Tau functions of KP solitons realized in Wiener space (Q2865926)

Consider the infinitely many non-linear differential equations with respect to \(u_j\), \(j=1, 2, \dots\) of infinitely many variables \(x_1, x_2, \dots\) obtained by Lax equations NEWLINE\[NEWLINE\frac{\partial L}{\partial x_k} + [L, (L^k)_+]=0,\qquad k=1, 2, \dots,NEWLINE\]NEWLINE in which the operator \(L= \partial + \sum_{j=1}^\infty u_j\partial^{-j}\). The family is called Kadomtsev-Petviashvili hierarchy since the KP equation which is a generalization of KdV equation to a two dimensional model, is reduced from these equations with \(k=2\) and \(k=3\). The functions \(u_j\) in the hierarchy are generated from a single tau function \(\tau\). The tau functions are characterized as a solution to a family of quadratic differential equations called Hirota equations. The authors first introduce Levy's stochastic area formula and present a generalization of it in which the second Levy formula plays a crucial role. Then they show that the generalized stochastic area formula is parametrized as a tau function of KP solitons and finally give a probabilistic interpretation of the reduction from KP to KdV solitons.