Polarity of almost all points for systems of nonlinear stochastic heat equations in the critical dimension

We study vector-valued solutions u(t,x)∈Rd to systems of nonlinear stochastic heat equations with multiplicative noise, ∂ ∂tu(t,x)=∂2 ∂x2u(t,x)+σ(u(t,x))W˙(t,x). Here, t≥0, x∈R and W˙(t,x) is an Rd-valued space–time white noise. say that a point z∈Rd polar if P{u(t,x)=z for some t>0 x∈R}=0. show that, in the critical dimension d=6, almost all points Rd are polar.