An Orsntein-Uhlenbeck (OU) process $U$ on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ is usually defined as solving the following SDE: \begin{align} \text{d}U_t&=\kappa(\mu-U_t)\text{d}t+\sigma\text{d}W_t, \quad\forall t>0 \\[3pt] U_0&=u\in\mathbb{R} \end{align} for some Brownian motion $W$ and parameters $\mu\in\mathbb{R}$ and $\kappa,\sigma\in\mathbb{R}_{\geq0}$.
Can we still define the OU process for the case $\kappa<0$?
wlog consider $\mu = 0$. If $\kappa< 0$ we can rewrite $\kappa=-\beta,\,\beta >0$. We get $dU_t=\beta U_t dt+\sigma dW_t$. Now if we solve the SDE: $$\begin{aligned}dU_t-\beta U_t dt&=\sigma dW_t\\ d(U_t e^{-\beta t})&=\sigma e^{-\beta t}dW_t\\ \implies U_t&=ue^{\beta t}+\sigma \int_0^te^{\beta(t-s)}dW_s \end{aligned}$$ We see that $U_t \sim \mathcal{N}(u e^{\beta t},\sigma^2(e^{2\beta t}-1)/(2\beta))$ so that $|E[U_t]|\to \infty$ (if $u\neq 0$), $V[U_t]\to \infty$ so it cannot be Ornstein-Uhlenbeck.