a solution of Eq. (\[spde-uniq\]) in the mild sense if the following two conditions are satisfied:

1.  $u \in \mathcal{B}^{\alpha,2}(0, T ; V)$ a.s.;

2.  $\{u_t\}$ is adapted to $\{\mathcal{F}_t\}$ and satisfies the following integral equation: $$\begin{aligned}
    \label{mild1}
    u_t=S_t u_{0}+\int_{0}^{t} S_{t-s} f(u_s) \mathrm{d}s+\int_{0}^{t} S_{t-s} \sigma(u_s) \mathrm{d} W_s+\int_{0}^{t} S_{t-s} g(u_s) \mathrm{d} B^H_s.\end{aligned}$$

The following Lemma \[itofbm2\] which will be proved in Appendix **B** provides the basic estimates needed to prove the pathwise unique solutions of Eq. (\[spde-uniq\]).

\[itofbm2\] Taking $\alpha<\alpha'<1-\beta$ and $\alpha<\beta$, there exists a constant $C>0$, such that for any $0\leq s<t \leq T$, $u,v\in V$, $$\begin{aligned}
    \mathcal{K}_1 (s, t) &:=&\Big|\int_{s}^{t} S_{t-r} g(u_r) \mathrm{d} B^H_r\Big|\cr
    &\leq& C \Lambda_{\alpha,B^H}^{0,t} \int_{s}^{t}\Big([(r-s)^{-\alpha}+(t-r)^{-\alpha}](1+|u_r|)+\int_{s}^{r}\frac{|u_r-u_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r,
    \cr
    \mathcal{K}_2(0,s) &:=&\Big|\int_{0}^{s} (S_{t-r}-S_{s-r})g(u_r) \mathrm{d} B^H_r\Big| \cr
    &\leq&   C\Lambda_{\alpha,B^H}^{0,t} (t-s)^\beta\int_{0}^s  [(s-r)^{-\beta} r^{-\alpha}+(s-r)^{-\alpha-\beta}](1+|u_r|)\mathrm{d}r\cr
    &&+C\Lambda_{\alpha,B^H}^{0,t} (t-s)^\beta\int_{0}^s  (s-r)^{-\beta}\Big(\int_{0}^{r}\frac{|u_r-u_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r,\cr
    \mathcal{K}_3(s,t) &:=&\Big|\int_{s}^{t} S_{t-r} (g(u_{r})-g(v_{r})) \mathrm{d} B^H_r\Big| \cr
    &\leq& C \Lambda_{\alpha,B^H}^{0,t} 
    \int_{s}^t[(r-s)^{-\alpha}+(t-r)^{-\alpha}]|u_r-v_r|
    \mathrm{d}r \cr
    &&+C  \Lambda_{\alpha,B^H}^{0,t}
    \int_{s}^t|u_r-v_r|\Big(\int_{s}^{r}\frac{|u_r-u_q|+|v_r-v_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r
    \cr
    &&+C  \Lambda_{\alpha,B^H}^{0,t}
\int_{s}^t\Big(\int_{s}^{r}\frac{|u_r-v_r-u_q+v_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r,
    \cr
    \mathcal{K}_4(0,s) &:=& \Big|\int_{0}^{s} (S_{t-r}-S_{s-r}) (g(u_{r})-g(v_{r})) \mathrm{d} B^H_r\Big|\cr
    &\leq& C  \Lambda_{\alpha,B^H}^{0,t} (t-s)^\beta \int_{0}^s [(s-r)^{-\beta} r^{-\alpha}+(s-r)^{-\alpha-\beta}]|u_r-v_r|\mathrm{d}r\cr
    &&+C   \Lambda_{\alpha,B^H}^{0,t} (t-s)^\beta \int_{0}^s (s-r)^{-\beta} |u_r-v_r| \Big(\int_{0}^{r}\frac{|u_r-u_q|+|v_r-v_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r\cr
    &&+C   \Lambda_{\alpha,B^H}^{0,t}  (t-s)^\beta \int_{0}^s (s-r)^{-\beta}\Big(\int_{0}^{r}\frac{|u_r-v_r-u_q+v_q|}{(r-q)^{1+\alpha}}\mathrm{d}q\Big)\mathrm{d}r.
    \end{aligned}$$

\[mixuniq\] Assume that $f,\sigma$ satisfy Lipschitz and linear growth conditions, and $g$ and $g^{\prime}$ satisfy (\[g1\]) and (\[g2\]), respectively. Then, for any initial value $u_{0} \in V_\beta,$ $\beta 