Suppose $(X(t),Y(t))$ $t\in[0,1]$ is a bivariate Gaussian process. We can assume that each component is continuously differentiable, but not necessarily stationary, and that the covariance kernels of $X$, $Y$ and the correlation function between $X$ and $Y$ are non-degenerate in the sense that $C_X(t,t)>0$, $C_Y(t,t)>0$ and $-1 < \rho_{XY}(t,t) <1$ for all $t\in [0,1]$.

My question is: is $P( \{ (X(t),Y(t) ) = 0 \mbox{ for some $t \in [0,1]$ }\})=0$? If so is there a simple proof?

Under the assumption that each component process is continuously differentiable, the number of zeros of each process must be countable, and so intuitively this is true since at the points where either process is zero, the probability that each component process is equal should be zero (since their distribution/covariance is non degenerate). I cannot turn this into a rigorous proof though, and perhaps the conjecture is even incorrect!