Regularizing Properties of (Non-Gaussian) Transition Semigroups in Hilbert Spaces

Abstract

Let $\mathcal {X}$ be a separable Hilbert space with norm $\left \|{\cdot }\right \|$ and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on $\mathcal {X}$, let $F:\mathcal {X}\rightarrow \mathcal {X}$ be a (smooth enough) function and let W(t) be a $\mathcal {X}$-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator $A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X})\subseteq \mathcal {X}\rightarrow \mathcal {X}$. We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation

$$ \left\{\begin{array}{ll} dX(t,x)=\left( AX(t,x)+F(X(t,x))\right)dt+ Q^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. $$

and in its associated transition semigroup

$$ P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_{b}(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; $$

where $B_{b}(\mathcal {X})$ is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of $\mathcal {X}$. More precisely there exists K := K(F, T) > 0 such that for every $\varphi \in B_{b}(\mathcal {X})$, $x\in \mathcal {X}$, t ∈ (0, T] and $h\in Q^{\alpha }(\mathcal {X})$ it holds

$$ |P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|. $$

Appendix A: Proof of Theorem 8

The aim of this section is to look for pathwise uniqueness of for Eq. 2.3.

Proof of Theorem 8

We first prove uniqueness. Let X₁(t, x), X₂(t, x) be two mild solutions of Eq. 2.3. Recall that by definition a mild solution solves

$$ X(t,x)=e^{tA}x+{{\int}_{0}^{t}}e^{(t-s)A}{\varPhi}(s,X(s,x))ds+{{\int}_{0}^{t}}e^{(t-s)A}Q^{\alpha}dW(s). $$

Hence, we have

$$ \begin{array}{@{}rcl@{}} \mathbb{E}[\left\|{X_{1}(t,x)-X_{2}(t,x)}\right\|^{2}]&=&\mathbb{E}\left[{\left\|{{{\int}_{0}^{t}}e^{(t-s)A}\left( {\varPhi}(s,X_{1}(s,x))-{\varPhi}(s,X_{2}(s,x))\right)ds}\right\|^{2}}\right]\\ &\leq& t\mathbb{E}\left[{{{\int}_{0}^{t}}\left\|{e^{(t-s)A}({\varPhi}(s,X_{1}(s,x))-{\varPhi}(s,X_{2}(s,x)))}\right\|^{2}ds}\right]\\ &\leq& t\mathbb{E}\left[{{{\int}_{0}^{t}}\left\|{{\varPhi}(s,X_{1}(s,x))-{\varPhi}(s,X_{2}(s,x))}\right\|^{2}ds}\right]\\ &\leq& tL_{{\varPhi}}^{2}{{\int}_{0}^{t}}\mathbb{E}\left[{\left\|{X_{1}(v,x)-X_{2}(v,x)}\right\|^{2}}\right]ds. \end{array} $$

From the Gronwall inequality and the same arguments of the proof of [20, Theorem 7.5] uniqueness follows.

The proof of existence is based on the contraction mapping theorem. We define the Volterra operator

$$V(Y)(t):=e^{tA}x+{{\int}_{0}^{t}}e^{(t-s)A}{\varPhi}(s,Y(s))ds+{{\int}_{0}^{t}}e^{(t-s)A}Q^{\alpha}dW(s),$$

in the space $\mathcal {X}^{2}([0,T])$, and first of all we show that V maps $\mathcal {X}^{2}[0,T]$ into itself. Indeed, for any $Y\in \mathcal {X}^{2}[0,T]$ we have

$$ \begin{array}{@{}rcl@{}} \left\|{V(Y)}\right\|_{\mathcal{X}^{2}[0,T]}^{2}&\leq& 3\|e^{\cdot A}x\|_{\mathcal{X}^{2}[0,T]}^{2}+3\left\|{{\int}_{0}^{\cdot} e^{(\cdot-s)A}{\varPhi}(s,Y(s))ds}\right\|_{\mathcal{X}^{2}[0,T]}^{2}\\ &&+3\left\|{{\int}_{0}^{\cdot} e^{(\cdot -s)A}Q^{\alpha}dW(s)}\right\|_{\mathcal{X}^{2}[0,T]}^{2}. \end{array} $$

(A.1)

We recall that $\|e^{tA}x\|_{\mathcal {X}^{2}[0,T]}^{2}\leq \left \|{x}\right \|^{2}$, for t > 0. Let $y\in \mathcal {X}$ be such that Eq. 2.2 holds, then

$$ \begin{array}{@{}rcl@{}} &&\left\|{{\int}_{0}^{\cdot} e^{(\cdot-s)A}{\varPhi}(s,Y(s))ds}\right\|_{\mathcal{X}^{2}[0,T]}^{2}\\ &=&\underset{t\in[0,T]}{\sup}\mathbb{E}\left[\left\|{{{\int}_{0}^{t}}e^{(t-s)A}{\varPhi}(s,Y(s))ds}\right\|^{2}\right]\\ &\leq& T \mathbb{E}\left[{\int}_{0}^{T}\left\|{{\varPhi}(s,Y(s))}\right\|^{2}ds\right]\\ &\leq& 2T\mathbb{E}\left[{\int}_{0}^{T}\left\|{{\varPhi}(s,Y(s))-{\varPhi}(s,y)}\right\|^{2}ds\right]\\ &&+2T\mathbb{E}\left[{\int}_{0}^{T}\left\|{{\varPhi}(s,y)}\right\|^{2}ds\right]\\ &\leq& 2TL^{2}_{\phi}\mathbb{E}\left[{\int}_{0}^{T}\left\|{Y(s)-y}\right\|^{2}ds\right]+2T\mathbb{E}\left[{\int}_{0}^{T}\left\|{{\varPhi}(s,y)}\right\|^{2}ds\right]\\ &\leq& 4T^{2}L^{2}_{\phi}\left\|{Y}\right\|^{2}_{\mathcal{X}[0,T]^{2}}+4T^{2}L_{{\varPhi}}^{2}\left\|{y}\right\|^{2}+2T{{\int}_{0}^{T}}\left\|{{\varPhi}(s,y)}\right\|^{2}ds. \end{array} $$

Since $Y\in \mathcal {X}^{2}[0,T]$ and recalling that Eq. 2.2 holds,

$$\left\|{{\int}_{0}^{\cdot} e^{(\cdot -s)A}{\varPhi}(s,Y(s))ds}\right\|_{\mathcal{X}^{2}[0,T]}^{2}<+\infty.$$

By [20, Theorem 4.36] and Hypothesis 1, the third summand in Eq. A.1 is finite. In the same way as the proof of uniqueness, we have

$$\left\|{V(Y_{1})-V(Y_{2})}\right\|_{\mathcal{X}^{2}[0,T]}^{2}\leq T^{2}L^{2}_{\phi} \left\|{Y_{1}-Y_{2}}\right\|_{\mathcal{X}^{2}[0,T]}^{2}.$$

So the existence follows by the contraction mapping theorem (using similar arguments as the one used in Theorem 19) and same arguments of proof of [20, Theorem 7.2]. □