## Limit theorems for large deviations and reaction-diffusion equations.(English)Zbl 0576.60070

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.
Reviewer: J.Biggins

### MSC:

 60J60 Diffusion processes 35K55 Nonlinear parabolic equations 60F10 Large deviations

### Keywords:

wave fronts; reaction-diffusion; Feynman-Kac formula
Full Text: