Fitting parameters of a Fokker-Planck-like equation with constraint

TL;DR

The paper analyzes a constrained stationary Fokker-Planck–type equation $\gamma u - \mu \nabla\cdot(\nabla\Phi u) - \Delta u = \mu S$ with constraint $\int \delta u\,dx = \ell$, establishing existence of a parameter $\mu\ge0$ for any $\ell\ge0$ under a confining potential $\Phi$ with Hessian bounds and smooth, compact data. It develops a robust operator-theoretic framework for $L_\mu$ and $L_\mu^*$, proving kernel characterizations, positivity, and well-posedness in weighted spaces, together with a dual formulation that enables monotonicity analysis. In radial symmetry and under compatibility conditions on $\Phi$, $S$, and $\delta$ (convexity/monotonicity assumptions), the constraint functional $\mathscr F(\mu)=\int \delta u_\mu$ is increasing, yielding existence and uniqueness of solutions for every $\ell$; large-$\mu$ asymptotics and Laplace-type analyses show $\mathscr F(\mu) \sim \mu \delta(0)\langle S\rangle/\gamma$, while counterexamples highlight the necessity of the hypotheses. Numerical experiments corroborate the theory, demonstrating monotonicity under the stated hypotheses and illustrating potential breakdowns when the compatibility conditions fail, with implications for tumor-immune modeling.

Abstract

We analyse a Fokker-Planck like equation, driven by a scalar parameter in order to reach an integral constraint. We exhibit criteria guaranteeing existence-uniqueness of a solution. We also provide counter-examples. This problem is motivated by an application to the immune control of tumor growth.

Figures (5)

• Figure 1: Solutions $r\mapsto u_\mu(r)$ for 10 equidistant values of $\mu$ in $[0,10]$. As $\mu$ increases the solution takes larger value at the origin and presents a stiffer profile for transient radius
• Figure 2: $\mu\mapsto \mathscr F(\mu)$ for $\mu$ up to $10^7$.
• Figure 3: Profile of $\mu\mapsto \mathscr F(\mu)$ for $\mu$ up to $100$ (left), up to $500$ (middle), and snapshots of the corresponding solution profiles $r\mapsto u_\mu(r)$ (right)
• Figure 4: Profile of $r\mapsto \delta(r)$ (top-left), profile of $\mu\mapsto \mathscr F(\mu)$ for $\mu$ up to $10^4$ (bottom-left) and $15\cdot 10^4$ (bottom-right), the solution profiles $r\mapsto u_\mu(r)$ for several $\mu$ up to $10^4$ (top-right)
• Figure 5: Profile of $r\mapsto \delta(r)$ (left), profile of $\mu\mapsto \mathscr F(\mu)$ for $\mu$ up to $2500$ (bottom-left) (bottom-right), snapshot on the corresponding solution profiles $r\mapsto u_\mu(r)$ (top-right)

Theorems & Definitions (9)

• Theorem 1
• Proposition 3.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Corollary 4.4
• Theorem 2
• Lemma 4.5
• Theorem 3