On the analyticity of stochastic flows (Q1282254)

Let \((C,W)\) be the \(d\)-dimensional Wiener space, where \(C\) denotes the space of \(R^d\)-valued continuous functions on \([0,\infty)\) starting at 0, \(W\) is the Wiener measure on \(C\). A stochastic differential equation on \(R^N\) is considered: \[ \begin{cases} dX(t)= \sum^d_{\alpha=1} V_\alpha \bigl(X(t)\bigr) dw^\alpha(t)+ V_0\bigl(X(t) \bigr)dt,\\ X(0)=x,\end{cases} \tag{1} \] where \(w\in C\), \(V_0,\dots, V_d\) are \(R^N\)-valued \(C^\infty\)-functions on \(R^N\) which and whose derivatives of all orders are bounded, \(dw^\alpha(t)\) denotes the Itô integral with respect to \(w^\alpha(t)\) under \(W\). It is shown that for real analytic \(V_\alpha\), \(\alpha=0,1, \dots,d\), there exists a \(C^{0,\omega} ([0, \infty) \times R^N;R^N)\)-valued Wiener functional \(\Phi\) on \(C\) such that \(X(t, x,w)\equiv \Phi(w) [t,x]\) solves (1). \((C^{0,\omega} ([0,\infty) \times R^N;R^N)\) denotes the space of continuous functions \(f:[0,\infty) \times R^N\to R^N\) such that \(f(t,*)\) is analytic for every \(t\in [0,\infty).)\) The authors consider the radius of convergence of the analytic function \(x\to X(t,x,w)\) and the problem of prolongation to an entire function on \(C^N\). There are studied two cases where the 1-dimensional flow \(x\to X(t,x,w)\) does not extend to an entire function on \(C^N\).