A support theorem for parabolic stochastic PDEs with nondegenerate Hölder diffusion coefficients

Abstract

AbstractIn this paper we work with parabolic SPDEs of the form $$\begin{aligned} \partial _t u(t,x)=\partial _x^2 u(t,x)+g(t,x,u)+\sigma (t,x,u)\dot{W}(t,x) \end{aligned}$$ ∂ t u ( t , x ) = ∂ x 2 u ( t , x ) + g ( t , x , u ) + σ ( t , x , u ) W ˙ ( t , x ) with Neumann boundary conditions, where $$x\in [0,1]$$ x ∈ [ 0 , 1 ] , $$\dot{W}(t,x)$$ W ˙ ( t , x ) is the space-time white noise on $$(t,x)\in [0,\infty )\times [0,1]$$ ( t , x ) ∈ [ 0 , ∞ ) × [ 0 , 1 ] , g is uniformly bounded, and the solution $$u\in \mathbb {R}$$ u ∈ R is real valued. The diffusion coefficient $$\sigma $$ σ is assumed to be uniformly elliptic but only Hölder continuous in u. Previously, support theorems for SPDEs have only been established assuming that $$\sigma $$ σ is Lipschitz continuous in u. We obtain new support theorems and small ball probabilities in this $$\sigma $$ σ Hölder continuous case via the recently established sharp two sided estimates of stochastic integrals.