The authors considered the nonlinear Stratonovich stochastic differential equation in Banach space \[ dx^\varepsilon(t)= b(x^\varepsilon(t))\,dt+ \sigma(x^\varepsilon(t))\,\varepsilon\circ dw(t),\quad x^\varepsilon(0)= 0, \] where the coefficients \(b\) and \(\sigma\) takes values in \(Y\) and in the space of bounded linear operators from \(X\) to \(Y\) denoted by \(L(X, Y)\), respectively; \(X\) and \(Y\) are real separable infinite-dimensional Banach spaces; \(b\) and \(\sigma\) satisfy suitable regularity conditions; \(w(t)\), \(0\leq t\leq 1\) is the \(X\)-valued Wiener process; \(\varepsilon\) is a small parameter.
Using the Banach space-space valued rough path theory of T. Lyons the authors have discussed the Freidlin-Wentzell type large deviation principle for \(x^\varepsilon\) and the asymptotic behavior of the Laplace type functional integral \[ \lim_{\varepsilon\to 0}\,E\Biggl[\exp\Biggl\{-{F(x^\varepsilon)\over \varepsilon^2}\Biggr\}\Biggr]. \] An application of the obtained results is shown for two examples, namely for heat processes on loop spaces and for the solutions of stochastic differential equations on M-type 2 Banach spaces.
For the entire collection see [Zbl 1113.60006].