Inferring diffusivity from killed diffusion

TL;DR

This work shows that diffusivity $D$ in an insulated domain can be consistently inferred from histograms of random binding events, even when only Poisson-count data are available. The key idea is that the binding-intensity density is governed by a stationary Schrödinger equation $\nabla\cdot(D\nabla u)-q u=-\phi$ with Neumann boundary conditions, yielding $\lambda(x)=n q(x)u_{D,q}(x)$ as the Poisson intensity. A Bayesian non-linear inverse problem is posed by placing a Gaussian-process prior on $D$ (via $D(x)=\tfrac14+\tfrac14 e^{W(x)}$) and pushing the prior through the PDE map to obtain the intensity $\Lambda_D$, enabling posterior computation with MCMC; consistency is proved in high dimensions under mild identifiability and regularity assumptions, with rates determined by the forward map and binning scheme. The analysis hinges on PDE regularity, spectral properties of Schrödinger operators, and novel concentration inequalities for high-dimensional Poisson data, and is complemented by numerical experiments illustrating posterior recovery of $D$ from synthetic data. The results provide a principled, scalable approach to diffusivity inference when trajectory data are unavailable or costly to obtain.

Abstract

We consider diffusion of independent molecules in an insulated Euclidean domain with unknown diffusivity parameter. At a random time and position, the molecules may bind and stop diffusing in dependence of a given `binding potential'. The binding process can be modeled by an additive random functional corresponding to the canonical construction of a `killed' diffusion Markov process. We study the problem of conducting inference on the infinite-dimensional diffusion parameter from a histogram plot of the `killing' positions of the process. We show first that these positions follow a Poisson point process whose intensity measure is determined by the solution of a certain Schrödinger equation. The inference problem can then be re-cast as a non-linear inverse problem for this PDE, which we show to be consistently solvable in a Bayesian way under natural conditions on the initial state of the diffusion, provided the binding potential is not too `aggressive'. In the course of our proofs we obtain novel posterior contraction rate results for high-dimensional Poisson count data that are of independent interest. A numerical illustration of the algorithm by standard MCMC methods is also provided.

Figures (4)

• Figure 1: Example of density reconstruction through MCMC, for $n=10^7$. The true diffusivity function is displayed on the left, and generates the random observation counts (centre) on each of the $K=36$ bins. The posterior mean computed via MCMC is diplayed on the right. More details about the implementation can be found in Sec. \ref{['subsec:numerics']}.
• Figure 2: Problem setting
• Figure 3: Posterior mean estimate $\bar{D}$ for the diffusivity function (left), along with the corresponding PDE solution $u_{\bar{D}, q}$(centre), and the true solution $u_{D_0,q}$ (right) for comparison.
• Figure 4: The second and third displays compare, respectively, the 'fit' of the histogram bin count $N(B_i)$ and the estimated posterior intensities $n\Lambda_{\bar{D}}(B_i)$ with the ground truth $n\Lambda_{D_0}(B_i)$. The numerically improved recovery also of the intensity $\Lambda_{D_0}$ by the Bayesian method when compared to the naive bin count is clearly visible in these images.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Proposition 3.1
• proof
• Proposition 3.2