| Abstract: |
| Walsh`s spider process is a diffusion on an infinite star-shaped graph $S_k$ consisting of a single vertex connected to $k$ half-lines. Away from the origin, it behaves as a one-dimensional Brownian motion along each ray, but exhibits a peculiar behavior at the origin. Roughly speaking, when the process reaches the central vertex, it instantaneously chooses an outgoing edge and starts a standard Brownian excursion on the $i$-th edge with probability $\alpha_i$.
We present two different approximation results for Walsh`s spider. First, we derive the process as the diffusive scaling limit of a deterministic transport model with random velocity exchanges. Deterministic motions on $k$ copies of $S_k$ are perturbed by two stochastic mechanisms: interactions at interfaces located at the graphs` centers, and random jumps between different copies of the same edge.
Second, we show that Walsh`s spider process can be approximated by a snapping-out Brownian motion with large permeability coefficients. This yields an interpretation of Walsh`s spider as a Brownian motion perturbed by the trace of a semi-permeable membrane located at the graph`s center. |
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