Non-local Boundary Value Problems, stochastic resetting and Brownian motions on graphs

TL;DR

The paper develops a rigorous framework for non-local boundary value problems (NLBVPs) on networks by unifying Caputo-Džrbašjan time derivatives $\mathfrak{D}^\Psi_t$ and Marchaud spatial derivatives $\mathbf{D}^\Phi_{x-}$ within Brownian motion on graphs. The core contribution is a probabilistic representation of NLBVP solutions on half-lines and star graphs, realized through time-changed reflecting Brownian motion $B^\bullet = B^+ + \mathsf{L}_{\gamma_t}$ and the time-change $S_t$, yielding explicit boundary dynamics and jump/reset structures. The paper proves two main results: a fractional-dynamics heat equation on the half-line (Theorem $\text{['fracdyntm']}$) and a corresponding solution on star graphs (Theorem $\text{['thm:Xpallino']}$), with detailed pathwise constructions, resolvent formulas, and vertex conditions. Applications span stochastic resetting, traffic-flow with delayed agents, and financial modeling, illustrating how non-local boundary phenomena capture memory, heavy-tailed waiting times, and structural breaks in networked systems. Overall, the work provides a versatile framework tying non-local boundary operators to stochastic processes on networks, enabling precise modeling of restart, delay, and jump phenomena in complex systems. $\upsilon(t,x)$ is represented as $\mathbb{E}_x[f(B^\bullet \circ S_t^{-1})]$, highlighting the central role of time-change and boundary-additive terms in shaping diffusion on graphs.

Abstract

We consider dynamic boundary conditions involving non-local operators. Our analysis includes a detailed description of such operators together with their relations with random times and random (additive) functionals. We provide some new characterizations for the boundary behaviour of the Brownian motion based on the interplay between non-local operators and boundary value problems. Our main focus is on Feller-Wentzell diffusions with jumps (resetting/restart). We first consider the instructive case of the real line, then we extend our results on star graphs with trapping points or repulsive vertices.

Figures (7)

• Figure 1: A possible path for $B^\bullet$. The case $\eta=0$. The process is pushed away from the boundary point, it never hits $x=0$ and it jumps randomly in $(0,\infty)$ according with $\mathsf{L}_{\gamma_t}$.
• Figure 2: A possible path for $B^+\circ \bar{V}^{-1}$. The case $\eta>0$ and $\Phi=Id$.
• Figure 3: A possible path for $B^\bullet \circ S^{-1}$. The case $\eta>0$. Since $\Psi \ne Id$ the random plateaus are given by $H^\Psi$ which is independent from $B^\bullet$.
• Figure 4: Two examples of independent Brownian motions with Poissonian resetting. The left plot shows the process on $\mathbb{R}$, while the right plot illustrates the process constrained to the positive half-line.
• Figure 5: A possible path for $B^\bullet_t$$0 \leq t \leq T$ (left) and $B^\bullet_{T-t}$ , $0\leq t \leq T$ reverse over time (right). From the path in Figure \ref{['Bpallino']} we obtain the paths which has been drown here.

Theorems & Definitions (27)

• Proposition 2.1
• proof
• Remark 1
• Theorem 3.1
• Remark 2
• Theorem 3.2
• proof
• Theorem 4.1
• proof
• Definition 4.2