The random multiplicative measures on $\mathbb{R}$ introduced in
Mandelbrot ([Mandelbrot 1996]) are a fundamental particular
case of a larger class we deal with in this paper. An element $\mu$
of this class is the vague limit of a continuous time
measure-valued martingale $\mu _{t}$, generated by multiplying
i.i.d. non-negative random weights, the $(W_M)_{M\in S}$, attached
to the points $M$ of a Poisson point process $S$, in the strip
$H=\{(x,y)\in \mathbb{R}\times\mathbb{R}_+ ; 0 < y\leq 1\}$ of the upper
half-plane.
We are interested in giving estimates for the dimension of such a
measure. Our results give these estimates almost surely for
uncountable families $(\mu ^{\lambda})_{\lambda \in U}$ of such
measures constructed simultaneously, when every measure $\mu^{\lambda}$
is obtained from a family of random weights
$(W_M(\lambda))_{M\in S}$ and $W_M(\lambda)$ depends smoothly
upon the parameter $\lambda\in U\subset\mathbb{R}$.
This problem leads to study in several sense the convergence, for
every $s\geq 0$, of the functions valued martingale $Z^{(s)}_t:
\lambda \mapsto \mu_{t}^{\lambda }([0,s])$. The study includes the
case of analytic versions of $Z^{(s)}_t(\lambda)$ where
$\lambda\in\mathbb{C}^n$. The results make it possible to show in certain
cases that the dimension of $\mu^{\lambda}$ depends smoothly
upon the parameter.
When the Poisson point process is statistically invariant by
horizontal translations, this construction provides the new
non-decreasing multifractal processes with stationary increments
$s\mapsto \mu ([0,s])$ for which we derive limit theorems, with
uniform versions when $\mu$ depends on $\lambda$.