Stochastic partial differential equation

A stochastic partial differential equation describes an unknown random field through spatial differential operators and random forcing. A model form is $du=(\kappa \Delta u+F(u))dt+G(u)d{W}_t$. The meaning of the noise, boundary data and solution is part of the specification; writing a formal derivative of Brownian motion does not define an ordinary function-valued forcing term.^[ref]

A simple exact example avoids the complications of noise at every spatial point. On $0<x<1$ with zero boundary values, considerThe solution is $u(t,x)=Y_t\sin (\pi x)$, whereHere $\kappa >0$. Spatial diffusion damps a single mode while noise continually excites it. More spatially irregular forcing can yield solutions that are distributions rather than pointwise functions, so regularity must be proved for the chosen dimension and noise.

A stochastic convolution propagates random forcing through a linear evolution. Let $S(t)$ be a strongly continuous semigroup on a Hilbert space $H$, let $W$ be a Cylindrical Wiener process on a Hilbert space $U$, and let $Q:U\to H$ be a bounded linear operator. The expressionis an Itô integral whenever its integrand is square integrable in the Hilbert-Schmidt operator norm. In particular,This Itô isometry tests whether the accumulated forcing is actually $H$-valued. The factor $S(t-s)$ evolves the contribution arriving at time $s$ through the remaining time $t-s$. For $du=Audt+Qd{W}_t$ with deterministic initial value $u_0$, the mild solution is $u(t)=S(t)u_0+Z(t)$.^[ref]

For $S(t)=e^{-at}$, $Q=\sigma$ and real Brownian motion $B$, with $a>0$,This is the zero-initial-value Ornstein-Uhlenbeck process: damping limits the variance despite continued forcing. With $a=0$, the limit is ${\sigma }^{2}t$ instead. These formulas follow by integrating ${\sigma }^2{e}^{-2a(t-s)}$ in the Itô isometry.

The word convolution describes the dependence on the time difference $t-s$. It does not mean an ordinary integral against a differentiable noise function; Brownian motion requires a stochastic integral.

A mild solution of an evolution equation satisfies an integrated equation built from its linear evolution, rather than requiring all derivatives in the differential equation to exist pointwise. Let $A$ generate a strongly continuous strongly continuous semigroup $S(t)$, and consider $du=(Au+F(u))dt+Qd{W}_t$. Whenever the integrals are defined, its mild formulation isThe final term is a stochastic convolution. The linear evolution propagates both the initial condition and each later contribution for the remaining time. This is the infinite-dimensional counterpart of variation of constants.^[ref]

A deterministic heat equation already shows why the definition is useful. Take $A=\kappa {\partial }_x^2$ on $L^2(\mathbb{R})$, zero forcing, and the discontinuous initial function $u_0=1_{[0,1]}$. The mild solution is its convolution with the heat kernel:It is smooth for $t>0$ and approaches $u_0$ in $L^2$ as $t$ decreases to zero, even though the initial function lacks the derivatives appearing in the classical equation.

The word mild does not mean approximate. It specifies how the equation is interpreted. Existence, uniqueness and whether a mild solution is also a classical or distributional solution depend on the generator, initial data, nonlinear term and noise.

A stochastic heat equation combines diffusive smoothing with random forcing. On the real line, a basic additive model iswhere $\xi$ is space-time white noise, formally specified by $\mathbb{E}[\xi (t,x)\xi (s,y)]=\delta (t-s)\delta (x-y)$. The noise is a random distribution, so this expression is interpreted through integration, not ordinary pointwise values. With zero initial data the mild solution isHere $W$ denotes the Gaussian noise integrated over space and time, and $G_{\kappa }$ is the heat kernel.^[ref]

The Itô isometry turns a rough random field into a tractable calculation:Thus the standard deviation at a fixed location grows like $t^{1/4}$ from zero. Diffusion reduces the influence of each earlier disturbance while fresh noise continually arrives.

Dimension changes the answer. In $d$ spatial dimensions the kernel's squared integral is proportional to $s^{-d/2}$. Its integral near zero is finite only for $d<2$. Consequently this additive white-noise model has pointwise finite-variance values in one dimension, whereas higher-dimensional versions require a distribution-valued interpretation or spatially smoother noise. The formal differential expression alone does not settle the solution's regularity.^[ref]

A regularity structure is an algebraic framework for local expansions that can include singular distributions as well as smooth functions. Its formal data are $(A,T,G)$: a locally finite set $A\subset \mathbb{R}$ containing zero and bounded below; a graded vector space $T={⨁}_{\alpha \in A}{T}_{\alpha }$ of Banach spaces, with $T_0$ the constants; and a group $G$ of linear maps fixing $1$ such thatThus changing an expansion's base point modifies a symbol only through lower-homogeneity terms. Martin Hairer introduced this framework to treat classes of singular stochastic partial differential equations.^[ref]

Ordinary polynomials provide a complete elementary example. In one dimension take $A=\{0,1,2,\dots \}$, $T_n$ spanned by $X^n$, and realize $X^n$ around $x$ as $(z-x{)}^n$. The function $f(z)=z^2$ then has abstract local expansion $a_x=x^2+2xX+X^2$. To transfer the expansion from $y$ to $x$, substitute $X\mapsto X+x-y$. For example,The expression on the left is the expansion at $y=2$ transported to $x=1$. In particular, ${\Gamma }_{12}{X}^2-X^2=-2X+1$ has strictly lower degree, illustrating the defining triangular property.

A model of a regularity structure adds concrete distributions and analytic bounds to these abstract symbols. Writing their realizations as ${\Pi }_x$, compatibility requires ${\Pi }_x{\Gamma }_{xy}={\Pi }_y$ and ${\Gamma }_{xy}{\Gamma }_{yz}={\Gamma }_{xz}$. For singular equations the symbols extend beyond monomials; constructing a model, proving reconstruction and controlling corrections are additional steps. The triple alone does not make arbitrary products of distributions meaningful.^[ref]

A model of a regularity structure realizes the abstract symbols of a regularity structure $(A,T,G)$ as concrete distributions, consistently across expansion points. It comprises continuous linear realization maps ${\Pi }_x:T\to {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ and maps ${\Gamma }_{xy}\in G$ satisfyingThus ${\Gamma }_{xy}$ transports an expansion from $y$ to $x$ without changing the represented distribution.^[ref]

Analytic bounds connect each symbol's assigned order $\alpha$ to its actual small-scale size. For ordinary isotropic scaling, put ${\phi }_x^{\lambda }(z)={\lambda }^{-d}\phi ((z-x)/\lambda )$. For $\tau \in T_{\alpha }$ one requiresThese bounds are uniform on each compact set of base points, below each fixed upper order, for $0<\lambda \le 1$ and smooth test functions supported in the unit ball with $C^r$ norm at most one; choose an integer $r>\max (0,-\inf A)$. Other scaling choices change the rescaled test functions and distance. Negative $\alpha$ allows singular distributions, whose small-scale averages can grow as $\lambda$ shrinks.^[ref]

In the one-dimensional polynomial model of a regularity structure, $({\Pi }_x{X}^n)(z)=(z-x{)}^n$ and ${\Gamma }_{xy}X=X+x-y$. For $x=1,y=2$,A change of variable $z=x+\lambda w$ in the test function pairing gives exactly ${\lambda }^n\int w^n\phi (w)dw$, explaining the analytic bound for the monomial $X^n$. The realization maps supply information absent from the abstract triple alone.

The polynomial model of a regularity structure realizes abstract monomials as ordinary coordinate differences around each base point. It is the example that turns a regularity structure into familiar Taylor polynomials. In $d$ dimensions, use a polynomial ring in symbols $X_1,\dots ,X_d$, graded by total degree. For a multi-index $k=(k_1,\dots ,k_d)$,Re-expansion uses ${\Gamma }_{xy}p(X)=p(X+x-y)$. Substitution immediately gives ${\Pi }_x{\Gamma }_{xy}={\Pi }_y$ and ${\Gamma }_{xy}{\Gamma }_{yz}={\Gamma }_{xz}$, the compatibility identities of a model of a regularity structure.^[ref]

For example, $f(z_1,z_2)=z_1{z}_2$ has local abstract expansion $a_x=2+2X_1+X_2+X_1{X}_2$ at $x=(1,2)$. At $y=(3,4)$ it has $a_y=12+4X_1+3X_2+X_1{X}_2$. Transporting the latter substitutes $X_1-2,X_2-2$:The coefficients change because the expansion point changes; the represented function is still $z_1{z}_2$.

The analytic scaling also follows by calculation. For a smooth test function and ${\phi }_x^{\lambda }(z)={\lambda }^{-d}\phi ((z-x)/\lambda )$, changing variables givesThus the assigned degree predicts exactly the power of $\lambda$ in the model bound. Anisotropic scaling can assign different weights to different coordinates instead; ordinary total degree is the isotropic choice. General regularity structures retain this organization of local expansions while admitting symbols that are not polynomials.