We consider a class of neutral stochastic partial differential equations with infinite delay in real separable Hilbert spaces. We derive the existence and uniqueness of mild solutions under some local Carathéodory-type conditions and also exponential stability in mean square of mild solutions as well as its sample paths. Some known results are generalized and improved.

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a$ and $b$ satisfying $a>-1,$ $-1<b<0,$ $\left|b\right|<1+a$. In this paper, motivated by the asymptotic property $$E[{(B_{s+\epsilon }^{a,b}-B_s^{a,b})}^2]=O({\epsilon }^{1+b})\nsim {\epsilon }^{1+a+b}=E[{(B_{\epsilon }^{a,b})}^2],\epsilon \to 0$$ for all $s>0$, we consider the generalized quadratic covariation ${\left[f(B^{a,b}),B^{a,b}\right]}^{(a,b)}$ defined by $${[f(B^{a,b}),B^{a,b}]}_t^{(a,b)}=\underset{\epsilon \downarrow 0}{lim}\frac{1+a+b}{{\epsilon }^{1+b}}{\int }_{\epsilon }^{t+\epsilon }\{f(B_{s+\epsilon }^{a,b})-f(B_s^{a,b})\}(B_{s+\epsilon }^{a,b}-B_s^{a,b})s^{b}ds,$$ provided the limit exists uniformly in probability. We construct a Banach space $\mathcal{\mathscr{H}}$ of measurable functions such that the generalized quadratic covariation exists in $L^2(\mathrm{\Omega })$ and the generalized Bouleau–Yor identity $${[f(B^{a,b}),B^{a,b}]}_t^{(a,b)}=-\frac{1}{(1+b)\mathrm{????}(a+1,b+1)}{\int }_{ℝ}f(x){\mathcal{ℒ}}^{a,b}(dx,t)$$ holds for all $f\in \mathcal{\mathscr{H}}$, where ${\mathcal{ℒ}}^{a,b}(x,t)={\int }_0^t\delta (B_s^{a,b}-x)d{s}^{1+a+b}$ is the weighted local time of $B^{a,b}$ and $\mathrm{????}(\cdot ,\cdot )$ is the Beta function.

Let Q ∈ C((0, 1]^{1 + m}) and let {ξ_{n, j}: 1 ⩽ j ⩽ n, n ⩾ 1} be a triangular array of independent zero-mean random variables. In this paper, we show that for every integer m ⩾ 1, the processes X^m_n(Q) = {X_n^m(t, Q), 0 ⩽ t ⩽ 1}, n ⩾ 1 defined by converges weakly in C[0, 1], as n → ∞, to a multiple Wiener integral under some suitable conditions on the function Q and array {ξ_{n, j}}. This convergence includes the approximating characterization of many popular self-similar processes such as fractional Brownian motion, sub-fractional Brownian motion, Rosenblatt process, and more general Hermite processes.