Space-time white noise is too irregular to be evaluated pointwise: it is a Gaussian random distribution indexed by square-integrable test functions. The stochastic heat equation becomes meaningful after the heat semigroup smooths this noise. Whether the smoothing is sufficient is an explicit integrability question, and for spatially white noise it singles out one spatial dimension for function-valued random-field solutions.
White noise as an isonormal process
Definition. Let \[ H=L^2(\mathbb R_+\times\mathbb R^d). \] Space-time white noise is an isonormal Gaussian process \[ W=\{W(\phi):\phi\in H\} \] such that \(W(\phi)\) is centered Gaussian and \[ \mathbb E[W(\phi)W(\psi)] = \langle\phi,\psi\rangle_H = \int_0^\infty\int_{\mathbb R^d} \phi(t,x)\psi(t,x)\,\mathrm{d} x\,\mathrm{d} t. \]
Proposition. If \(A_1,\ldots,A_n\) are pairwise disjoint measurable space-time sets of finite Lebesgue measure, then \[ W(A_j):=W(1_{A_j}) \] are independent and \(W(A_j)\sim N(0,|A_j|)\).
Proof. The vector is jointly Gaussian, and \[ \operatorname{Cov}(W(A_i),W(A_j)) =|A_i\cap A_j|=0 \] for \(i\ne j\). Orthogonality implies independence for Gaussian vectors.
The formal symbol \(\dot W(t,x)\) denotes the distributional derivative of this random measure. There is no random function with independent finite-variance values at every space-time point.
Walsh stochastic integration
Theorem (Walsh isometry). For predictable simple space-time processes \(H\), define the integral by summing coefficients times white-noise increments over predictable rectangles. The definition extends uniquely to predictable \(H\) satisfying \[ \mathbb E\int_0^t\int_{\mathbb R^d} |H(s,y)|^2\,\mathrm{d} y\,\mathrm{d} s<\infty, \] and \[ \mathbb E\left| \int_0^t\int_{\mathbb R^d} H(s,y)W(\mathrm{d} s,\mathrm{d} y) \right|^2 = \mathbb E\int_0^t\int_{\mathbb R^d} |H(s,y)|^2\,\mathrm{d} y\,\mathrm{d} s. \]
Proof. For simple integrands, expand the square. Cross terms from disjoint future rectangles vanish by conditional centering and independent white-noise increments; diagonal terms equal coefficient squares times rectangle volume. This is an isometry from simple predictable functions into \(L^2(\Omega)\). Density and Hilbert-space completion define the general integral exactly as in Brownian stochastic integration.
The linear stochastic convolution
Theorem. For \[ \partial_tu=\frac12\Delta u+\dot W,\qquad u(0,\cdot)=u_0, \] the mild candidate is \[ u(t,x) = (P_tu_0)(x) +\int_0^t\int_{\mathbb R^d} p_{t-s}(x-y)W(\mathrm{d} s,\mathrm{d} y). \] The stochastic integral is a finite-variance random variable at each \((t,x)\) if and only if \(d=1\).
Proof. The Walsh isometry requires \[ \int_0^t\int_{\mathbb R^d} p_{t-s}(x-y)^2\,\mathrm{d} y\,\mathrm{d} s<\infty. \] Translation invariance and a Gaussian integral give \[ \|p_r\|_{L^2(\mathbb R^d)}^2=(4\pi r)^{-d/2}. \] Therefore the variance is \[ \int_0^t(4\pi r)^{-d/2}\,\mathrm{d} r, \] finite near zero exactly when \(d<2\), hence for integer dimensions only when \(d=1\).
Remark. In \(d\ge2\), the equation can still have distribution-valued solutions. What fails is a pointwise random-field solution with finite second moment. This distinction is analogous to white noise itself being a distribution rather than a function.
The semilinear stochastic heat equation
Theorem. In \(d=1\), consider \[ \partial_tu=\frac12\Delta u+b(u)+\sigma(u)\dot W \] with bounded continuous initial data and globally Lipschitz \(b,\sigma\). Then there is a unique predictable mild solution \[ \begin{aligned} u(t,x)=&(P_tu_0)(x) +\int_0^tP_{t-s}\{b(u(s,\cdot))\}(x)\,\mathrm{d} s\\ &+\int_0^t\int_{\mathbb R} p_{t-s}(x-y)\sigma(u(s,y)) W(\mathrm{d} s,\mathrm{d} y), \end{aligned} \] with bounded second moments on compact time intervals.
Proof outline. Define Picard iterates by inserting the preceding iterate into the drift and noise terms. Lipschitz continuity, Jensen, and the Walsh isometry give \[ \sup_x\mathbb E|u^{n+1}(t,x)-u^n(t,x)|^2 \le C\int_0^t \{1+(t-s)^{-1/2}\} \sup_y\mathbb E|u^n(s,y)-u^{n-1}(s,y)|^2\,\mathrm{d} s. \] The kernel is locally integrable. Iterated convolution or an exponentially weighted norm makes the map contractive, giving existence. The same estimate for two solutions gives uniqueness. Linear-growth estimates provide uniform second-moment bounds.
Spatial and temporal regularity
Proposition. For the one-dimensional additive equation with regular initial data, the stochastic convolution has a modification locally H?lder continuous in space of every order \(\alpha<1/2\) and in time of every order \(\gamma<1/4\).
Proof outline. The Walsh isometry and heat-kernel Fourier transform give \[ \mathbb E|u(t,x)-u(t,y)|^2\le C|x-y|, \] and \[ \mathbb E|u(t,x)-u(s,x)|^2\le C|t-s|^{1/2}. \] Gaussian higher moments are powers of the second moment. Kolmogorov-Chentsov then yields every spatial exponent below \(1/2\) and temporal exponent below \(1/4\).
Spatially colored noise
Theorem (Dalang condition). Suppose the noise is white in time and has spatial spectral measure \(\mu\). A function-valued stochastic heat convolution exists under the condition \[ \int_{\mathbb R^d} \frac{\mu(\mathrm{d}\xi)}{1+\|\xi\|^2}<\infty. \] For spatial white noise, \(\mu\) is Lebesgue measure, and the condition again reduces to \(d=1\).
Proof. Plancherel and \(\widehat p_t(\xi)=e^{-t\|\xi\|^2/2}\) turn the stochastic-convolution variance into \[ \int_{\mathbb R^d} \int_0^t e^{-s\|\xi\|^2}\,\mathrm{d} s\,\mu(\mathrm{d}\xi) = \int_{\mathbb R^d} \frac{1-e^{-t\|\xi\|^2}}{\|\xi\|^2}\,\mu(\mathrm{d}\xi), \] with the value \(t\) at \(\xi=0\). This is finite exactly under the displayed integrability condition up to equivalent behavior at zero and infinity.
The stochastic heat equation is thus not merely a deterministic PDE with a random right side. Its meaning depends on a stochastic integral, and existence is governed by a quantitative balance between noise roughness and semigroup smoothing.