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For a reductive group $G$ over a non-Archimedean local field (e.g $GL_n(\mathbb{Q}_p )$ ), Jacquet's Whittaker function is essentially proportional toa character of an irreducible representation of the Langlands dual group$G^\vee( \mathbb{C} )$ ( a Schur function if $G = GL_n( \mathbb{Q}_p )$). Wepropose a probabilistic approach to this claim, known as theShintani-Casselman-Shalika formula, when the group $G$ has at least oneminuscule cocharacter in the coweight lattice.  Our presentation goes along the following lines. Thanks to a minuscule randomwalk $W^{(z)}$ on the coweight lattice and a related random walk on the Borelsubgroup, we establish a Poisson kernel formula for the non-ArchimedeanWhittaker function. The expression and its ingredients are similar to the onepreviously obtained by the author in the Archimedean case. A simplemanipulation reduces the problem to evaluating the probability of $W^{(z)}$never exiting the Weyl chamber. Then, an implementation of the reflectionprinciple forces the appearance of the Weyl character formula and thereforeretrieves characters of $G^\vee\left( \mathbb{C} \right)$.  The construction of the random walk on the Borel subgroup requires some care.It is extracted from a spherical random walk whose increments have adistribution that can be understood as elements from the spherical Heckealgebra.

Non-Archimedean Whittaker functions as characters: a probabilistic approach to the Shintani-Casselman-Shalika formula