arXiv subjects

D. A. Bignamini

Publications and source records attributed to D. A. Bignamini.

3 recordsLinked to original sources

arXiv2021

Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces

We consider the stochastic differential equation $$ \left\{ \begin{array}{lc} dX(t)=[AX(t)+F(X(t))]dt+C^{1/2}dW(t), & t>0;\\ X(0)=x \in \mathcal{X}; \end{array}\right. $$ where $\mathcal{X}$ is a Hilbert space, $\{W(t)\}_{t\geq 0}$ is a $\mathcal{X}$-valued cylindrical Wiener process, $A, C$ are suitable operators on $\mathcal{X}$ and $F:{\rm Dom}\,(F)\subseteq \mathcal{X}\to \mathcal{X}$ is a smooth enough function. We establish a logarithmic Harnack inequality for the transition semigroup $\{P(t)\}_{t\geq 0}$ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are also shown.

math.PR↗

arXiv2021

On generators of transition semigroups associated to semilinear stochastic partial differential equations

Let $\mathcal{X}$ be a real separable Hilbert space. 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 $\{W(t)\}_{t\geq 0}$ be a $\mathcal{X}$-valued cylindrical Wiener process. For $α\in [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2α-1}:Q^{1-2α}(\mathcal{X})\subseteq\mathcal{X}\rightarrow\mathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation \begin{gather} \left\{\begin{array}{ll} dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ Q^αdW(t), & t>0;\\ X(0,x)=x\in \mathcal{X}, \end{array} \right. \end{gather} and in its associated transition semigroup \begin{align} P(t)φ(x):=E[φ(X(t,x))], \qquad φ\in B_b(\mathcal{X}),\ t\geq 0,\ x\in \mathcal{X}; \end{align} where $B_b(\mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $\mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(\mathcal{X},ν)$, where $ν$ is the unique invariant probability measure of \eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincaré inequalities and we study the maximal Sobolev regularity for the stationary equation \[λu-N_2 u=f,\qquad λ>0,\ f\in L^2(\mathcal{X},ν);\] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(\mathcal{X},ν)$.

math.PR↗

arXiv2021

Regularizing properties of (non-Gaussian) transition semigroups in Hilbert spaces

Let $\mathcal{X}$ be a separable Hilbert space with norm $\|\cdot\|$ 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 $α\in [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2α-1}:Q^{1-2α}(\mathcal{X})\subseteq \mathcal{X}\rightarrow \mathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation \begin{gather*} \left\{\begin{array}{ll} dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ Q^αdW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. \end{gather*} and in its associated transition semigroup \begin{align*} P(t)φ(x):=\mathbb{E}[φ(X(t,x))], \qquad φ\in B_b(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; \end{align*} 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 $φ\in B_b(\mathcal{X})$, $x\in \mathcal{X}$, $t\in(0,T]$ and $h\in Q^α(\mathcal{X})$ it holds \[|P(t)φ(x+h)-P(t)φ(x)|\leq Kt^{-1/2}\|Q^{-α}h\|.\]

math.AP↗