{n,k}\varphi_{n,k}$ and $u_{\mathrm{in}}=\sum_{k=1}^n\alpha_{n,k}\varphi_{n,k}$, \[ D_t=M_t+C_t,\quad t > 0, \quad\text{with}\quad \begin{cases} M_t=\frac{1}{2}\sum_{k=1}^n\lambda_{n,k}r_{n,k}(t)\alpha_{n,k}^2\\ C_t=\frac{1}{2}\sum_{k=1}^ng(r_{n,k}(t)) \end{cases} \] where $r_{n,k}(t)=\mathrm{e}^{-2t\lambda_{n,k}}$ and $g(u)=-u-\log(1-u)$ are both non-negative quantities. In this decomposition, $M_t$ is the mean term that depends on $u_{\mathrm{in}}$, while $C_t$ is the covariance term which is universal in the sense that it does not depend on $u_{\mathrm{in}}$.
In order to explore the long time behavior while stabilizing the spectral gap, we take \[ t_n=(n+1)^2s \] for a fixed $s > 0$. For each fixed $k\geq1$, using $\lim_{n\to\infty}(n+1)^2\lambda_{n,k}=(\pi k)^2$ we obtain \[ r_{n,k}(t_n)=\mathrm{e}^{-2t_n\lambda_{n,k}} \xrightarrow[n\to\infty]{}\mathrm{e}^{-2\pi^2k^2s}. \] Moreover the mean term is negligible for $|u_{\mathrm{in}}|\le c\sqrt{n}$. Indeed, \[ 0\leq M_{t_n} \leq \frac12 |u_{\mathrm{in}}|^2\max_{1\le k\le n}\lambda_{n,k}\mathrm{e}^{-2t_n\lambda_{n,k}} =\frac{|u_{\mathrm{in}}|^2}{2(n+1)^2}\max_{1\le k\le n}a_{n,k}\mathrm{e}^{-2sa_{n,k}}, \] where $a_{n,k}:=(n+1)^2\lambda_{n,k}\geq0$. Since $\max_{a\geq0}(a\mathrm{e}^{-2sa})=\frac{1}{2\mathrm{e}s}$, we get, \[ 0\leq M_{t_n}\leq \frac{|u_{\mathrm{in}}|^2}{2(n+1)^2}\frac{1}{2\mathrm{e}s} \leq \frac{c^2n}{4\mathrm{e}s(n+1)^2}\xrightarrow[n\to\infty]{}0 \] uniformly over $|u_{\mathrm{in}}|\leq c\sqrt n$. For the covariance term, using that $g(u)=-u-\log(1-u)\sim \frac{u^2}{2}$ as $u\searrow0$, the series below is finite for every $s > 0$, and by dominated convergence \[ C_{t_n}\xrightarrow[n\to\infty]{} C(s) :=\frac12\sum_{k=1}^{\infty} g\bigl(\mathrm{e}^{-2\pi^2k^2s}\bigr)\in(0,\infty). \] Therefore, we have obtained that for $t_n\sim n^2s$, $s > 0$, and $|u_{\mathrm{in}}|\leq c\sqrt{n}$, $c > 0$, we have \[ D_{t_n}=M_{t_n}+C_{t_n}\xrightarrow[n\to\infty]{} C(s), \] and the limit is universal, in other words independent of $u_{\mathrm{in}}$ in the regime $|u_{\mathrm{in}}|\le c\sqrt n$. The function $s\mapsto C(s)$ is decreasing and continuous on $(0,\infty)$, with $C(s)\to+\infty$ as $s\searrow0$ and $C(s)\to0$ as $s\to\infty$. In particular, on the diffusive time scale $t\asymp n^2$ the relaxation in relative entropy has a nontrivial profile rather than a cutoff.
Higher dimensions and continuous setting. The analysis above works essentially the same way for a $d$-dimensional domain $\Lambda\subset\mathbb{Z}^d$, $d\geq1$. We identify the functions or fields $\Lambda\mapsto\mathbb{R}$ with $\mathbb{R}^{\Lambda}$ and then with $\mathbb{R}^n$ where $n=|\Lambda|$. The discrete Laplacian is here \[ \Delta=J-2dI,\quad\text{with}\quad J_{xy}=\mathbf{1}_{|x-y|_1=1},\quad x,y\in\Lambda. \] The GFF is here again $\mathcal{N}(0,(-\Delta)^{-1})$, with density proportional to \[ u\in\mathbb{R}^\Lambda\mapsto \exp\Bigl(-\frac{1}{2}\langle(-\Delta)u,u\rangle\Bigr) =\exp\Bigl(-\frac{1}{4}\sum_{\substack{x,y\in\Lambda\cup\partial\Lambda\\|x-y|_1=1}}(u_x-u_y)^2\Bigr) \] where $u_x=0$ if $x\in\partial\Lambda=\{y\not\in\Lambda:\exists x\in\Lambda,|x-y|_1=1\}$. The covariance $(-\Delta)^{-1}$ is the Green function of $\Delta$ on $\Lambda$ with zero Dirichlet boundary conditions. Since $\Delta(-\Delta)^{-1}=-I$, it follows that for all fixed $y\in\Lambda$, the function $f_y=(-\Delta)^{-1}(\cdot,y)$ solves the Dirichlet problem $\Delta f_y=-\mathbf{1}_y$ on $\Lambda$ with boundary condition $f_y=0$ on $\partial\Lambda$. The Kakutani probabilistic representation of the solution of the Dirichlet problem gives then \[ (-\Delta)^{-1}_{xy} =f_y(x) =\tfrac{1}{2d}\mathbb{E}_x\Bigl(\sum_{k=0}^{T_{\partial\Lambda}-1}\mathbf{1}_{\{X_k=y\}}\Bigr), \quad x,y\in\Lambda \] where ${(X_k)}_{k\in\mathbb{N}}$ is the discrete time simple random walk on $\mathbb{Z}^d$ with transition kernel $P=\frac{1}{2d}J=\frac{1}{2d}\Delta+I$ and generator $P-I=\frac{1}{2d}\Delta$, and $T_{\partial\Lambda}=\inf\{k\in\mathbb{N}:X_k\in\partial\Lambda\}$ is the hitting time of the boundary. The prefactor $1/(2d)$ can be avoided by passing to continuous time. Indeed, using the continuous time random walk ${(X_t)}_{t\in\mathbb{R}_+}$ with generator $\Delta$ gives \[ (-\Delta)^{-1}_{xy} =\mathbb{E}_x\Bigl(\int_0^{T_{\partial\Lambda}}\mathbf{1}_{\{X_t=y\}}\mathrm{d}t\Bigr) \quad\text{where this time}\quad T_{\partial\Lambda}=\inf\{t\in\mathbb{R}_+:X_t\in\partial\Lambda\}. \]
At each point of the space domain $\Lambda\subset\mathbb{Z}^d$, we put a Gaussian real random variable, and these are correlated in such a way that their covariance matrix is $(-\Delta)^{-1}$. This describes a field of Gaussian real random variables.
The extension to continuous setting can be done at the cost of analytic technicalities. More precisely, the continuous GFF on $[0,1]$ becomes a normal distribution on continuous functions defined on $[0,1]$ pinned at $0$ and $1$, which can be identified with a Brownian Bridge (BB). The continuous GFF on $[0,1]^d$, $d > 1$, is no longer a random function, it is a random distribution, which can be constructed like BB by regularizing random series in $L^2$ and using the Kolmogorov-Chentsov continuity theorem.
Further reading.
- [W] John B. Walsh
An introduction to stochastic partial differential equations
École d'Été de Probabilités de Saint-Flour XIV 1984
Lecture Notes in Mathematics 1180, Springer (1986) - [S] Daniel A. Spielman
Spectral and Algebraic Graph Theory
Book draft - Yale University lecture notes (2025) - [BCL] Jeanne Boursier, Djalil Chafaï, and Cyril Labbé
Universal cutoff for Dyson Ornstein Uhlenbeck process
Probability Theory and Related Fields 185 1-2 449-512 (2023) - [CF] D. Chafaï and M. Fathi
On cutoff via rigidity for high dimensional curved diffusions
Comptes Rendus Mathématique 363 1103--1121 (2025) - On this blog
Spectral gap concentration in high dimension
(2025-02-07)