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## Section: New Results

### Tempered multistable measures and processes

Participants : Jacques LÃ©vy VÃ©hel, Lining Liu.

This year, we concentrated on the following points:

• Define a new type of multistable processes called tempered multistable processes.

• Study the short time and long time behaviors of tempered multistable processes.

• Compare the multistable Lévy processes defined by finite-dimensional distributions (characteristic functions), Poisson representation and series representation.

The idea of the construction of tempered multistable measure and processes comes from the paper [63] . The interest of such processes is that they may be chosen to have moments of all orders. In addition, they are martingales. This will allow to construct stochastic (partial) differential equation driven by tempered multistable measures, which may be used to describe certain physical phenomena.

The characteristic function of a termpered multistable process $X\left(t\right)$ is

$\begin{array}{ccc}\hfill 𝔼\left(expiyX\left(t\right)\right)& =& exp\left\{\frac{1}{2}{\int }_{0}^{t}\Gamma \left(-\alpha \left(x\right)\right)\left[{\left(1-\frac{iy}{\theta }\right)}^{\alpha \left(x\right)}+{\left(1+\frac{iy}{\theta }\right)}^{\alpha \left(x\right)}-2\right]{\theta }^{\alpha \left(x\right)}dx\right\}.\hfill \end{array}$

We have investigated the long time and short time behaviors this process:

Short time behavior:

Let $\alpha$: $ℝ\to \left[a,b\right]\subseteq \left(0,2\right)$ be continuous. Let $u\in ℝ$ and suppose that as $v\to u$,

 $|\alpha \left(u\right)-\alpha \left(v\right)|=o\left(\frac{1}{|log|u-v||}\right).$ (18)

Then when $h\to 0$,

 ${h}^{-1/\alpha \left(t\right)}\left[X\left(t+hu\right)-X\left(t\right)\right]\to {Y}_{\alpha \left(t\right)}\left(u\right)$ (19)

in finite-dimentional-distributions, where

${Y}_{\alpha \left(t\right)}\left(u\right)=\int {\mathbf{1}}_{\left[0,u\right]}\left(z\right)d{M}_{\alpha \left(t\right)}\left(z\right),$

and ${M}_{\alpha \left(t\right)}$ is an $\alpha \left(t\right)$ stable measure. In an other word, $X\left(t\right)=M\left[0,t\right]$ is $1/\alpha \left(t\right)$-localisable at $t$ with local form ${Y}_{\alpha \left(t\right)}$.

Long time behavior:

Let $\alpha$: $ℝ\to \left[a,b\right]\subseteq \left(0,2\right)$ be continuous and ${lim}_{s\to \infty }\alpha \left(s\right)\to \alpha$. Then for $h\to \infty$

 ${h}^{-1/2}\left[X\left(t+hu\right)-X\left(t\right)\right]\to \Gamma \left(2-\alpha \right)B\left(u\right)$ (20)

in finite-dimensional-distributions, where $B$ is standard Brownian motion.

Let us now describe our work on the multistable LÃ©vy motion. For $0 and $\alpha :ℝ\to \left[a,b\right]$, the multistable Lévy motion ${M}_{c}$ defined by finite-dimensional distributions (characteristics function) is the process such that

 $𝔼\left(exp\left(i\sum _{j=1}^{d}{\theta }_{j}{M}_{c}\left({t}_{j}\right)\right)\right)=exp\left(-\int |\sum _{j=1}^{d}{\theta }_{j}{\mathbf{1}}_{\left[0,{t}_{j}\right]}{\left(s\right)|}^{\alpha \left(s\right)}ds\right);$ (21)

There also exist a Poisson representation of multistable LÃ©vy process ${M}_{p}$:

 ${M}_{p}\left(t\right)=\sum _{\left(X,Y\right)\in \Pi }{C}_{\alpha \left(X\right)}{\mathbf{1}}_{\left[0,t\right]}\left(X\right){Y}^{<-1/\alpha \left(X\right)>},$ (22)

where $\left(X,Y\right)$ be the random point of the Poisson process $\Pi$, $t>0$, ${Y}^{<-1/\alpha \left(X\right)>}=$sign${\left(Y\right)|Y|}^{-1/\alpha \left(X\right)}$ and

 ${C}_{\alpha \left(X\right)}={\left(\frac{1}{\Gamma \left(1-\alpha \left(X\right)\right)cos\left(\frac{\pi }{2}\alpha \left(X\right)\right)}\right)}^{1/\alpha \left(X\right)};$ (23)

Finally, the series representation of multistable LÃ©vy motion ${M}_{s}$ is

 ${M}_{s}\left(t\right)=\sum _{i=1}^{\infty }{C}_{\alpha \left({U}_{i}\right)}{\gamma }_{i}{\Gamma }_{i}^{-1/\alpha \left({U}_{i}\right)}{\mathbf{1}}_{\left({U}_{i}\le t\right)},$ (24)

where ${\left\{\Gamma \right\}}_{i\ge 1}$ is a sequence of arrival times of a Poisson process with unit arrival time, ${\left\{U\right\}}_{i\ge 1}$ is a sequence of i.i.d random variables with uniform distribution on $\left[0,t\right]$, ${\left\{\gamma \right\}}_{i\ge 1}$ is a sequence of i.i.d random variables with distribution $ℙ\left({\gamma }_{i}=1\right)=ℙ\left({\gamma }_{i}=-1\right)=1/2$. All three sequences ${\left\{\Gamma \right\}}_{i\ge 1}$, ${\left\{U\right\}}_{i\ge 1}$ and ${\left\{\gamma \right\}}_{i\ge 1}$ are independent, and

 ${C}_{\alpha \left({U}_{i}\right)}={\left(\frac{1}{\Gamma \left(1-\alpha \left({U}_{i}\right)\right)cos\left(\frac{\pi }{2}\alpha \left({U}_{i}\right)\right)}\right)}^{1/\alpha \left({U}_{i}\right)}.$ (25)

We have proved that these three definitions yield the same process in law.