Nagasawa’s paired-diffusion construction
Start by watching what changes when a Brownian walk must reach an exact final position. Then explore how two positive heat factors define an evolving density and its associated drift.
Scope. These are one-dimensional Brownian examples, with exact endpoint pinning followed by Gaussian endpoint weights. They illustrate the diffusion side of Nagasawa’s paired-factor construction; this page is not a derivation or numerical solution of the paper’s Wheeler–DeWitt equation. The factors remain real. The displayed log ratio is a real field; identifying it with a Schrödinger phase requires the convention and constants of the chosen formulation.
Pin a Brownian walk to a final position
Both panels start every particle at the same position. On the left, particles are free to finish anywhere. On the right, every particle must reach the chosen destination at time \(T\): this conditioned process is a Brownian bridge. Play the paths and watch the spread first grow, then shrink as the deadline approaches. Move the destination to see how the endpoint changes the whole path.
Loading the pinned Brownian walk…
The shaded bands contain 95% of the probability at each time, rather than 95% of complete paths. For ordinary Brownian motion the standard deviation is \(\sqrt{2Dt}\). For the pinned walk it is \(\sqrt{2Dt(1-t/T)}\), which vanishes at both endpoints. The dashed lines show the expected position, not an individual particle’s trajectory.
At a current position \(x\), the pinned process has the forward drift
\[ b_{\mathrm{pin}}(t,x)=\frac{x_T-x}{T-t},\qquad t<T. \]
This is the expected velocity toward the destination: a larger gap or a shorter remaining time produces a larger drift. Random motion continues throughout the interval. At \(T\) the position is fixed at \(x_T\); the drift expression is not evaluated there.
An ordinary path is \(X_t=x_0+\sqrt{2D}\,W_t\). Using the same Brownian sample, the corresponding pinned path is
\[ B_t=x_0+(x_T-x_0)\frac{t}{T} +\sqrt{2D}\left(W_t-\frac{t}{T}W_T\right). \]
This gives the exact Brownian-bridge distribution at the sampled times. The endpoint condition changes the whole trajectory, rather than moving only its last point. The two panels use this coupling to compare ordinary and pinned paths. Changing the destination keeps the same random samples; New paths draws a fresh set. Paths are sampled over the whole interval, then revealed by playback; lines between sample times are visual interpolation.
Gaussian endpoint weights
The next example allows a range of starting and ending positions through Gaussian weights. Its terminal factor favors an endpoint region rather than pinning every particle to one exact location.
Set the horizon \(T\), diffusion strength \(D\), and constant baseline drift \(a\), then play the construction or move directly through normalized time. The factors are separate boundary inputs, but their product and the drift define one process. In this second widget, the blue and amber path sections show opposite views of the same sampled trajectories.
Loading the interactive bridge…
Equations for Gaussian endpoint weights
The reference process is \(dX_t=a\,dt+\sqrt{2D}\,dW_t\), with generator \(L=a\,\partial_x+D\,\partial_{xx}\). Its forward factor \(\hat\phi\) and backward factor \(\phi\) obey
\[ \partial_t\hat\phi=L^*\hat\phi =D\,\partial_{xx}\hat\phi-a\,\partial_x\hat\phi, \qquad \partial_t\phi=-L\phi =-D\,\partial_{xx}\phi-a\,\partial_x\phi. \]
The widget evolves Gaussian endpoint weights exactly: \(\hat\phi(0,x)=\mathcal N(m_0,\sigma_0^2)\) and \(\phi(T,x)=\mathcal N(m_T,\sigma_T^2)\). At time \(t\), their centers are \(m_0+at\) and \(m_T-a(T-t)\), and their variances are
\[ q_0(t)=\sigma_0^2+2Dt,\qquad q_T(t)=\sigma_T^2+2D(T-t),\qquad \rho_t(x)=\frac{\hat\phi(t,x)\phi(t,x)} {\int_{\mathbb R}\hat\phi(t,y)\phi(t,y)\,dy}. \]
The forward process shown in the path panel uses the h-transform drift
\[ b(t,x)=a+2D\,\partial_x\log\phi(t,x) =a-\frac{2D\,[x-(m_T-a(T-t))]}{q_T(t)}. \]
Thus the \(a\) term comes from the baseline drift already present in \(L\); the h-transform adds the gradient term. With a fixed nonzero \(a\), reflection \(x\mapsto -x\) also sends \(a\mapsto -a\) and reflects both factor centers. Paths use exact Gaussian transitions on a display grid, not an Euler time-step scheme. Their blue section runs forward from \(0\) to \(t\); the amber dashed section retraces those same paths backward from \(T\) to \(t\).
The time field has no preset upper limit: any positive finite \(T\) that the browser’s numeric representation can handle is accepted. Rendering cost and floating-point precision still apply, as they do to any browser calculation.