Consider the operator $\mathcal{L}$ acting on the function $f:\{0,1\}\mapsto \mathbb{R}$ defined as following: $$\mathcal{L}f(x)=f(1-x)-f(x)$$ This is the infinitesimal generator of a continuous time random walk $X_t$ on the space state $\{0,1\}$ with jump rate equal to 1. We can define the semigroup $P_t$ associated with the operator $\mathcal{L}$ as $$P_t=\exp{\mathcal{L}t}$$ and this would have a nice interpretation with random walk, i.e. $$P_t f(x) = E_x[f(X_t)]=P(X_t=0|X_0=x)f(0)+P(X_t=1|X_0=x)f(1)$$ I would like to know if there is a probabilistic interpretation of the semigroup $P^{*}_{t}=\exp{-\mathcal{L}t}$. Is this just a time reversal?
I am quite puzzled by this since $f(0)P_t f(0) + f(1) P_t f(1) \rightarrow 0$ and $f(0)P^{*}_t f(0) + f(1) P^{*}_t f(1) \rightarrow \infty$.
Thanks in advance for your answers.
$-\mathcal{L}$ is not a generator of a Markov process, since it does not satisfy the maximum principle. Indeed, let $f(0) = 0$ and $f(1) = -1$. Then $x=0$ is a point of maximum, but $-\mathcal Lf(0) = 1>0$.