Let $X_t$ be an $\mathbb{R}^d$-valued stochastic process on the probability space $( \Omega, \mathcal{F}, P)$. Given $x \in \mathbb{R}^d$, I am interested in finding a probability measure $P^{x}$ on $(\Omega, \mathcal{F} )$ such that $$ P^x ( X_t \in B ) = P ( X_t + x \in B), \quad B \in \mathcal{B}(\mathbb{R^d}). $$
- Does there there exist a probability measure $P^x$ satisfying the above-mentioned relation?
- If such a measure exists, is it unique? If it is not unique, what additional assumptions are needed to make it unique?
We can consider the distribution of $X_t +x$ on $( \mathbb{R}^d, \mathcal{B}(\mathbb{R^d}))$ given by the convolution $P_{ X_t } * \delta_x$, but what can we say about a corresponding probability measure on $( \Omega, \mathcal{F})$? Could Kolmogorov extension theorem be helpful?
Yes, such a measure exists, but it is not unique.
Existence. Consider the mapping $f\colon \Omega\to\Omega$ that sends the sample path $(\omega_t)$ to $(\omega_t-x)$. This mapping is measurable, so one can form the pushforward measure $f_{\star}P$, and verify (by definition of the pushforward) that $f_{\star}P(X_t\in B)=P(X_t+x\in B)$. Thus, we may set $P^x=f_{\star}P$.
Non-Uniqueness. For example, consider a Brownian motion $(B_t\colon t\in[0,1])$ and let $(X_t\colon t\in[0,1])$ be a fractional Brownian motion with Hurst parameter $H=\tfrac14$, so that for all $t\in[0,1]$, we have that $X_t$ is a Gaussian random variable with mean $0$ and variance $\sqrt{t}$. (Its covariance is equal to $$ \mathbb EX_tX_s=\frac{\sqrt{t}+\sqrt{s}-\sqrt{t-s}}{2}, $$ although that is not very important to this answer.) Then, the process $(B'_t\colon t\in[0,1])$ given by $B'_t=t^{1/4}X_t$ has the property that $B'_t$ is normally distributed with mean $0$ and variance $t$, just like $B_t$, whereas the processes $(B'_t\colon t\in[0,1])$ and $(B_t\colon t\in [0,1])$ have different distributions since the former has negatively correlated increments yet the latter has independent increments. This shows that even in the case $x=0$ of your question, the measure is not unique.