Limit theorems for large deviations and reaction-diffusion equations
Consider the differential equation, for u(t,x) with \(t\geq 0\) and \(x\in R^ r\), \(u_ t=Lu+f(x,u)\) with the initial condition \(u(0,x)=g(x)\), where L is a differential operator governing a Markov diffusion process \(\{X_ t\}\). The Feynman-Kac formula gives \(u(t,x)=E_ xg(X_ t)\exp \{\int^{t}_{0}c(X_ s,u(t-s,X_ s))ds\}\) where \(uc(x,u)=f(x,u)\). Introducing the small parameter \(\epsilon\) with \(u^{\epsilon}(t,x)=u(t\epsilon,x\epsilon)\) yields \(u_ t^{\epsilon}=L^{\epsilon}u^{\epsilon}+\epsilon^{-1}f(x,u)\), but the initial condition is now taken as \(u^{\epsilon}(0,x)=g(x)\). To deduce the behaviour of \(u^{\epsilon}\) as \(\epsilon\) \(\downarrow 0\) the corresponding Feynman-Kac formula is used, and the evaluation of the asymptotics of this involves the large deviations of the Markov processes governed by \(L^{\epsilon}\) as \(\epsilon\) \(\downarrow 0\). In this way conditions for \(u^{\epsilon}(t,x)\) to converge to (essentially) a 0-1 step function are obtained and the evolution with t of the boundary between the two values is considered, in particular its advance can in some cases be represented using a velocity field. Numerous variations on these basic ideas are also discussed.
scientific article; zbMATH DE number 3921665