Degenerate parabolic stochastic partial differential equations

M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4301
Proof. In order to prove this claim, we denote by σ i the ith row of σ . Let us fix test functions
ψ 1 ∈ C ∞ (T N ), ψ 2 ∈ Cc ∞(R) and define θ(ξ) = ξ
−∞ ψ 2(ζ ) dζ . We denote by ⟨·, ·⟩ x the duality
between the space of distributions over T N and C ∞ (T N ). It holds
σ i v
∇g, ψ 1 ψ 2 = − div(σ i ψ 1 ), ψ 2 (ξ) dξ = − div σ i
ψ 1 , θ(v)
x
If the following was true
= σ i ∇θ(v), ψ 1
x .
−∞
σ i ∇θ(v) = θ ′ (v) σ i ∇v in D ′ (T N ), (12)
we would obtain
σ i ∇g, ψ 1 ψ 2
=
θ ′ (v) σ i ∇v, ψ 1
x = σ i ∇v δ v=ξ , ψ 1 ψ 2
and the proof would be complete.
Hence it remains to verify (12). Towards this end, let us consider an approximation to the
identity on T N , denoted by (ϱ τ ). To be more precise, let ˜ϱ ∈ Cc ∞(RN ) be a nonnegative
symmetric function satisfying R N ˜ϱ = 1 and supp ˜ϱ ⊂ B(0, 1/2). This function can be easily
extended to become Z N -periodic, and let this modification be denoted by ¯ϱ. Now it is correct to
define ϱ = ¯ϱ ◦q −1 , where q denotes the quotient mapping q : R N → T N = R N /Z N , and finally
ϱ τ (x) = 1
τ N ϱ x
.
τ
Since the identity (12) is fulfilled by any sufficiently regular v, let us consider v τ , the
mollifications of v given by (ϱ τ ). We have
σ i ∇θ(v τ ) −→ σ i ∇θ(v) in D ′ (T N ).
In order to obtain convergence of the corresponding right hand sides, i.e.
θ ′ (v τ ) σ i ∇v τ −→ θ ′ (v) σ i ∇v in D ′ (T N ),
we employ similar arguments as in the commutation lemma of DiPerna and Lions (see
[10, Lemma II.1]). Namely, since σ i (∇v) ∈ L 2 (T N ) it is approximated in L 2 (T N ) by its
mollifications [σ i ∇v] τ . Consequently,
θ ′ (v τ ) σ i ∇v τ −→ θ ′ (v) σ i ∇v in D ′ (T N ).
Thus, it is enough to show that
θ ′ (v τ ) σ i ∇v τ − σ i ∇v τ
−→ 0 in D ′ (T N ). (13)
It holds
σ i (x)∇v τ (x) − σ i ∇v τ (x)
= v(y) σ i
(x)(∇ϱ τ )(x − y) dy + v(y)div y σ i (y)ϱ τ (x − y) dy
T N T
N
= − v(y) σ i (y) − σ i (x)
(∇ϱ τ )(x − y)dy + v(y)div σ i (y) ϱ τ (x − y)dy.
T N T N
x