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.