Time-invariance and spatial-invariance of a stochastic process

Question

Many stochastic processes have independent and stationary increments, i.e. let $(X_t)_{t\ge 0}$ be a stochastic process on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, then $X_t - X_s \overset{d}{\sim} X_{t-s}$ (stationary increments) and for every $n\in \mathbb{N}$ and $0=t_0 < t_1 < \dots < t_{n}$ it holds that $(X_{t_{k+1}}-X_{t_k})_{k=0,\dots,n-1}$ are independent random variables (independent increments).

Sometimes it is said that the process $(X_t)_{t\ge 0}$ is time-invariant and spatially-invariant or i also read time-homogeneous and spatially-homogeneous. Do these latter notions have a concrete definition and which property (stationary or independent increments) corresponds to which notion and why?

My guess is that stationary increments is related to the time-invariance, since it says that the distribution of the process does not depend on the actual point in time, but only on the distance of the increment which one considers.

Can anyone shed some light on this elementary issue? // So far and if I understand it right, time-homogeneous or temporally-homogeneous and spatially-homogeneous aims at the properties of the stochastic process as a Markov process.

Answer 1

• time-invariant/time-homogeneous: You are right, this property refers to $(X_t)_{t \geq 0}$ considered as a Markov process. A process is called a time-homogeneous Markov process if $$\mathbb{E}(f(X_t) \mid \mathcal{F}_s)= \mathbb{E}^{X_s}(f(X_{t-s}))$$ In order to prove this in the case of the Lévy process $(X_t)_{t \geq 0}$ you actually need the independence of the increments as well as the stationarity of the increments. The independence implies that $(X_t)_{t \geq 0}$ is a Markov process and the stationarity of the increments the time-homogeneity.
• spatially homogeneous: This means that $$\mathbb{E}^x(f(X_t)) = \mathbb{E}^0(f(x+X_t))$$ i.e. the Lévy process started at $x$ behaves like the Lévy process started at $0$ translated by $x$.

Actually, one can show that any spatially homogeneous time-homogenous Markov process (with cadlag sample paths) is a Lévy process.