Stochastic wave equations and regularity

On (0,pi), write the wave equation with Dirichlet Laplacian as dU=Vdt and $dV=\Delta_DUdt+dW$, with zero initial displacement and velocity and unit cylindrical forcing in the velocity equation. In normalized sine coordinates, the kth displacement is the Brownian integral with response $\sin(k(t-r))/k$, while its velocity uses $\cos(k(t-r))$. These formulas define scalar coordinates first; a state-space solution requires summing their covariance contributions.

The deterministic wave group acts naturally on paired Sobolev spaces with one derivative of difference between displacement and velocity. It conserves an appropriate quadratic energy rather than regularizing arbitrary initial data as an analytic heat semigroup does. An inverse frequency in the forced displacement response is useful, but it is not exponential high-frequency damping. Nonlinear wave equations, boundary forcing and spatially correlated noise can require other spaces or methods; the present coordinate calculation is a controlled linear case.

The Itô isometry gives $\operatorname{Var}(U_k)=t/(2k^2)-\sin(2kt)/(4k^3)$ and $\operatorname{Var}(V_k)=t/2+\sin(2kt)/(4k)$. At any fixed positive time, displacement variance has order k inverse squared, whereas velocity variance approaches a nonzero constant. Thus the displacement is L²-valued in one dimension, but the velocity is not. A finite collection of coordinate velocities cannot reveal this infinite tail obstruction.

Using the spectral convention from the heat lesson, the paired state belongs in mean square to H_A^s times H_A^(s−1) exactly for s less than one-half on the interval. The expected ordinary energy in H0¹ times L² diverges for unit white forcing. On a smooth bounded d-dimensional Dirichlet domain, Weyl counting gives the analogous fixed-time threshold s less than one minus d/2. These are Sobolev trace calculations, not a claim about pointwise path regularity or a nonlinear random-field existence theorem in every dimension.

For one unit-forced mode, expected velocity squared plus k squared times displacement squared equals t; the sine corrections cancel. The physical energy with an extra factor one-half therefore has mean t/2. An undamped wave mode has no stationary covariance balancing this growth. Adding damping changes the stochastic dynamics and can permit equilibrium, but it is a new equation. A deterministic animation with bounded amplitude does not contradict energy growth under repeated random forcing.

Regularity conclusions also depend on where the noise enters. Spatial coloring reduces the covariance of high modes, while forcing the displacement equation instead changes the response operators. A numerical method that evolves only a few smooth modes will always show finite velocity energy at that cutoff. To claim convergence, examine covariance tails in the intended paired norm and control the time approximation separately. Oscillatory cancellation inside an ordinary integral cannot be assumed to cancel the nonnegative squared responses appearing in stochastic variance.