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Let $U,H$ be two separable Hilbert spaces and $T>0$. We consider an SDE which evolves in the Hilbert space $H$ of the form \begin{align} dX(t)=AX(t)dt+\widetilde{\mathscr L}B(X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $B:H\to H$ is a bounded and $\theta$-H\"older continuous function, for some suitable $\theta\in(0,1)$, and $\widetilde{\mathscr L}:H\to H$ and $G:U\to H$ are linear bounded operators. We prove that, under suitable assumptions on the coefficients, the weak mild solution to the equation depends on the initial datum in a Lipschitz way. This implies that pathwise uniqueness holds true. Here, the presence of the operator $\Lambda$ plays a crucial role. In particular the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler--Bernoulli Beam equation upto dimension $3$ even in the hyperbolic case., Comment: arXiv admin note: text overlap with arXiv:2308.05415

In a separable Hilbert space, we study supercontractivity and ultracontractivity properties for a transition semigroups associated with a stochastic partial differential equations. This is done in terms of exponential integrability of Lipschitz functions and some logarithmic Sobolev-type inequalities with respect to invariant measures. The abstract characterization results concerning the improving of summability can be applied to transition semigroups associated to a stochastic reaction-diffusion equations.

In an infinite dimensional separable Hilbert space $X$, we study the realizations of Ornstein-Uhlenbeck evolution operators $\pst$ in the spaces $L^p(X,\g_t)$, $\{\g_t\}_{t\in\R}$ being the unique evolution system of measures for $\pst$ in $\R$. We prove hyperconctractivity results, relying on suitable Log-Sobolev estimates. Among the examples we consider the transition evolution operator of a non autonomous stochastic parabolic PDE.

In this paper, we prove pathwise uniqueness for stochastic differential equations in infinite dimension. Under our assumptions, we are able to consider the stochastic heat equation up to dimension $3$, the stochastic damped wave equation in dimension $1$ and the stochastic Euler-Bernoulli damped beam equation up to dimension $3$. We do not require that the so-called {\it structure condition} holds true., Comment: Added a section about weak existence, which combined with pathwise uniqueness, implies strong existence

In this paper we study two notions of differentiability introduced by P. Cannarsa and G. Da Prato (see [28]) and L. Gross (see [56]) in both the framework of infinite dimensional analysis and the framework of Malliavin calculus.

We consider stochastic reaction-diffusion equations with colored noise and prove Schauder type estimates, which will depend on the color of the noise, for the stationary and evolution problems associated with the corresponding transition semigroup, defined on the Banach space of bounded and uniformly continuous functions.

We prove Schauder type estimates for solutions of stationary and evolution equations driven by weak generators of transition semigroups associated to a semilinear stochastic partial differential equations with values in a separable Hilbert space.

Let $\mathcal{X}$ be a real separable Hilbert space. Let $C$ be a linear, bounded and positive operator on $\mathcal{X}$ and let $A$ be the infinitesimal generator of a strongly continuous semigroup on $\mathcal{X}$. Let $\{W(t)\}_{t\geq 0}$ be a $\mathcal{X}$-valued cylindrical Wiener process on a filtered (normal) probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\geq 0},\mathbb{P})$. Let $F:D(F)\subseteq\mathcal{X}\rightarrow\mathcal{X}$ be a smooth enough function. Under suitable conditions on $A$, $C$ and $F$ the following semilinear stochastic partial differential equation \begin{gather*} \begin{cases} dX(t,x)=\big(AX(t,x)+F(X(t,x))\big)dt+ \sqrt{C}dW(t), & t>0;\\ X(0,x)=x\in \mathcal{X}, \end{cases} \end{gather*} has a unique generalized mild solution $\{X(t,x)\}_{t\geq 0}$. We consider the transition semigroup defined by \begin{align*} P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_b(\mathcal{X}),\ t\geq 0,\ x\in \mathcal{X}. \end{align*} If $\mathcal{O}$ is an open set of $\mathcal{X}$, we consider the stopped semigroup defined by \begin{equation*} P^{\mathcal{O}}(t)\varphi(x):=\mathbb{E}\left[\varphi(X(t,x))\mathbb{I}_{\{\omega\in\Omega\; :\;\tau_x(\omega)> t\}}\right],\quad \varphi\in B_b(\mathcal{O}),\; x\in\mathcal{O},\; t>0 \end{equation*} where $\tau_x$ is the stopping time defined by \begin{equation*} \tau_x=\inf\{ s> 0\; : \; X(s,x)\in \mathcal{O}^c \}. \end{equation*} We will study the infinitesimal generators of $P(t)$ and $P^{\mathcal{O}}(t)$ in $L^2(\mathcal{X},\nu)$ and $L^2(\mathcal{O},\nu)$ respectively, where $\nu$ is the unique invariant measure of $P(t)$. We will focus on investigating how these two semigroups are related to the operator formally defined by \begin{equation*} N\varphi(x):=\frac{1}{2}\mbox{Tr}[C\nabla^2\varphi(x)]+\langle Ax+F(x), \nabla\varphi(x) \rangle. \end{equation*}