A family of diffusion fields on the plane is considered fulfilling the following stochastic integral equation: \[ x_{st}=x+\int^{s}_{0}\int^{t}_{0}a(u,v,x_{uv})\,du\,dv + \varepsilon \int^{s}_{0}\int^{t}_{0}b(u,v,x_{uv})\,dw_{uv},\quad x\in \mathbb R^ n,\; s,t\in [0,T]. \] The action functional is obtained in a similar form as for diffusions on the line where the second derivative \(\delta^ 2\phi /\partial s\partial t\) and integration on \([0,T]\times [0,T]\) replaces the corresponding \(d\phi/dt\) and integration on \([0,T]\).