Existence of non-smooth processes satisfying only one smoothness assumption

The conjecture is false: its second smoothness assumption cannot hold for any evolutionary process.

Fix a time $t$, and write $N_t=\mu_t(I_t)\in(0,\infty)$. For every $h>0$, the relative fitness is defined by

$$U_{t,t+h}=\frac{W_{t,t+h}}{\mathbb E_t[W_{t,t+h}]}, \qquad \mathbb E_t[f]=\frac1{N_t}\int_{I_t}f\,d\mu_t.$$

Consequently $\mathbb E_t[U_{t,t+h}]=1$, or equivalently

$$\int_{I_t}U_{t,t+h}\,d\mu_t=N_t.$$

However, the relative-fitness-density assumption in Definition 8.1(2), equation (8.4), requires the existence of the uncentered limit

$$\Upsilon_t=L^2(\mu_t)\!\operatorname{-}\lim_{h\downarrow0}\frac{U_{t,t+h}}h.$$

Whenever $U_{t,t+h}\in L^2(\mu_t)$, Cauchy–Schwarz gives

$$\left\|\frac{U_{t,t+h}}h\right\|_{L^2(\mu_t)} \ge \frac{\left|\int_{I_t}U_{t,t+h}\,d\mu_t\right|}{h\sqrt{\mu_t(I_t)}} =\frac{\sqrt{N_t}}h\longrightarrow\infty.$$

If $L^2$ is instead normalized by $\mathbb E_t$, the corresponding lower bound is $1/h$, with the same conclusion. If $U_{t,t+h}\notin L^2$, the assumption already fails. Thus condition (2) is impossible for every admissible process. In particular, there cannot exist a process satisfying condition (2) without satisfying the other two conditions, contrary to Conjecture 8.3.

There is also an explicit counterexample to the preceding claim that any two conditions imply the third. On the one-point probability space take the identity transition kernel and the constant observable $X_t=0$. Then the expectation derivative and local-change density both exist and equal zero, so conditions (1) and (3) hold, whereas $U_{t,t+h}=1$ and condition (2) demands the nonexistent limit $\lim_{h\downarrow0}1/h$. The centered expression $(U_{t,t+h}-1)/h$ would avoid this obstruction, but it is not the density specified in equation (8.4).