Abstract

Stochastic partial differential equations are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining, approximating, and studying the properties of a few examples of such equations. We focus mainly on the dynamical $\Phi ^4$ equation, the KPZ equation, and the parabolic Anderson model, as well as a few other equations which arise mainly in physics.

1. Introduction

Partial differential equations (PDEs) and randomness are ubiquitous constructions used to model both mathematical and physical phenomena. For instance, PDEs have been used for centuries to describe the building block laws of physics, and to model aggregate macroscopic phenomena, such as heat conduction, diffusion, electro-magnetic dynamics, interface and fluid dynamics. Randomness has become a default paradigm for modeling systems with uncertainty or with many complicated or chaotic microscopic interactions.

Combining these two approaches leads to the study of stochastic PDEs (SPDEs) in which the coefficients or forcing terms in PDEs are described via certain random processes. While SPDEs have become increasingly important in applications, there remain many fundamental mathematical challenges in their study—in particular, showing how they arise from microscopic particle based models remains a major source of research problems and has seen some radical progress in the past decade.

The purpose of this article is to introduce a few important classes of SPDEs and to describe how they arise and the mathematical challenges that go along with demonstrating that. Though this article will mainly focus on nonlinear systems, we will start our investigation in Section 2 in the simpler and more classical setting of linear SPDEs which are very well understood. In Section 3 we turn our attention to nonlinear SPDEs and introduce our two main examples (the dynamical $\Phi ^4$ equation and the KPZ (Kardar–Parisi–Zhang) equation) along with a host of other important SPDEs which arise in physics. Our discussion in this section is heuristic and ignores some of the serious mathematical challenges which arise when one tries to make sense of what it means to “solve” an SPDE. This challenge is addressed in Section 4. In the course of making sense of SPDEs, there are often renormalizations which arise (effectively changing the equation). Section 5 describes how these renormalizations have physical meaning and arise in certain discrete approximation schemes for the continuum equations. Finally, Section 6 seeks to demonstrate how these SPDEs (in particular, the KPZ equation) arise as universal limits from microscopic systems.

Before proceeding to our main text, one disclaimer. Our aim is to make this material approachable to nonexperts. As such, we will not state precise theorems or give proofs but rather will attempt to provide some intuition behind results and the challenges which accompany proving them. An interested reader can find much more detail and precision in the works cited or, can consult other survey articles such as GP18bGub18CW17Hai15aHai14a.

2. A first (linear) SPDE

We will start our discussion on linear SPDEs with the stochastic heat equation which is driven by a random additive noise term $\xi$:

$$\begin{equation} \partial _t u(t,x) = \partial _x^2 u(t,x) +\xi (t,x), \tag{2.1}{} \end{equation}$$

where $\xi$ is the so-called space-time white noise. It will take a bit of work to define this noise and make sense of what it means to solve this equation. However, before going down that route, we will first address the question of what sort of physical system does this model? In particular, we will explain heuristically how this equation arises from a simple microscopic model of polymers in liquid.

Consider modeling a polymer chain (e.g., composed of DNA or proteins) in a liquid. A simple model involves describing the polymer by a string of $N$ beads that are linked together sequentially by springs and that are subject to kicking by noise, as shown in Figure 2.1, where $N=13$:

Imagine that each bead of the polymer is kicked by the surrounding liquid molecules. In our simplified model,⁠¹ As usual, one always has to make various simplifying assumptions in order to describe a complicated physical system via a mathematically analyzable model. It is natural to ask whether having random kicking leads to a reasonable microscopic model. After all, the liquid itself is governed by certain physical laws of motion for its particles. Such concerns arose early in the development of Brownian motion as the model for a single tracer particle moving in a liquid; see Bru68 for a nice historical review. We do not provide further justification for this as a reasonable microscopic model here.^✖ we describe such a system via the following equations of motion for the position $r_i \in \mathbf{R}^3$ of the $i$th bead:

$$\begin{equation} dr_i (t)= (r_{i+1} (t)- r_i (t))dt + (r_{i-1} (t) - r_i (t)) dt + w_i(t)\quad (t\in \mathbf{R}_+,i=1,\dots ,N) \tag{2.2}{} \end{equation}$$

where $w_i (t)$ is random kicking at time $t$, and where a boundary condition is given by fixing $r_0-r_1\equiv 0$ and $r_{N+1}-r_N\equiv 0$ for all time. Equation 2.2 means the following.

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The linear drift terms $(r_{i+1} - r_i)dt$ and $(r_{i-1} - r_i) dt$ arise from assuming a linear spring force between the $i$th bead with its neighboring (in the sense of label number) beads. Without the kicking term $w_i(t)$, equation 2.2 would simply be a coupled system of $N$ ordinary differential equations.

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The term $w_i(t)$ represents the random kicking that is experienced by the $i$th bead at time $t$. We make the simplifying assumption that the kicking is “overdamped”,⁠² Essentially, this means that the kicks occur instantaneously in time and do not result in any inertia. This effectively decouples the various kicks.^✖ and we model the kicks in terms of random jumps in the location of the $r_i(t)$. Namely, for each particle $r_i$ there is a random sequence of kicking times⁠³ It is natural to assume the gaps between times $t_{i,j}-t_{i,j-1}$ are chosen according to independent exponential random variables of mean 1. In this case, the times are distributed as a Poisson point process of intensity 1.^✖ $t_{i,1}<t_{i,2}<\cdots$. At the kicking time $t_{i,j}$, we update $r_i \mapsto r_i + w_i(t_{i,j})$, where $w_i(t_{i,j})$ is an $\mathbf{R}^3$-valued random variable. We assume that the $w_i$ are statistically isotropic (i.e., their distribution is invariant under rotation) and are all independent and identically distributed (i.i.d.). Note that the resulting $r_i(t)$ process is piecewise continuous, with jumps occurring at the kicking times.

The question with which we are concerned is what happens to the polymer when its length grows, and possibly space and time are scaled accordingly. By default one might expect that as $N$ increases, the complexity of studying this system goes likewise. However, it turns out that there is a very tractable continuum limit for the evolution of our polymer model. That is to say, in the scaling limit, things simplify! In fact, this limit is quite robust and (up to some scaling constants) is not affected by various changes in the microscopic model, such as how we model the kicking (e.g., different distribution on the $w_i$ or on the kicking times). This robustness can, itself, be seen as evidence that the microscopic model may be reasonable.

With the aim of demonstrating a continuum limit of our model, think of $N$ as large and define

$$\begin{equation} u_N(t,x) \mathrel {≔}\frac {1}{\sqrt {N}} r_{[xN]} (N^2 t)\in \mathbf{R}^3, \tag{2.3}{} \end{equation}$$

where $x\in [0,1]$ encodes the label $i$ via $i=[xN]$ (the closest integer to $xN$). The linear drift term in 2.2 is in fact a discrete Laplacian, so under our diffusive scaling (that is, scaling $x$ by $N$ and $t$ by $N^2$) it approximates a continuum Laplacian $\partial _x^2$. Thus, for large $N$ one can expect that the following relation approximately holds:

$$\begin{equation} \partial _t u_N(t,x) \approx \partial _x^2 u_N(t,x) + N^{\frac 32} w_{[Nx]}(N^2 t)\;. \tag{2.4}{} \end{equation}$$

In the scaled coordinates, the kicking starts to add up. Namely, in a time-space region of size $dt \times dx$, there are roughly $N^2 dt \times N dx$ kicks. By the central limit theorem the sum of $N^2\times N$ i.i.d. random variables divided by $N^{\frac 32}$ converges to a Gaussian random variable. Thus, on each time-space region, the kicking adds up to a Gaussian random variable with variance equal to the area of the region. Different regions have covariance given by the area of their overlap. This limit is called space-time (or time-space, given our ordering of variables) white noise and is denoted by $\xi (t,x)$. This heuristic leads us to the following type⁠⁴ This type of convergence result was first proved by Funaki Fun83 in the slightly different setting where the $r_i$ are driven by Brownian motions. In the present setting, we do not know if a precise result of this sort has been proved (though we have no doubt that it can be). We are suppressing coefficients which may (depending on the nature of the discrete noise) arise in the limiting equation.^✖ of limit as $N\to \infty$,

$$\begin{equation} u_N(t,x)\to u(t,x), \tag{2.5}{} \end{equation}$$

where $\partial _t u(t,x) = \partial _x^2 u(t,x)+\xi (t,x),$ and where $\xi (t,x)$ is space-time white noise.

Equation 2.5 is our first example of an SPDE—it is called the linear stochastic heat equation with additive noise. Even in this simple linear example we encounter an equation which requires some work to make sense of because of the noise. For convenience of our exposition, from this point forward, we will think of $x$ as a spatial variable (although in our example it actually stands for the parametrization of the limiting polymer length); and although $u$ and $\xi$ are $\mathbf{R}^3$-valued, the three components are completely independent (decoupled), so it will be convenient to simply consider equation 2.5 as $\mathbf{R}$-valued instead of $\mathbf{R}^3$-valued in the rest of this paper.

Let us look at equation 2.5 more closely, with spatial dimension $d$ now being arbitrary (recall $d=1$ corresponds with the above polymer example),

$$\begin{equation} \partial _t u(t,x) = \Delta u(t,x) + \xi (t,x), \qquad x\in [0,1]^d \subset \mathbf{R}^d\;, \tag{2.6}{} \end{equation}$$

with, for instance, periodic boundary condition.

Consider the case $d=0$, where 2.6 becomes the stochastic (ordinary) differential equation $\partial _t u(t) = \xi (t)$. The $d=0$ white noise $\xi (t)$ is defined to be the derivative of Brownian motion so that $u(t) = \int _0^t \xi (s)$ is a Brownian motion. Of course, Brownian motion is famously almost nowhere differentiable, so $\xi$ is not defined as a function. Rather, $\xi (t)$ can be defined as a random distribution in a suitable negative regularity space (such as a negative Sobolev space). Since the Hölder regularity of Brownian motion is $1/2-$ (meaning any exponent below $1/2$), its derivative is said to have regularity $-1/2-$. There are other ways to define $\xi (t)$. For instance, if we restrict to a periodic $t\in [0,T]$, then $\xi (t) = \sum _{k\in \mathbf{Z}} N_k e_k(t)$, where the $N_k$ are i.i.d. Gaussian random variables, and the $\{e_k\}_{k\in \mathbf{Z}}$ constitute an orthonormal basis of $L^2([0,T])$. Alternatively, one can define $\xi (t)$ via the machinery of Gaussian processes (see, for instance Jan97) wherein it suffices to specify its mean and variance. By definition, $\xi$ is mean zero, and since $\xi$ is a distribution as mentioned above, its covariance

$$\begin{equation*} \mathbf{E}\big (\xi (t)\xi (t')\big ) =\delta (t-t') \end{equation*}$$

(here $\mathbf{E}$ represents the expectation value operator and $\delta$ is a Dirac delta function) must be interpreted in a distributional sense as well. For a smooth test function $f$, one defines the stochastic integral $\xi (f)=\int f(t)\xi (t)$. Then $\xi$ is defined by the property that $\mathbf{E}\big (\xi (f)\big )=0$ for all test functions $f$, and for test functions $f$ and $g$,

$$\begin{equation} \mathbf{E} \big (\xi (f)\xi (g)\big ) =\langle f,g \rangle _{L^2}. \tag{2.7}{} \end{equation}$$

The random distribution $\xi$ is defined from this information using Kolmogorov’s continuity theorem.

For general dimension $d\geq 1$, space-time white noise can be defined via analogous methods. As a Gaussian process, $\xi (t,x)$ is a random distribution with covariance

$$\begin{equation} \mathbf{E}\big (\xi (t,x)\xi (t',x')\big ) =\delta (t-t')\delta (x-x'), \tag{2.8}{} \end{equation}$$

where the last $\delta$ is the Dirac delta function on $d$-dimensional space. Its action on space-time test functions $f$ and $g$ has covariance given by 2.7. where the $\langle \cdot ,\cdot \rangle _{L^2}$ is $L^2$ product over space-time.

As the dimension $d$ increases, the regularity of $\xi$ decreases. We will work with spaces of space-time distributions (for $\alpha < 0$) or functions (for $\alpha \geq 0$) denoted by $\mathcal{C}^{\alpha }$. These are essentially equivalent to the Besov spaces $\mathcal{B}^{\alpha }_{\infty ,\infty }$ in harmonic analysis, and their precise definitions can be given via wavelets in Hai14b, Eq. (3.2). The smaller $\alpha$ corresponds to less regular (or more singular) functions or distributions. A well-known result is that the space-time white noise $\xi \in \mathcal{C}^{\alpha }$ for $\alpha <-\frac {d+2}{2}$.⁠⁵ As we will primarily work with parabolic equations, these spaces $\mathcal{C}^{\alpha }$ have a built-in parabolic scaling between time and space wherein time regularity is doubled. For instance a $\mathcal{C}^{2}$ function has second continuous spatial derivatives and first continuous time derivative. Extending the situation discussed earlier for ($d=0$) white noise, we have that space-time white noise $\xi \in \mathcal{C}^{\alpha }$ for $\alpha <-\frac {d+2}{2}$. The case $d=0$ has $\xi \in \mathcal{C}^{-1-}$ which, given the doubling of time regularity, corresponds to the $-\frac 12-$ regularity discussed above.^✖

Having made sense of $\xi$, it remains to understand what it means to solve equation 2.6. For a linear equation as 2.6, the meaning of solution is not hard to define; essentially one only needs to give a suitable meaning to the inverted linear differential operator acting on the noise $\xi$: given an initial data $u(0,x)=u_0(x)$, the solution to 2.6 is defined by

$$\begin{equation} u= (\partial _t - \Delta )^{-1} \xi + e^{t\Delta } u_0\;. \tag{2.9}{} \end{equation}$$

Here $e^{t\Delta }$ is the heat semigroup so that $e^{t\Delta } u_0 (x)\mathrel {≔}\int _{[0,1]^d} P(t,x-y) u_0(y)dy$ solves the classical (deterministic) heat equation starting from $u_0$, with heat kernel $P(t,x)\mathrel {≔}(4\pi t)^{-\frac {d}{2}} e^{-\frac {|x|^2}{4t}}$. The expression $(\partial _t - \Delta )^{-1} \xi$ also acts as an integral operator on $\xi$ via

$$\begin{equation} (\partial _t - \Delta )^{-1} \xi (t,x)\mathrel {≔}\int _0^t \int _{[0,1]^d} P(t-s,x-y)\xi (ds,dy), \tag{2.10}{} \end{equation}$$

which is the space-time convolution of the heat kernel $P(t,x)$ with $\xi$. Just like $\xi$ itself, $(\partial _t - \Delta )^{-1} \xi$ is also a well-defined random distribution.

To get a bit more flavor of “solution theories” of stochastic PDEs, we list some well-known properties for 2.6.

(P1)

The solution, in the above sense, is obviously unique, since the difference of two solutions would solve a deterministic heat equation with zero initial condition which must be zero.

(P2)

With the aforementioned regularity of $\xi$, by standard parabolic PDE theory, in particular the Schauder estimate which states that the operator $(\partial _t - \Delta )^{-1}$ increases regularity by $2$, one has $u\in \mathcal{C}^{\alpha }$ for any $\alpha < -\frac {d+2}{2} +2 =\frac {2-d}{2}$. In particular, $u$ is (almost surely) a random continuous function in $d=1$, and a random distribution in $d\ge 2$. So the limiting polymer parametrized by $x\in [0,1]$ in the above example is a random continuous curve.

(P3)

The random distribution $(\partial _t - \Delta )^{-1} \xi$ has Gaussian probability law. This is because $\xi$ is Gaussian and any linear combination of Gaussian random variables is still Gaussian.

(P4)

Equation 2.6 has an invariant measure called the Gaussian free field. This is a Gaussian random field on $[0,1]^d$ with covariance given by the Green’s function of the Laplace $\Delta$. Being invariant means that if the initial data $u_0$ is random and distributed as Gaussian free field, then $u(t,\cdot )$ has the same law of Gaussian free field for all $t>0$. On the other hand starting from arbitrary $u_0$, the law of $u(t,\cdot )$ will approach that of the Gaussian free field as $t\to \infty$. We refer to She07 for a nice review of the Gaussian free field.

(P5)

Equation 2.6 is scaling invariant in any dimension $d$, namely,$$\begin{equation*} \widetilde{u}(t,x)\mathrel {≔}\lambda ^{\frac {d-2}{2}} u(\lambda ^2 t, \lambda x)\quad \text{ satisfies } \partial _t \widetilde{u}(t,x) = \Delta \widetilde{u}(t,x) +\widetilde{\xi }(t,x), \end{equation*}$$

where $\widetilde{\xi }(t,x)\mathrel {≔}\lambda ^{\frac {d+2}{2}} \xi (\lambda ^2 t,\lambda x) \stackrel {law}{=}\xi (t,x)$. The last scaling relation of the white noise can be seen from its covariance 2.8 recalling that the Dirac $\delta$ on $n$-dimensional space has scaling dimension $-n$. Note that the scaling taken in 2.3 and 2.4 was precisely the one that leaves the limit equation invariant.

So far a reader who is new to the area of SPDEs should have acquired the following message: the solution theories of SPDEs share some of the same fundamental challenges as in the study of classical PDEs. These include showing that solutions exist (or can be defined) both locally or globally, are unique within certain regularity classes, and arise as a scaling limits for various approximation schemes. The rest of this article will focus on recent progress on these challenges for nonlinear SPDEs. Before doing so, let us briefly remark on another important challenge present both for PDEs and SPDEs—explicitly representing solutions via formulas.

Linear PDEs always admit explicit solutions. Linear SPDEs, as we saw above, have solutions which are random Gaussian processes with explicit mean and covariance (computable from the equation explicitly). Most nonlinear PDEs do not admit explicit solutions; those that do are generally related to the area of integrable systems. Likewise, most nonlinear SPDEs do not admit explicit descriptions for the probability distribution of their solutions. There are, however, a few special SPDEs (such as the KPZ equation discussed below) which can be explicitly solved in this sense. The study of such SPDEs fall under the area of integrable probability or exactly solvable systems; see for instance Cor14BP15 and references therein. We will not pursue this direction further in this article.

3. Nonlinear SPDEs

Linear systems are often insufficient to effectively model many interesting phenomena. Indeed, as we will now see, nonlinear SPDEs arise in a number of important areas of physics (and many other directions that we will not discuss here). Such nonlinear systems, however, are generally much more challenging to work with. Before coming to that, let us start with a few examples.

Consider a piece of magnet that is being heated up, as in Figure 3.1. As the temperature $T$ increases, the magnetic field produced by the magnet weakens, and at a critical temperature $T_c$, known as the Curie temperature,⁠⁶ The Curie temperature is named after Pierre Curie who first experimentally demonstrated that certain magnets lost their magnetism entirely at a critical temperature.^✖ the magnetic field disappears. Though various magnet materials have different microscopic structures, a common physical explanation for magnetism is that it comes from the alignment of the magnetic moments of many of the atoms in the material. As a simplified mathematical model one can imagine that a magnet is made up of millions of tiny arrows (or spins) with directions oscillating over time. Below the Curie temperature, i.e., $T<T_c$, the spins tend to align in order to minimize an interaction energy (which energetically prefers alignment), which causes a macroscopic magnetization (shown in the bottom-right picture); above the Curie temperature $T>T_c$, the spin configurations are much more disordered due to strong thermal fluctuation,⁠⁷ In statistical mechanics, thermal fluctuations are random deviations of a system from its low energy state. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches absolute zero. Thermal fluctuations are a basic manifestation of the temperature of systems.^✖ and as a result the magnetic fields cancel out (shown in the bottom-left picture).

A general mantra in statistical physics holds that “interesting scaling limits arise at critical points”. In particular, here we would like to understand what happens to the spin system when the temperature $T$ approaches $T_c$ while time and space are tuned accordingly. Near criticality, the spins start to oscillate more and more drastically and the small scale disorder starts to propagate to larger and larger scales. The resulting magnetic field fluctuations are believed to be described by the nonlinear SPDE,

$$\begin{equation*} \partial _t \Phi = \Delta \Phi - \Phi ^3 + \xi , \tag{$\Phi ^4$}{} \end{equation*}$$

when $\Phi =\Phi (t,x)$ and the spatial dimension of $x$ is $1,2$, or $3$. This is called the dynamical $\Phi ^4$ equation since the deterministic part arises from the gradient of an energy $\int \frac 12 |\nabla \Phi |^2+ \frac 14 \Phi ^4\,dx$. We will return to this equation later in Section 5.1 and describe how it arises from a particular model of magnets.

As another example, we consider a model for interface growth, where each point of the interface randomly grows up or drops down over time, with a trend to locally smooth out the interface (like the spring force in the polymer example in Section 2). Such systems are ubiquitously found in nature—for instance, the left image in Figure 3.2 shows the end result of an interface grown in the ocean from a volcanic eruption.

We are interested in modeling the evolution of such interfaces.

To drastically simplify the situation, let us assume our interface is a one-dimensional function $H(t,x)$ for $x\in \mathbf{R}$. The simplest scenario is that the upward growth and downward drop of $H$ occurs equally likely, though randomly. In this case, it turns out that the interface behaves similarly to the one-dimensional version of the polymer in Section 2 whose beads are kicked by isotropic random force (see the right image in Figure 3.2). Thus, the large scale fluctuation should be given by the linear SPDE 2.5.

In the asymmetric scenario where the interface is more likely to grow up than to drop down, one expects to see nontrivial fluctuation described by equation 2.5 perturbed by a nonlinearity. In particular, the asymmetry should not be too strong or it will overwhelm the local smoothing (the $\partial _x^2 H$ term) and randomness (the $\xi$ term), and not be too weak or it will not change the limiting equation. This critical tuning is called weakly asymmetric and results (under the same sort of scaling as in the symmetric case) in the following SPDE description for fluctuation of $H(t,x)$:

$$\begin{equation*} \partial _t H = \partial _x^2 H + (\partial _x H)^2 + \xi . \tag{KPZ}{} \end{equation*}$$

Due to the asymmetry, the interface establishes an overall height shift. So, for the above limit, we must recenter into this moving frame. The KPZ equation was first proposed by Kardar, Parisi, and Zhang in KPZ86; see the nice review KS91b for more background. We will return to this equation later in Section 6 and describe why it arises from various models of interface growth.

3.1. Some other important nonlinear SPDEs

Besides the $\Phi ^4$ and KPZ equations discussed above, there are a number of physically important equations—some of which we briefly review now. The reader is warned that it is a formidable challenge to define the meaning of a solution to nonlinear SPDEs driven by very singular noises. We postpone this important issue until Section 3.2.

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Stochastic Navier–Stokes equation (with spatial dimensions $d=2,3$ of particular physical interest),$$\begin{equation} \partial _t \vec{u}+\vec{u}\cdot \nabla \vec{u} = \Delta \vec{u}-\nabla p+\vec{\zeta }, \quad \operatorname {div} \vec{u} = 0, \tag{3.1}{} \end{equation}$$

where $p$ is the pressure and $\vec{\zeta }$ is a $d$-vector valued noise. When the noise $\vec{\zeta }$ is taken to be singular (for instance each component of $\vec{\zeta }$ is an independent space-time white noise), it models motion of fluid with randomness arising from microscopic scales, and in this case we refer to DPD02ZZ15 for well-posedness results.

We remark that while this article focuses on singular noises, when modeling large scale random stirring of the fluid, the noise $\vec{\zeta }$ is often assumed to be smooth (it is called colored noise in contrast with white noise), and in fact the most important case is that the equation is driven by only a small number of random Fourier modes. In these situations the long-time behavior is of primary interest, and various dynamical system questions such as ergodicity and mixing are studied. There is a vast literature on this topic, and we only refer to the book KS12 and the survey articles Mat03Fla08Kup10.

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Stochastic heat equation with multiplicative noise in one spatial dimension,$$\begin{equation} \partial _t u = \Delta u + f(u)\xi , \tag{3.2}{} \end{equation}$$

where $f$ is some continuous function. The Itô solution theory successful for stochastic ordinary differential equations (ODEs) can extend to this stochastic PDE; see for instance the lecture notes Wal86. The specialization $f(u)=u$, i.e.,$$\begin{equation} \partial _t u = \Delta u + u\xi , \tag{3.3}{} \end{equation}$$

has a significant connection to the KPZ equation: one can formally check that if $H$ solves KPZ, then the Hopf–Cole transform $u\mathrel {≔}e^H$ solves 3.3. Other choices of $f$ are$$\begin{equation*} f(u)=\alpha \sqrt {u} \qquad \text{and} \qquad f(u)=\alpha \sqrt {u(1-u)} \qquad (\alpha \in \mathbf{R}), \end{equation*}$$

which (along with $f(u)=u$) arise in modeling population dynamics and genetics; see for instance Daw72Fle75.

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Nonlinear parabolic Anderson model in spatial dimensions $d=2,3$,$$\begin{equation} \partial _t u = \Delta u + f(u)\zeta , \tag{3.4}{} \end{equation}$$

where $f$ is a continuous function and $\zeta$ is a noise which typically is assumed to be spatially independent (i.e., white) but constant in time. This models the motion of mass through random media. The assumption of constant in time noise is consistent with the regime where the mass is assumed to move much faster than the time scale in which the media changes. We refer to GIP15HL18 and references therein for well-posedness results.

The parabolic Anderson model, especially the linear case ($f(u)=u$), is a simple model which exhibits intermittency over long time; for the study of long time behavior, one often considers the spatial-discrete equation with $\zeta$ being independent noises on lattice sites; see, for instance, the reviews CM94 and K16 for further discussion and references regarding long time behaviors of the parabolic Anderson model.

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Dynamical sine-Gordon equation,$$\begin{equation} \partial _t u = \Delta u + \sin (\beta u) +\xi , \qquad (t,x)\in \mathbf{R}_+\times \mathbf{T}^2. \tag{3.5}{} \end{equation}$$

This equation describes the natural dynamics of a class of two-dimensional systems that exhibits the Berezinskiĭ–Kosterlitz–Thouless (BKT) phase transition Ber70KT73Jos13, such as two-dimensional Coulomb gas and certain condensed matter materials.⁠⁸ These include thin disordered super-conducting granular films. The phase transition is from bound vortex–antivortex pairs at low temperatures to unpaired vortices and antivortices at some critical temperature.^✖ See MS77FS81 for earlier studies of the model in equilibrium. Here $\beta ^2$ represents the inverse temperature, and $\beta ^2=8\pi$ is the BKT critical point. See HS16CHS18 for the construction of local solutions of this dynamic.

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Random motion of a curve in an $N$-dimensional manifold $M$ driven by $m$ independent space-time white noises (see Hai16BGHZ19RWZZ18),$$\begin{equation} \partial _t h^\alpha = \partial _x^2 h^\alpha + \sum _{\beta ,\gamma =1}^N \Gamma _{\beta \gamma }^\alpha (h) \partial _x h^\beta \partial _x h^\gamma + \sum _{i=1}^m \sigma ^\alpha _i (h) \xi _i \qquad \alpha \in \{1,2,\ldots , N\} \;, \tag{3.6}{} \end{equation}$$

where $h$ is a map from an interval to $M$, $\Gamma _{\beta \gamma }^\alpha$ are the Christoffel symbols for the Levi-Civita connection, and $\{\sigma _i \}_{i=1}^m$ is a collection of vector fields on the manifold. This is a non-Euclidean generalization of 2.5.

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Stochastic Yang–Mills flow in spatial dimensions $d\le 4$$$\begin{equation} \partial _t A = - d^*_{A} F_A + \xi , \tag{3.7}{} \end{equation}$$

where the deterministic part (without $\xi$) is the Yang–Mills gradient flow introduced in DK90 which is extensively studied in geometry (see the monograph Fee14). Here, in the setting of differential geometry, one fixes a Lie group, $A$ is a connection (or a Lie algebra valued 1-form), $F_A$ is the curvature of $A$, $d_{A}$ is the covariant derivative operator, and $d^*_{A}$ is its adjoint. The noise $\xi =\xi _1dx_1 + \cdots + \xi _d dx_d$ is a 1-form with each component $\xi _i$ being an (independent copy of) Lie algebra valued space-time white noise. See She18 for some initial progress in $d=2$ in the case that the Lie group is Abelian. Note that 3.7 is not a parabolic equation, and one usually adds an additional term $- d_A d^*_{A} A$ on the right-hand side to obtain a parabolic equation,$$\begin{equation} \partial _t A = - d^*_{A} F_A - d_A d^*_{A} A + \xi , \tag{3.8}{} \end{equation}$$

which is gauge equivalent with the original equation (the Donaldson–De Turck trick).

The study of geometric equations with randomness such as 3.6 and 3.7 is of general interest. Equation 3.7 is motivated by the problem of quantization of the Yang–Mills field theory; see also the next item.

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Stochastic quantization. This refers to a large class of singular SPDEs arising from Euclidean quantum field theories defined via Hamiltonians (or actions, energy, etc.). They were introduced by Parisi and Wu in PW81. Given a Hamiltonian $\mathcal{H}(\Phi )$, which is a functional of $\Phi$, one considers a gradient flow of $\mathcal{H}(\Phi )$ perturbed by space-time white noise $\xi$:$$\begin{equation} \partial _t \Phi = - \frac {\delta \mathcal{H}(\Phi )}{\delta \Phi } + \xi \;. \tag{3.9}{} \end{equation}$$

Here $\frac {\delta \mathcal{H}(\Phi )}{\delta \Phi }$ is the variational derivative of the functional $\mathcal{H}(\Phi )$; for instance, when $\mathcal{H}(\Phi )=\frac 12\int (\nabla \Phi )^2 dx$ is the Dirichlet form, $\frac {\delta \mathcal{H}(\Phi )}{\delta \Phi } =-\Delta \Phi$ and 3.9 boils down to the stochastic heat equation 2.6. Note that $\Phi$ can be also multicomponent fields, with $\xi$ being likewise multicomponent. The aforementioned $\Phi ^4$ equation, sine-Gordon equation, and stochastic Yang–Mills flow all belong to this class of stochastic quantization equations, each corresponding to a Hamiltonian $\mathcal{H}$.

The significance of these stochastic quantization equations 3.9 is that given a Hamiltonian $\mathcal{H}(\Phi )$, the formal measure$$\begin{equation} \tfrac {1}{Z}e^{-\mathcal{H}(\Phi )} D\Phi \tag{3.10}{} \end{equation}$$

is formally an invariant measure⁠⁹ Being invariant means that if the initial condition of 3.9 is random with probability law given by 3.10, then the solution at any $t>0$ will likewise be distributed according to this same probability law. For readers familiar with stochastic ODEs, one simple example is given by the Ornstein–Uhlenbeck process, $dX_t=-\frac 12 X_tdt+dB_t$, where $B_t$ is the Brownian motion, and its invariant measure is the (one-dimensional) Gaussian measure $\frac {1}{\sqrt {2\pi }}e^{-\frac {X^2}{2}}dX$.^✖ for equation 3.9. Here $D\Phi$ is the formal Lebesgue measure and $Z$ is a normalization constant. We emphasize that 3.10 is only a formal measure because, among several other reasons, there is no Lebesgue measure $D\Phi$ on an infinite-dimensional space and it is a priori not clear at all if the measure can be normalized. These measures arise from Euclidean quantum field theories. In their path integral formulations, quantities of physical interest are defined by expectations with respect to these measures. The task of constructive quantum field theory is to give precise meaning or constructions to these formal measures; see the book Jaf00.

Given the very recent progress of SPDEs, a new approach to construct the measure of the form 3.10 is to construct the long-time solution to the stochastic PDE 3.9 and average the distribution of the solution over time. This approach has been shown to be successful for the $\Phi ^4$ model in $d\le 3$ in a series of very recent works, which starts with MW17b on the torus $\mathbf{T}^3$ where a priori estimates were obtained to rule out the possibility of finite time blowup. Then GH18bGH18a established a priori estimates for solutions on the full space $\mathbf{R}^3$ yielding the construction of $\Phi ^4$ quantum field theory on the whole $\mathbf{R}^3$, as well as verification of some key properties that this invariant measure must satisfy as desired by physicists, such as reflection positivity. See also AK17. Similar uniform a priori estimates are obtained by MW18 using maximum principle.

•

Random (nonlinear) Schrödinger equation,$$\begin{equation} i \frac {du}{dt} = \Delta u +\lambda |u|^2 u + u \xi , \tag{3.11}{} \end{equation}$$

where $u$ is complex valued, and $\xi$ is a real valued spatial white noise. The linear case $\lambda =0$ is a model for Anderson localization (a complex version of 3.4; see the recent works AC15GKR18). In the nonlinear case, it describes the evolution of nonlinear dispersive waves in a totally disordered medium, with $\lambda >0$ corresponding to the focusing case and $\lambda < 0$ to the defocusing case; see Con12GGF$^+$12 for its physical background and DW18DM17GUZ18 for recent mathematical results.

•

(Nonlinear) stochastic wave equation,$$\begin{equation} \partial _t^2 u - \Delta u + F(u) = \xi , \tag{3.12}{} \end{equation}$$

with given initial data $(u,\partial _t u)|_{t=0}$. The linear case ($F=0$) in $d=1$ spatial dimension, as Walsh explained in Wal86, describes “a guitar left outdoors during a sandstorm. The grains of sand hit the strings continually but irregularly.” If $\xi (dt,dx)$ is the random measure of the number of grains hitting in $(dt,dx)$ (centered by subtracting the mean), then $\xi$ should be space-time white noise since the numbers hitting over different time intervals or string portions will be essentially independent. The position $u(t,x)$ of the string should satisfy 3.12 with $F=0$.

Equation 3.12 with nonzero $F$ are investigated in earlier works by AHR96OR98 in spatial dimensions $d= 2,3$, and they proved that with just a function $F$ (satisfying some nice properties) the solution to 3.12 is trivial, namely the same with the solution for $F=0$; the reason for this triviality will be clear in the next subsection.

More recently, GKO18b obtained nontrivial solutions with $F$ given (formally!—see the next subsection) by $F(u)=\lambda u^k$ in $d=2$, and $F(u)=u +u^2$ in $d=3$ in GKO18a. GUZ18 then studied a stochastic wave equation with $F(u)=u^3$ and multiplicative noise ($u\xi$ on the right-hand side) in $d=2,3$.

Remark 3.1.

Note that these nonlinear SPDEs are generally not scaling invariant, unlike the linear stochastic heat equation 2.6, which is scaling invariant in any dimension $d$ (recall property (P5) at the end of Section 2). For instance, for the KPZ equation, $\widetilde{H}(t,x)\!\mathrel {≔}\! \lambda ^{\frac {d-2}{2}} H(\lambda ^2 t, \lambda x)$ will satisfy $\partial _t \widetilde{H} = \partial _x^2 \widetilde{H} + \lambda ^{\frac {2-d}{2}}(\partial _x \widetilde{H})^2 + \widetilde{\xi }$, where $\widetilde{\xi }$ is a new white noise and thus not invariant unless $d=2$. (Indeed, any choice of three scaling components for $t,x,H$ cannot make four terms invariant.) This will turn out to be important for defining solutions to these equations; see Remark 4.2. Also, as we will see in Sections 5 and 6, it would not be possible to derive these equations (that are not scaling invariant) as limits of scaling certain physical models, unless the physical models have a weak asymmetry, a long interaction range, or a weak intensity of noise, which sets an additional scale.

3.2. Challenge of solution theory

Solution theories for SPDEs have been developed since the 1970s. Earlier progress was recorded by the books written in the 1980s such as KR82Wal86, and more recent books such as CR99PR07DKM$^+$09DPZ14LR15. However, many very important equations, including some of those listed above, remained poorly understood—that is, until very recently.

The difficulty in building solution theories to nonlinear SPDEs is that often these equations are too singular; namely, the solution (if it exists) would have low enough regularity so that certain parts of the equation do not a priori make sense. Indeed, recall that for the linear equation 2.6 in $d$ spatial dimensions, the solution is almost surely an element of $\mathcal{C}^{\alpha }$ for $\alpha <\frac {2-d}{2}$, which is continuous when $d=1$ and is a distribution when $d\ge 2$. Since with nonlinear terms the solutions are not expected to be more regular, the “$\Phi ^3$” term in the $\Phi ^4$ equation when $d\ge 2$ is a priori meaningless because distributions in general cannot be multiplied. Similarly, for the KPZ equation, if $d\ge 1$, $\partial _x H$ is distribution valued and thus the term “$(\partial _x H)^2$” does not have a clear meaning. For this type of singular SPDE, it is challenging to even interpret what one means by a solution.⁠¹⁰ The same issue exists for the dispersive equations. For instance, the solution to the stochastic wave equation 3.12 in spatial dimensions $d\ge 2$ is distributional, and this is exactly the reason that the triviality result by AHR96OR98 for the nonlinear problem should be expected. Indeed, their proofs are based on a Colombeau distribution machinery.^✖

Starting in the 1980s, the idea of renormalization entered the study of SPDEs JLM85AR91DPD03. Recently, this idea has received far-reaching generalization in work by Hairer Hai14b, Gubinelli, Imkeller, and Perkowski GIP15, and many subsequent works. The idea is to subtract terms with infinite constants from the nonlinearities. Taking the $\Phi ^4$ equation as an example with spatial dimension $d\le 3$, one needs to consider the renormalized $\Phi ^4$ equation,

$$\begin{equation} \partial _t \Phi = \Delta \Phi - (\Phi ^3 -\infty \Phi ) + \xi \;. \tag{3.13}{} \end{equation}$$

This precisely means the following. Since the origin of the problem is the singularity of the driving noise $\xi$, one starts by regularizing $\xi$. For instance, we take a space-time convolution of $\xi$ with a mollifier $\varphi _\varepsilon$ that is a smooth function of space and time with support of size $\varepsilon$ so that $\varphi _\varepsilon \to \delta$ (the Dirac delta distribution) as $\varepsilon \to 0$. Now we consider the $\Phi ^4$ equation driven by $\xi _\varepsilon$,

$$\begin{equation} \partial _t \Phi _\varepsilon = \Delta \Phi _\varepsilon - \Phi _\varepsilon ^3 + \xi _\varepsilon , \qquad \text{where } \xi _\varepsilon = \xi * \varphi _\varepsilon . \tag{3.14}{} \end{equation}$$

For any $\varepsilon >0$, due to the smoothness of the noise, we can solve the above PDE in the classical sense, and $\xi _\varepsilon \in \mathcal{C}^\infty$ implies $\Phi _\varepsilon \in \mathcal{C}^\infty$. As $\varepsilon \to 0$, $\xi _\varepsilon \to \xi$, but $\Phi _\varepsilon$ do not converge to any nontrivial limit!

The idea is that before the limit, one should insert renormalization terms (also often called counter-terms in the context of quantum field theory⁠¹¹ In fact the corresponding quantum field theory requires a renormalization $\int \frac 12 |\nabla \Phi |^2+\frac 14 \Phi ^4 - \frac {C_\varepsilon }{2} \Phi ^2 \,d x$ for dimensions $2$ and $3$ which is well known in physics.^✖)

$$\begin{equation} \partial _t \Phi _\varepsilon = \Delta \Phi _\varepsilon - ( \Phi _\varepsilon ^3 - C_\varepsilon \Phi _\varepsilon ) + \xi _\varepsilon , \tag{3.15}{} \end{equation}$$

where $C_\varepsilon$ diverges as $\varepsilon \to 0$ at a suitable rate. If the sequence of constants $C_\varepsilon$ is suitably chosen, the sequence of smooth solutions $\Phi _\varepsilon$ of 3.15 will converge to a nontrivial limit as $\varepsilon \to 0$:

$$\begin{equation*} \Phi = \lim _{\varepsilon \to 0} \Phi _\varepsilon . \end{equation*}$$

This is what we mean by a solution to the (renormalized) $\Phi ^4$ equation. Note that we do not attempt to make sense of a limiting equation 3.13, but we construct $\Phi$ via by a limit procedure, the limit of solutions to a sequence of regularized and renormalized equations 3.15.

The same renormalization procedure applies to the KPZ equation (in one spatial dimension) and many of the other singular SPDEs listed above.

This discussion prompts several questions:

•

Why does $\Phi _\varepsilon$ converge, and how does one choose suitable constants $C_\varepsilon$ to make this convergence happen? Is the resulting limit unique, or does it depend on the mollification? This is essentially the question of “well-posedness” which will be addressed in Section 4.

•

Why are we allowed to “change” the equation by inserting new terms that are not negligible—in fact infinite? We address this renormalization question in Section 5, in which we will see that the SPDEs such as $\Phi ^4$ arise as scaling limits of physical systems and the renormalization will turn out to have physical meanings in these systems.

•

How robust are these singular SPDEs under different approximation schemes? This is a universality question, meaning that one singular SPDE should be able to serve as the continuum large scale description of a class of systems which may have different small scale details. We discuss this in Section 6.

These questions are of course entangled in many ways. In terms of approximations and convergence, the procedure described as in 3.14 is the simplest way of approximation and approaching a limit, but scaling limits of physical models are essentially also ways of obtaining the limits. In terms of uniqueness, one expects to get the same SPDE limit not only for different choices of mollifications in 3.14 but also via scaling limits of perhaps apparently very different models, which is what universality means. Section 5 below will focus on the meaning of renormalization in physical models, and Section 6 will provide more detailed discussions on deriving an SPDE from these physical models, and of course in all these endeavors one needs to first understand the meaning of a solution as discussed in Section 4.

4. Well-posedness of singular SPDEs

We discuss how to choose suitable renormalization constants so that one can obtain a nontrivial limit for solutions to renormalized equations. Our exposition consists of two parts.

•

Starting from the 1990s, solutions to renormalized singular SPDEs have been constructed; see for instance AR91DPD03. Here we present an elegant argument due to DPD03 which illustrates a simple example of renormalization, plus a standard Picard iteration (fixed point) PDE argument. This argument, despite its simplicity, yields solutions to several singular (but not too singular) equations, such as the $\Phi ^4$ equation in two spatial dimensions.

•

The above argument fails for more singular SPDEs, such as $\Phi ^{4}$ in three spatial dimensions and the KPZ equation in one spatial dimension. This motivates us to turn to a more robust approach—the theory of regularity structures introduced in Hai14b. We will also mention some alternative theories or methods, such as paracontrolled distributions GIP15 or renormalization groups Kup16.

To focus our discussion in this section, we will work with the $\Phi ^4$ equation.

4.1. A PDE argument and renormalization

Consider the $\Phi ^4$ equation, where the spatial variable takes values in the two-dimensional torus. As explained above, we take a sequence of mollified noises $\xi _\varepsilon$ and consider the mollified equation 3.14.

Write $u_\varepsilon \mathrel {≔}(\partial _t - \Delta )^{-1} \xi _\varepsilon$ for the stationary solution⁠¹² This corresponds to dropping the term involving initial data in 2.9 and integrating time in 2.10 from $-\infty$ instead of $0$. Stationarity means that the distribution of $u_{\varepsilon }$ does not depend on $t$. This assumption will be convenient when performing moment calculations, such as 4.3. Namely, the moments will not depend on space-time points.^✖ to the mollified linear stochastic heat equation 2.6 $\partial _{t} u_{\varepsilon } = \Delta u_{\varepsilon } + \xi _{\varepsilon }$. The key observation is that the most singular part of $\Phi$ is $u$, so if we write

$$\begin{equation} \Phi _\varepsilon = u_\varepsilon + v_\varepsilon , \tag{4.1}{} \end{equation}$$

we can expect the remainder $v_\varepsilon$ to converge in a space of better regularity. Subtracting this linear equation from 3.14 gives

$$\begin{equation} \partial _{t} v_{\varepsilon } = \Delta v_{\varepsilon }-(v_{\varepsilon } + u_{\varepsilon })^{3} = \Delta v_{\varepsilon }-v_{\varepsilon }^{3} -3u_{\varepsilon }v_{\varepsilon }^{2} -3u_{\varepsilon }^{2} v_{\varepsilon } -u_{\varepsilon }^{3} \;. \tag{4.2}{} \end{equation}$$

This equation looks more promising since the rough driving noise $\xi _{\varepsilon }$ has dropped out. This manipulation has not solved the problem of multiplying distributions, since the limit of $u_\varepsilon$ is still a distribution valued in two spatial dimensions (as we discussed earlier—see the fact (P2) in the end of Section 2). However $u_{\varepsilon }$ is a rather concrete object since it is Gaussian distributed ((P3) in the end of Section 2). This makes it possible to study the behavior of $u_{\varepsilon }^{2}$ and $u_{\varepsilon }^{3}$ via probabilistic methods.

As an illustration, consider the expectation

$$\begin{equation} \mathbf{E} [u_{\varepsilon }(t,x)^{2}] = \int P(t-s,x-y) P(t-r,x-z) \varphi _\varepsilon ^{(2)}(s-r,y-z) \,dsdydrdz, \tag{4.3}{} \end{equation}$$

where $\varphi _\varepsilon ^{(2)}$ is the convolution of $\varphi _\varepsilon$ introduced in 3.14 with itself and $P$ is the heat kernel introduced in 2.10. Due to the singularity of the heat kernel $P$ at the origin, this integral diverges like $O(\log \varepsilon )$ as $\varepsilon \to 0$ in two spatial dimensions. Denoting $C_\varepsilon \mathrel {≔}\mathbf{E} [u_{\varepsilon }(t,x)^{2}]$ (which does not depend on $(t,x)$ by stationarity of $u_\varepsilon$), this calculation indicates that in 4.2 we should subtract $C_\varepsilon$ from $u_{\varepsilon }^{2}$, and subtract⁠¹³ The factor $3$ arises from three ways of choosing two powers of $u_\varepsilon$ from the cubic term $u_\varepsilon ^3$. A Wick theorem allows one to compute expectation of a product of arbitrarily many Gaussian variables, and in two dimensions the Wick renormalized power $\,\colon \!\! u^{n}\! \colon = \lim _{\varepsilon \to 0} C_\varepsilon ^{\frac {n}{2}} H_n(u_\varepsilon /\sqrt {C_\varepsilon })$.^✖ $3C_\varepsilon u_\varepsilon$ from $u_{\varepsilon }^{3}$.

This amounts to considering the renormalized $\Phi ^4$ equation

$$\begin{equation} \partial _t \Phi _\varepsilon = \Delta \Phi _\varepsilon - ( \Phi _\varepsilon ^3 - 3C_\varepsilon \Phi _\varepsilon ) + \xi _\varepsilon \;. \tag{4.4}{} \end{equation}$$

These renormalized powers of $u_\varepsilon$ do converge to nontrivial limits. In fact, thanks to Gaussianity of $u_\varepsilon$, given a smooth test function $f$ one can explicitly compute any probabilistic moment of $(u_{\varepsilon }^{2}-C_\varepsilon ) (f)$ and prove its convergence. By choosing $f$ from a suitable set of wavelets or Fourier basis, one can apply a version of Kolmogorov’s theorem⁠¹⁴ This is a version of Kolmogorov’s theorem (formulated differently as the classical Kolmogorov theorem) which is adapted to prove convergences in the $\mathcal{C}^\alpha$ spaces in the present context.^✖ to prove that $u_{\varepsilon }^{2}-C_\varepsilon$ and $u_{\varepsilon }^{3}-C_\varepsilon u_\varepsilon$ converge in $\mathcal{C}^\alpha$ for any $\alpha <0$. We denote these limits $\,\colon \!\! u^2\! \colon$ and $\,\colon \!\! u^3\! \colon$. They are elements of $\mathcal{C}^\alpha$ for any $\alpha <0$.

To summarize, we have found that the renormalization constants can be found through explicit moment calculations/expectations.

Passing 4.2 to the limit, we get

$$\begin{equation} \partial _{t} v =\Delta v - \Big (v^{3}+3u v^{2} +3\,\colon \!\! u^{2}\! \colon v + \,\colon \!\! u^{3}\! \colon \Big )\;. \tag{4.5}{} \end{equation}$$

We can prove local well-posedness of this equation as a classical PDE, by a standard fixed point argument. For this, we use a classical result in harmonic analysis under the name Young’s theorem⁠¹⁵ Note that the original form of Young’s theorem is only for one-dimensional functions of finite $p$-variations, and the version of Young’s theorem we are referring to here can be found in Hai14b, Proposition 4.14 and BCD11, Theorems 2.47 and 2.52, for instance.^✖ which states that if $f \in \mathcal{C}^{\alpha }$, $g \in \mathcal{C}^{\beta }$, and $\alpha +\beta >0$, then $f\cdot g \in \mathcal{C}^{\min (\alpha , \beta )}$. Thus if we assume that $v \in \mathcal{C}^{\beta }$ for, say, $\beta =1$, then the worst term in the parenthesis in 4.5 has regularity $\alpha$. By the classical Schauder estimate, which states that the heat kernel improves regularity by $2$, the fixed point map

$$\begin{equation} v \mapsto (\partial _t - \Delta )^{-1} \Big (v^{3}+3u v^{2} +3\,\colon \!\! u^{2}\! \colon v + \,\colon \!\! u^{3}\! \colon \Big ) \tag{4.6}{} \end{equation}$$

is well defined. Namely, it maps a generic element $v \in \mathcal{C}^{\beta }$ to a new element which is again in $\mathcal{C}^{\beta }$ for $\beta =1$. This is since for $\alpha <0$ sufficiently close to $0$ one has $\mathcal{C}^{\alpha +2}\subset \mathcal{C}^{\beta }$. With a bit of extra effort, one can show that over a short time interval the fixed point map is contractive and thus has a fixed point in $\mathcal{C}^{\beta }$, and this fixed point $v$ is the solution. (The sharp result is $\mathcal{C}^{\beta }$ for any $\beta <2$.) To conclude, one has $\Phi = u+ v$, which is the local solution to the renormalized $\Phi ^4$ equation in two spatial dimensions.

The above argument was first used by Da Prato and Debussche in DPD03, and it applies to other equations, for instance the stochastic Navier–Stokes equation 3.1 with space-time white noise on a two-dimensional torus DPD02. Let us mention another, somewhat surprising application, that is the dynamical sine-Gordon equation 3.5 in two spatial dimensions in the regime $\beta ^2\in (0,4\pi )$. The renormalized equation reads

$$\begin{equation*} \partial _{t}u_\varepsilon =\tfrac {1}{2}\Delta u_\varepsilon + C_\varepsilon \sin (\beta u_\varepsilon )+\xi _\varepsilon \;, \qquad \beta \in \mathbf{R}\;, \end{equation*}$$

where $C_\varepsilon$ is a renormalization constant which diverges like $O(\varepsilon ^{-\frac {\beta ^2}{4\pi }})$. By writing $u_\varepsilon = \phi _\varepsilon + v_\varepsilon$ with $\phi _\varepsilon \mathrel {≔}(\partial _t - \Delta )^{-1} \xi _\varepsilon$, one finds that

$$\begin{equation*} \partial _{t} v_\varepsilon =\tfrac {1}{2}\Delta v_\varepsilon + C_\varepsilon \sin (\beta \phi _\varepsilon ) \cos (\beta v_\varepsilon ) + C_\varepsilon \cos (\beta \phi _\varepsilon ) \sin (\beta v_\varepsilon ) \;. \end{equation*}$$

HS16 proved that $C_\varepsilon e^{\pm i \beta \phi _\varepsilon } =C_\varepsilon ( \cos (\beta \phi _\varepsilon ) \pm i \sin (\beta \phi _\varepsilon ))$ converges to a nontrivial limit in $\mathcal{C}^{-\frac {\beta ^2}{4\pi }}$, so $\beta ^2\in (0,4\pi )$ is precisely⁠¹⁶ Recall that, for the fixed point argument to work, we must have that the regularity plus 2 is at least 1.^✖ the regime where the above classical PDE argument applies. Note that the constant $C_\varepsilon$ can be again found by calculating the expectation of $e^{\pm i \beta \phi _\varepsilon }$, i.e., the characteristic function of the Gaussian random variable $\phi _\varepsilon$.

The same idea (but with a slightly different transformation than 4.1) applies to the linear parabolic Anderson model in HL15:

$$\begin{equation*} \partial _t u_\varepsilon = \Delta u_\varepsilon + u_\varepsilon \left(\zeta _\varepsilon -C_\varepsilon \right), \qquad (t,x)\in \mathbf{R}_+\times \mathbf{T}^2, \end{equation*}$$

where $\zeta _\varepsilon$ is regularized spatial white noise on $\mathbf{T}^2$. With a transformation $v_\varepsilon =u_\varepsilon e^{Y_\varepsilon }$, where $\Delta Y_\varepsilon =\zeta _\varepsilon$, one can simply check

$$\begin{equation*} \partial _t v_\varepsilon = \Delta v_\varepsilon + v_\varepsilon \left( |\nabla Y_\varepsilon |^2 -C_\varepsilon \right) - 2 \nabla Y_\varepsilon \cdot \nabla v_\varepsilon \;. \end{equation*}$$

Again, $Y_\varepsilon$ is a Gaussian process, and with $C_\varepsilon \mathrel {≔}\mathbf{E}|Y_\varepsilon |^2 = -\frac {1}{2\pi } \log \varepsilon +O(1)$, $|\nabla Y_\varepsilon |^2 -C_\varepsilon$ converges to a nontrivial limit, and the equation for $v$ is shown to be locally well-posed by standard PDE methods as above. (In fact HL15 constructed a global solution on $\mathbf{R}_+\times \mathbf{R}^2$, making use of the linearity of the equation.) DW18 studies the stochastic Schrödinger equation 3.11 on $\mathbf{T}^2$, in which a transformation similar to that in HL15 can be applied.

This type of strategy has also been applied to stochastic hyperbolic equations. Consider the stochastic nonlinear wave equations,

$$\begin{equation*} \partial _t^2 u - \Delta u \pm u^k = \xi , \qquad (x,t)\in \mathbf{R}_+\times \mathbf{T}^2, \end{equation*}$$

with given initial data $(u,\partial _t u)|_{t=0}$, where $\xi$ is the space-time white noise on $\mathbf{R}_+\times \mathbf{T}^2$, and $k\ge 2$. GKO18b adopts the above Da Prato–Debussche trick to write $u$ as a linear part plus a remainder. Such an idea previously appears in the context of deterministic dispersive PDEs with random initial data in earlier work of McKean McK95 and Bourgain Bou96. The proof in GKO18b is based on a fixed point argument for the remainder equation (as above), but with the Schauder estimates replaced by Strichartz estimates for the wave equations. The key point is to use function spaces where the wave equation allows for a gain in regularity. This gain is sufficient to prove that the remainder has better regularity than the linear solution and gives a well-defined nonlinearity for which suitable local-in-time estimates can be established.

4.2. Regularity structures and paracontrolled distributions

The above argument fails for the $\Phi ^4$ equation in three spatial dimensions. As dimension increases, the space-time white noise (and thus the solution) becomes more singular. To see how this problem rears its head, consider the term $\,\colon \!\! u^{2}\! \colon v$ in 4.5. One can show⁠¹⁷ In three spatial dimensions $\xi \in \mathcal{C}^{-\frac 52 -}$, therefore $u\in \mathcal{C}^{-\frac 12 -}$. From this one can show that $\,\colon \!\! u^{2}\! \colon \in \mathcal{C}^{-1-}$ (the rigorous proof of this fact is done by moment analysis).^✖ that in three spatial dimensions, as a space-time distribution, $\,\colon \!\! u^{2}\! \colon \in \mathcal{C}^\alpha$ for any $\alpha <-1$. Thus, to multiply $\,\colon \!\! u^{2}\! \colon$ with $v$, we would have to formulate the fixed point map 4.6 for $v\in \mathcal{C}^\beta$ with $\beta >1$. The product $\,\colon \!\! u^{2}\! \colon v$ would then lie in $\mathcal{C}^\alpha$ for any $\alpha <-1$. Unfortunately, $(\partial _t - \Delta )^{-1}$ only provides two more degrees of regularities, and thus the fixed point map 4.6 will not bring an element in $\mathcal{C}^\beta$ back to the same space.

A natural idea is to go one step further in the expansion 4.1. In view of the equation 4.2, we define a second order perturbative term, $w_\varepsilon \mathrel {≔}(\partial _t - \Delta )^{-1} \,\colon \!\! u_\varepsilon ^{3}\! \colon$, and rewrite the expansion as

$$\begin{equation} \Phi _\varepsilon = u_\varepsilon - w_\varepsilon +R_\varepsilon \;. \tag{4.7}{} \end{equation}$$

It turns out that one can prove that $w_\varepsilon$ converges in $\mathcal{C}^\alpha$ for $\alpha <1$ to a limit $w$. It remains to see whether $R_\varepsilon$ converges to a limit $R$ with even better regularity. Using 4.4, it is straightforward to derive an equation for $R_\varepsilon$:

$$\begin{equation} R_\varepsilon = (\partial _t - \Delta )^{-1} \Big ( -3(u_\varepsilon ^{2}-C_\varepsilon ) R_\varepsilon - 3 (u_\varepsilon ^{2}-C_\varepsilon ) w_\varepsilon +\cdots \Big )\;. \tag{4.8}{} \end{equation}$$

There should be eight terms in the parenthesis, but we have only written down the two terms that are important for our discussion; the other terms (in “$\cdots$”) can be treated by the standard PDE argument as in the two-dimensional case above. It turns out that even after this higher order expansion 4.7, the above PDE fixed point argument still does not work because of the two terms written on the right-hand side of 4.8.

For the second term, $(u_\varepsilon ^{2}-C_\varepsilon ) w_\varepsilon$, we have $u_\varepsilon ^{2}-C_\varepsilon \to \,\colon \!\! u^{2}\! \colon \in \mathcal{C}^{-1-}$ and $w_\varepsilon \to w\in \mathcal{C}^{\frac 12-}$. This is below the borderline of applicability of Young’s theorem. It is not hard to overcome this difficulty. In fact, in three spatial dimensions, the term $(u_\varepsilon ^{2}-C_\varepsilon ) w_\varepsilon$ requires further renormalization to converge to a nontrivial limit. Since this term is nothing but a convolution of several heat kernels and Gaussian noises, one can again carry out a moment analysis to find a suitable renormalization constant $\overline{C}_\varepsilon$, which turns out to diverge logarithmically such that

$$\begin{equation} (u_\varepsilon ^{2}-C_\varepsilon ) w_\varepsilon - \overline{C}_\varepsilon u_\varepsilon \qquad \text{converges in $\mathcal{C}^{-\frac 12-}$}\;. \tag{4.9}{} \end{equation}$$

This amounts to renormalizing the $\Phi ^4$ equation in three spatial dimensions as

$$\begin{equation} \partial _t \Phi _\varepsilon = \Delta \Phi _\varepsilon - ( \Phi _\varepsilon ^3 - (3C_\varepsilon +\overline{C}_\varepsilon )\Phi _\varepsilon ) + \xi _\varepsilon \;, \tag{4.10}{} \end{equation}$$

where $C_\varepsilon \sim \varepsilon ^{-1}$ and $\overline{C}_\varepsilon \sim |\log \varepsilon |$. See also Remark 4.1.

The first term, $-3(u_\varepsilon ^{2}-C_\varepsilon ) R_\varepsilon$, is of the same nature as $\,\colon \!\! u^{2}\! \colon v$ in 4.6, so we suffer exactly the same vicious circle of difficulty as in the discussion for $\,\colon \!\! u^{2}\! \colon v$; namely, the fixed point argument does not close. In fact, higher order expansions beyond 4.7 will always end up with such a term so the same problem will remain. This is the real obstacle. The idea of regularity structures (which overcomes this obstacle) is that the solutions to the two equations,

$$\begin{equation} R= (\partial _t - \Delta )^{-1} \big ( -3 \,\colon \!\! u^{2}\! \colon R \big )\;, \qquad \widetilde{R} = (\partial _t - \Delta )^{-1} \big (-3 \,\colon \!\! u^{2}\! \colon \big ), \tag{4.11}{} \end{equation}$$

should have the same small scale behavior, because $\,\colon \!\! u^{2}\! \colon$ is more singular than $R$ and it is the factor $\,\colon \!\! u^{2}\! \colon$ that dominates the small scale roughness. (Here we have ignored all the other terms in 4.8, which have better regularities than that of $\,\colon \!\! u^{2}\! \colon R$, in order to focus on the main issue of the problem.) This a priori knowledge that $R$ should locally look like $\widetilde{R}$ can be formulated as that when the space-time points $z$ and $z_0$ are close, one should expect that⁠¹⁸ Equation 4.12 is reminiscent of a Taylor expansion $F(z)=F(z_0)+F'(z_0)(z-z_0)+\cdots$, where one approximates a differentiable function by Taylor polynomials. Here we approximate $R$ by $\widetilde{R}$ which is also an object that is simply a convolution of heat kernels with white noises. Taylor polynomials are special examples of regularity structures, and the theory of regularity structures is a generalization of Taylor expansion.^✖

$$\begin{equation} R(z)-R(z_0) = R(z_0) (\widetilde{R}(z)-\widetilde{R}(z_0)) + O(|z-z_0|) \;. \tag{4.12}{} \end{equation}$$

Namely, the local increment of $R$ is approximately the same as the local increment of $\widetilde{R}$—up to multiplying a factor $R(z_0)$ which depends on the base point $z_0$; the reason that this multiplicative factor should be $R(z_0)$ is clear from the structure of equation 4.11.

Since $\widetilde{R}$ is again a concrete object, which is simply convolutions of heat kernels with powers of Gaussians, it is easy to prove $\widetilde{R}\in \mathcal{C}^{1-}$ by analyzing its moments as before. Thus if $R$ satisfies 4.12, that is, locally looks like $\widetilde{R}$, one has $R\in \mathcal{C}^{1-}$ as well. The converse is not true; the set of $R$ satisfying 4.12 is a strictly smaller set than $\mathcal{C}^{1-}$. The key is to formulate a fixed point problem in the space of all functions $R$ that have prescribed local expansion 4.12 (rather than in standard function spaces such as $\mathcal{C}^\alpha$). The aforementioned vicious term $\,\colon \!\! u^{2}\! \colon R$, which could not be defined for arbitrary $R\in \mathcal{C}^{1-}$, can now be defined if $R$ locally looks like $\widetilde{R}$, because $\,\colon \!\! u^{2}\! \colon \widetilde{R}$ is again simply a concrete combination of Gaussian processes and heat kernels! It turns out that the fixed point argument closes in the space of functions having such prescribed local expansions, and the fixed point $R$ together with 4.7 yields the solution to the $\Phi ^4$ equation in three dimensions.

The above idea of solving stochastic equations in a space of functions or distributions that have prescribed local approximations by certain canonical objects to some extent had its precursor in the simpler setting of stochastic ordinary differential equations, which is called rough path theory (see Lyo98 or the book FH14) in particular a formulation by Gub04. Constructing the solution to the $\Phi ^4$ equation on a three-dimensional torus was the first example of the theory built in Hai14b. The review articles Hai15aHai15b have more detailed pedagogical explanations on the theory and the application to this equation.

Remark 4.1.

We have found the renormalization of $u_{\varepsilon }^{2}$, $u_{\varepsilon }^{3}$ and $u_\varepsilon ^{2} w_\varepsilon$ and proved convergence of the renormalized objects by moment analysis. Analyzing moments of these random objects are the only probabilistic component of Hai14b. As the equation in question becomes more singular, the number of such random objects to be studied increases, and it is tedious or even impossible to analyze each of them by hand. CH16 develops a blackbox that provides systematic and automatic treatment for renormalization and moment analysis for these perturbative objects arising from general singular SPDEs. Moreover, there are also algebraic aspects for the renormalization procedure (so-called renormalization groups), which has been systematically treated in BHZ16. Finally, there is a question regarding what the renormalized equation (e.g., 4.10) will look like after renormalizing these random objects, and this is answered in BCCH17.

Hairer’s theory has been applied to provide solutions to other very singular SPDEs, for instance, a generalized parabolic Anderson model (a generalization of 3.4),

$$\begin{equation*} \partial _t u = \Delta u + \sum \nolimits _{ij} f_{ij}(u) \partial _i u \partial _j u+ g(u)\xi , \end{equation*}$$

where $f$ and $g$ are sufficiently regular functions. The well-posedness of the KPZ equation in one spatial dimension was solved in Hai13 using the theory of controlled rough paths Gub04, which can now be viewed as a special case of regularity structures; see the book FH14 for rough paths, regularity structures, and applications to KPZ.

Other applications include (but are not limited to) the stochastic Navier–Stokes equation 3.1 with white noise on the three-dimensional torus ZZ15, the stochastic heat equation with multiplicative noise 3.2 HP15HL18, the dynamical sine-Gordon equation 3.5 on two-dimensional torus for arbitrary $\beta ^2 < 8\pi$ HS16CHS18, the stochastic quantization of Abelian gauge theory/stochastic gauged Ginzburg–Landau equation by She18, and the random motion of string in manifold 3.6 BGHZ19.

Besides Hairer’s theory Hai14b, some alternative methods have also been introduced. The paracontrolled distribution method of Gubinelli, Imkeller, and Perkowski GIP15 is based on a similar idea of controlling the local behavior of solutions, but it is implemented in a different way by using Littlewood–Paley theory and paraproducts BCD11. See GP18b for a review on paracontrolled distribution. The paracontrolled distribution method has been also successfully applied to, for instance, the KPZ equation GP17bHos18 in $d=1$ (and more recently the construction of a solution on the entire real line instead of a circle PR18), a multi-component coupled KPZ equation FH17, and the $\Phi ^4$ equation CC18b in $d=3$ (and more interestingly, its global solution by MW17b).

The paracontrolled distribution method has not only allowed us to prove well-posedness results for stochastic PDEs, but it also resulted in the construction of other singular objects that could not be made sense of before. For instance, AC15 constructed the Anderson Hamiltonian (i.e., Schrödinger operator) on the two-dimensional torus, formally defined as $\mathscr{H}=-\Delta +\xi$, where $\xi$ is a singular potential such as white noise. As another example, CC18a proved existence and uniqueness of solution for stochastic ordinary differential equations $dX_t =V(t,X_t)dt +dB_t$ with distributional valued drift $V$, where $B$ is a $d$-dimensional Brownian motion, and this is achieved via the study of the generator of the above stochastic ordinary differential equations given by $\partial _t+\frac 12 \Delta +V\cdot \nabla$. CC18a also managed to make sense of a singular polymer measure on the space of continuous functions formally given by $\mathbb{Q}_T(d\omega )=Z_0^{-1}\exp \left(\int _0^T\xi (\omega _s)ds\right)\mathbb{W}_T(d\omega )$, where $\mathbb{W}$ is the Wiener measure (i.e., the Gaussian measure for Brownian motions) on $C([0, T ], \mathbf{R}^d)$ for $d = 2, 3$, $\xi$ is a spatial white noise on the $d$-dimensional torus independent of $\mathbb{W}$, and $Z_0$ is an (infinite) renormalization constant.

We will discuss another application of the paracontrolled distribution method on the scaling limit problem with a bit more detail in Section 5.2.

In the line of this paracontrolled distribution approach, BB16 provided a semigroup approach, which has been applied to the generalized parabolic Anderson model 3.4 on a potentially unbounded two-dimensional Riemannian manifold.

Another method based on renormalization group flow was introduced by Kupiainen Kup16, which for instance has been applied to prove local well-posedness for a generalized KPZ equation KM17 introduced by H. Spohn Spo14 in the context of stochastic hydrodynamics.

With all these alternative methods, the theory of regularity structures is by far the most systematic and general approach; for instance it has developed the blackbox theorems as mentioned in Remark 4.1 which makes the implementation of this theory very automatic, and it can deal with equations which are extremely singular (that is, very close or even arbitrarily close to criticality, see Remark 4.2) such as the random string in a manifold 3.6 or the dynamical sine-Gordon equation 3.5 for arbitrary $\beta ^2 < 8\pi$.

Remark 4.2 (Subcriticality of stochastic PDE).

The methods developed in Hai14b, BCD11, and Kup16 are all for subcritical semilinear stochastic PDEs. For stochastic PDEs with white noise, the equation being subcritical means that the nonlinear term has better regularity than the linear terms; namely, small scale roughness is dominated by the linear solution. For instance, for the $\Phi ^4$ equation in three spatial dimensions, the term $\Phi ^3$ has regularity $-\frac 32-$ while $\Delta \Phi$ and $\xi$ have regularities $-\frac 52-$. Subcriticality often depends on spatial dimensions: KPZ, 3.2, and 3.6 are subcritical in $d<2$, while $\Phi ^4$, the parabolic Anderson model 3.4, the Navier–Stokes equation 3.1 with space-time white noise, and the stochastic Yang–Mills heat flow 3.8 are subcritical in $d<4$. The dynamical sine-Gordon equation 3.5 however is subcritical for $\beta ^2 < 8\pi$.

The stochastic PDEs being discussed here in supercritical regimes (i.e., above the aforementioned criticalities) are not expected to have nontrivial meanings of solutions. We only expect to get Gaussian limit, although the Gaussian variances may be nontrivial; the reader is referred to MU18, Theorem 1.1 for a flavor of such a result for the KPZ equation in $d\ge 3$.

Critical dimensions are much more subtle. We refer to CD18CSZ17bCSZ18Gu18 for the very new progress on the KPZ equation in $d=2$.

Remark 4.3.

We remark that although we have focused on semilinear equations in our expositions, the methods developed in Hai14bBCD11 have also extended to quasilinear equations; see OW18FG19BDH19GH17b.

4.3. A brief discussion on weak solutions

The solutions to SPDEs that we have discussed so far are called strong solutions, as opposed to the weak solutions that we will now briefly discuss.⁠¹⁹ Not all equations that admit weak solutions admit strong solutions. A famous stochastic differential equation example is called Tanaka’s equation; see KS91a, Example 3.5.^✖ Let us immediately point out that the weak solutions in the stochastic context have nothing to do with the weak solutions in deterministic PDE theory; one sometimes adopts the terminology “probabilistically weak solutions”.

For a strong solution, one starts with a probability space on which the noise is defined and then builds a mapping from that probability space and the initial data space to a space of functions (or distributions) that satisfies the prescribed equation⁠²⁰ As we saw, making sense of what it means to satisfy the equation often takes significant work and involves regularizations and renormalizations. There are also some measurability assumptions which should be imposed on strong solutions so future noise cannot effect the evolution before its time.^✖ with probability $1$ (i.e., for almost every point in the probability space). Though subtle, it is important to understand that a strong solution to an SPDE need not be function valued (as we saw, in some instances it is distribution valued, living in some spaces of negative regularity).

For a standard PDE, a weak solution requires that the equation holds when tested against a suitable class of functions. For SPDEs, the analogue of this involves treating solutions statistically as probability measures on the solution space, rather than as random variables supported on the probability space on which the noise is defined. Roughly, a weak solution means that we can define some noise (with the right distribution and measurability assumptions satisfied) so that the canonical process⁠²¹ The canonical process is the random variable whose probability space is defined as the solution space equipped with the probability measure of the proposed solution.^✖ on the solution space, along with the noise satisfying the desired equation.⁠²² Here is a, hopefully, more intuitive explanation for this different notion of strong versus weak. Imagine that human life were governed by an SPDE. Then, a strong solution would tell us how each individual’s life would unfold, given the knowledge of all of the randomness which befalls them, in addition to the world around them. A weak solution is statistical—it tells us that people with certain characteristics have certain probabilities of having their life unfold in various ways. Given such a prescription of probabilities, how can be verify that this is, indeed, a weak solution to the “SPDE of life”? Well, we need to demonstrate that there exists randomness which would, in fact, result in the aforementioned probabilities. Then, we would need to verify that the randomness is distributed in the way that the SPDE of life claims (e.g., space-time white noise). While this is all a bit tongue-in-cheek, we hope it helps explain the difference.^✖ As we will see, martingale problems provide a very convenient way to demonstrate that a weak solution solves an SPDE (instead of demonstrating the existence of a suitable noise as above).

Let us illustrate these ideas in the simplest setting of stochastic differential equations (SDEs). Let $\xi (t)$ denote temporal white noise. Like its space-time counterpart, this can be defined in various ways (e.g., as a series with random coefficients). Consider the SDE $dX_t = \xi (t) dt$, which in integrated form reads $X_t = X_0 + \int _0^t \xi (ds)$ (let us assume that $X_0=0$ for simplicity). Once integration with respect to white noise is understood, this defines a solution map (and hence a strong solution) from $(\xi ,X_0)$ to the full trajectory of $X_t$ for $t\geq 0$. One checks that $t\mapsto X_t$ is continuous and that its marginal distributions are Gaussian with covariance of $X_s$ and $X_t$, given by the minimum of $s$ and $t$. This, in fact, implies that the distribution of the function $t\to X_t$ is a Wiener measure—that is, the distribution of Brownian motion. If instead of $X_t$ we had another Brownian motion $\widetilde{X}_t$ (for instance, we could have $\widetilde{X}_t = -X_t$ or just an independent Brownian motion), then $\widetilde{X}_t$ would be a weak solution, but not a strong solution. This is because $X_t$ and $\widetilde{X}_t$ have the same distribution, even if they are not “driven” by the same noise.

The martingale problem provides an alternative characterization to the Gaussian description above for Brownian motion. The Lévy characterization theorem says that $X_t$ is distributed as a Brownian motion if it is almost surely continuous and both $X_t$ and $X_t^2-t$ are martingales.⁠²³ In fact, local martingales.^✖ A measure on $X_t$ that satisfies this is said to satisfy the martingale problem characterizing Brownian motion.

What does it mean that $X_t$ (or $X_t^2-t$) is a martingale? Roughly speaking, this means that given the history of $X_t$ up to time $t$, the expected value of its future location is exactly $X_t$. This is like a fair gambling system in which your future expected profit is always zero. Martingales are essentially a particular class of centered noise.

Martingale problems exist for general classes of SDEs and are often very useful for proving convergence results. For instance, to show that a discrete time Markov chain converges to an SDE (e.g., a random walk converges to Brownian motion), one can demonstrate that the discrete chain satisfies a discrete version of the SDE’s martingale problem. Then, provided one can demonstrate compactness of the measures (on the evolution of the Markov chain), all limit points must satisfy the limiting SDE’s martingale problem. This generally proves uniqueness of the limit points and, hence, convergence.

Weak solutions to linear SPDEs can also be characterized in terms of martingale problems. Let us describe how this works for the multiplicative noise stochastic heat equation 3.3, recalled here:⁠²⁴ This equation also admits a strong solution which can be written as a chaos series of multiple stochastic integrals against space-time white noise $\xi$.^✖

$$\begin{equation} \partial _t u(t,x) - \Delta u(t,x) = u(t,x)\xi (t,x) \qquad (t,x)\in \mathbf{R}_+ \times \mathbf{R}\;. \tag{4.13}{} \end{equation}$$

Let us write $u_t(x) = u(t,x)$ and think of $u$ as a measure on $C(\mathbf{R}_+,C(\mathbf{R}))$ (continuous maps from $t\in \mathbf{R}_+$ to continuous spatial functions). For any test function $\varphi \in C^\infty (\mathbf{R})$, write $u_t(\phi )=\int u_t(x)\phi (x)dx$. With this notation, define the processes

$$\begin{equation} N_t(\varphi ) \mathrel {≔}u_t(\varphi )- u_0(\varphi ) - \int _0^t u_s(\Delta \varphi )\,ds \quad \text{and}\quad Q_t(\varphi ) \mathrel {≔}N_t(\varphi )^2 - \int _0^t (u_s^2,\varphi ^2 )\,ds. \tag{4.14}{} \end{equation}$$

We say that $u$ satisfies the martingale problem for the multiplicative noise stochastic heat equation if both $N_t$ and $Q_t$ are (local) martingales for all test functions $\phi$. Any $u$ that satisfies this is a weak solution; see for instance BG97, Definition 4.10. Just as martingale problems are useful in proving convergence of Markov chains to SDEs, so too can they be used in SPDE convergence proofs; see Sections 6.2 and 6.3 for some examples where this type of martingale problem has been used for such a purpose.

It is generally hard to formulate a martingale problem characterization for weak solutions to singular nonlinear SPDEs. For the KPZ equation (in one spatial dimension) $H$, one can use the Hopf–Cole transform and define $H$ as a Hopf–Cole solution to the KPZ equation if $u=e^H$ is a solution to the multiplicative noise stochastic heat equation. This notion of solution agrees with those discussed earlier in this text. However, such linearizing transformations are uncommon, and this should be thought of as a rather useful trick, not a general theory.

Remarkably, for the stochastic Burgers equation (which is formally the equation satisfied by the spatial derivative of the KPZ equation $\partial _x H$)

$$\begin{equation} \partial _t u = \tfrac { 1}{2}\partial _x^2 u - \tfrac {1}{2} \partial _x (u^2) +\partial _x \xi , \tag{4.15}{} \end{equation}$$

GJ14 found a way to formulate a martingale problem characterization and GP17a (with a slightly improved formulation) matches the solution to this martingale problem to the Hopf–Cole solution (see 3.3) whereby showing uniqueness of the solution to this martingale problem; they call it an energy solution. There were some limitations of this notion of solution, namely it only works for particular types of initial data; very recently, however, GP18c has generalized the notion of energy solution to system configurations with finite entropy with respect to stationarity; and Yan18 has extended this method to include more general initial data such as flat initial data. It has proved to be quite useful in demonstrating convergence results, as we explain later in Section 6.4. Finally, let us mention that very recently GP18a developed a martingale approach for a class of singular stochastic PDEs of Burgers type, including fractional and multicomponent Burgers equations.

Let us end this discussion by mentioning (without any explanation) another powerful approach to defining weak solutions of SPDEs—Dirichlet forms. For example, for the $\Phi ^4$ equation in two spatial dimensions, before Da Prato and Debussche constructed their strong solution in DPD03, the paper AR91 constructed a weak solution via Dirichlet forms (which involves significant functional analysis). For comprehensive discussion on this topic we refer to the book FOT11 and the references therein.

5. Renormalization in physical models

Let us take stock of what we have learned so far. In Section 2 we observed that (at least in the linear case) SPDEs arise as scaling limits for microscopic models of physical systems. In Section 3 we introduced a number of nonlinear SPDEs and claimed that they model various interesting physical systems. However, before trying to justify that claim, we had to confront the challenge of well-posedness. Namely, how to make sense of what a “solution” means to these equations. Section 4 surveyed the main techniques for doing this.

In defining a solution, we mollified (or smoothed) the noise and defined the solution through a limit transition. From that perspective, it is reasonable to hope that the same methods can be applied to other types of regularizations of the noise, or equation—for instance to show that discrete systems converge to the SPDEs. We will address this further in Section 6.

In Section 4 we found that besides regularizing the noise, we also had to introduce certain renormalizations to our equations for them to admit limits. At first glance this tweaking of the equation seems a bit crooked. In this section we will explain how these renormalizations have concrete physical meaning, thus justifying our definitions. For instance, a diverging renormalization constant may relate to a tuning for a microscopic system of the overall scale, reference frame, temperature, or other physically meaningful parameters. We will focus our discussion on two systems: the dynamic $\Phi ^4$ equation and parabolic Anderson model. For the KPZ equation, we save this discussion until Section 6, where we will also highlight the notion of universality.

5.1. $\Phi ^4$ equation

Let us consider the example of a magnet near its critical temperature in Section 3. Many mathematical models have been proposed to describe various behaviors of magnetic systems. Here we investigate one particular example called the Kac–Ising model.⁠²⁵ The Ising model was introduced in 1920 by Lenz and named after his student Ising who showed that in one spatial dimension it did not admit any phase transition. That original model involves nearest-neighbor pair interactions of $-1$ and $+1$ spins. There are many generalizations of this model besides the long-range Kac–Ising model we consider here. For instance, one can consider the higher spin versions with $q$ different types of spins and interactions which award equal spins along edges. This is known as the $q$-state Potts model.^✖

We define the model in two dimensions. Denote by $\Lambda _{N}=\mathbf{Z}^d/(2N+1)\mathbf{Z}^d$ a large two-dimensional discrete torus of “radius” $N=\varepsilon ^{-1}$ (we introduce $\varepsilon$ here to later take to zero), which represents the space in which our magnetic material lives. Each site $k\in \Lambda _{N}$ is decorated with a spin $\sigma (k)$ which for simplicity is assumed to take values in $\{ -1, +1 \}$. Denote by $\Sigma _{N} = \{ -1, +1 \}^{\Lambda _{N}}$ the set of spin configurations on $\Lambda _{N}$. For a spin configuration $\sigma = (\sigma (k), \, k \in \Lambda _N )\in \Sigma _{N}$, we define the Hamiltonian as

$$\begin{equation} H(\sigma ) \mathrel {≔}- \frac 12 \sum _{k, j \in \Lambda _{N}} \kappa _{\gamma }(k-j) \sigma (j) \sigma (k) \;, \tag{5.1}{} \end{equation}$$

where, for $\gamma \in (0,1$), $\kappa _{\gamma }(k)$ is a nonnegative function⁠²⁶ A concrete choice of the interaction kernel $\kappa _\gamma$ is to set $\kappa _\gamma (k) = c_\gamma \, \gamma ^d \, \mathfrak{R}(\gamma k)$, where $\mathfrak{R}\colon \mathbf{R}^d \to \mathbf{R}$ is a smooth, nonnegative function with compact support and $c_\gamma$ is chosen to ensure that $\sum _{k \in \Lambda _N} \kappa _\gamma (k)=1$.^✖ supported on $|k|\le \gamma ^{-1}$ which integrates to $1$. Then for any inverse temperature $\beta >0$ we can define the Gibbs measure $\lambda _{\gamma }$ on $\sigma \in \Sigma _{N}$ as

$$\begin{equation*} \lambda _{\gamma } (\sigma ) \mathrel {≔}\frac {1}{Z(\beta )}\exp \Big ( - \beta H(\sigma ) \Big )\; , \end{equation*}$$

where $Z(\beta ) \mathrel {≔}\sum _{\sigma \in \Sigma _{N}} \exp \big ( - \beta H(\sigma ) \big )$ makes $\lambda _{\gamma }$ into a probability measure.

The measure $\lambda _{\gamma }$ is known as an equilibrium measure since the probability of finding a configuration $\sigma$ is proportional to the exponential of the energy of that configuration. It is also known as equilibrium because it arises as the equilibrium (or stationary, or invariant) measure for various simple, local stochastic dynamics on the configuration space. We will consider one such example known as Glauber dynamics Gla63. For $j\in \Lambda _N$, let $\sigma ^j \in \Sigma _N$ denote the spin configuration that coincides with $\sigma$ except the spin at position $j$ is flipped: $\sigma ^j(j) = -\sigma (j)$. The Glauber dynamic is the following continuous time Markov process: For each $j\in \Lambda _N$, the configuration $\sigma$ is updated to $\sigma ^j$ at rate⁠²⁷ For those not used to continuous time Markov processes, let us explain this more precisely. Starting from some configuration $\sigma$, to each $j\in \Lambda _N$ we associate independent exponentially distributed random variables of rate $c_{\gamma }(\sigma ,j)$ (i.e., with mean inverse to the rate). One then compares all of these random variables, and for the $j$ whose random variable is minimal, the configuration updates to $\sigma ^j$. The time at which this occurs is the value of the associated random variable. From that point on, one repeats the whole story, choosing new exponential random variables with rates given by the updated $\sigma ^j$. Due to the “memoryless” property of exponential random variables (i.e., for $X$ exponential of rate 1, conditional on $X>x$, the law of $X-x$ is that of a rate 1 exponential random variable), this constructs a continuous time Markov process.^✖ $c_\gamma (\sigma , j)$ where

$$\begin{equation} c_\gamma (\sigma , j) \mathrel {≔}\frac {\lambda _{\gamma }(\sigma ^j)}{\lambda _{\gamma }(\sigma ) + \lambda _{\gamma }(\sigma ^j)}\;. \tag{5.2}{} \end{equation}$$

Once an update occurs for some $j$, all rates are recalculated relative to the new configuration. It is standard to show that the measure $\lambda _{\gamma }$ is the unique invariant measure for the Glauber dynamic, meaning that for any starting measuring, eventually the measure will converge in distribution to $\lambda _{\gamma }$. Likewise, if started according to $\lambda _{\gamma }$, then the distribution at any later time will still be distributed according to that measure.

Figure 5.1 illustrates the dynamics where, for each fixed time $t$, one has a spin configuration $\sigma (t)$. We would like to observe a scaling limit of the system from a large distance scale of $\varepsilon ^{-1}$ and a long time scale of $\alpha ^{-1}$. At larger scales, the $\sigma$ oscillating between $\pm 1$ would yield a field in a very weak topology; it will be more convenient to consider an averaged field⁠²⁸ Using the interaction kernel $\kappa _\gamma$ to average out the field $\sigma$ is merely a matter of convenience, and it will lead to a clean form of 5.5. Also, convergence of $\sigma$ follows a fortiori.^✖

$$\begin{equation} h_\gamma (\sigma (t),k)\mathrel {≔}\sum _{\ell \in \Lambda _N} \kappa _\gamma (k - \ell ) \sigma (t,\ell ). \tag{5.3}{} \end{equation}$$

We may also abuse notation and write $h_\gamma (t,k) = h_\gamma (\sigma (t),k)$, suppressing the explicit dependence on $\sigma (t)$.

Remark 5.1.

In order to prove an SPDE limit, we first write down a discretized SPDE. Generally, this involves a coupled system of stochastic differential equations driven by martingales (recall the brief discussion from Section 4.3). Without delving deeply into details of the theory of Markov processes, let us illustrate this with a simple example. Our applications of this general idea will be more involved, though we will avoid going into details there also. Consider a continuous time random walk $x(t)$ on $\mathbf{Z}$ where a left jump (by 1) occurs at rate $\ell$ and a right jump (by 1) occurs at rate $r$. For any function $f:\mathbf{Z}\to \mathbf{R}$, the expected value $u(t,y)= \mathbf{E}_y\big [f\big (x(t)\big )\big ]$ (where $\mathbf{E}_y$ means the expected value assuming initial data $x(0)=y$) satisfies the system of ODEs $\tfrac {\partial }{\partial t} u(t,y) = Lu(t,y)$, where the operator $L$ acts on functions $g(y)$ as $\big (Lg\big )(y) \mathrel {≔}\ell \big (g(y-1)-g(y)\big ) + r \big (g(y+1)-g(y)\big )$. Without taking the expectations,