Is there a random walk in the plane which is everywhere differentiable? I imagine like brownian motion but smoothed out on small distances
2026-04-05 14:15:38.1775398538
Differentiable random walk
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You can "smooth out" a brownian motion in 2D (but it could as well work in nD) $P_0P_1 \cdots P_{n}$ represented in blue by taking a cubic spline approximation as shown on the following graphics where the different (Bezier) arcs making the spline curve are all colored in a slightly different way in order to understand how they have been made.
In this way one obtains a continuously differentiable curve
I don't describe the theory of cubic splines here. I let the reader consult one of the numerous sites where one can find information about them. It suffices to say that the $k$th cubic Bezier arc is defined by points $C_k, A_k, B_k, C_{k+1}$, where $A_k:=\tfrac23 P_k+\tfrac13 P_{k+1}$, $B_k:=\tfrac13 P_k+\tfrac23 P_{k+1}$, and $C_k:=\tfrac12 B_{k-1}+\tfrac12 A_{k}$ ; with particular cases for the initial and final Bezier curves.