Superdiffusive Limits for the Fast-Slow Systems and MPGD
Without loss of generality, we set q=1 in ( 7 ) and consider the following family of recursions:

\left\{\begin{aligned} x^{(m)}_{k+1}&=x^{(m)}_{k}+m^{-1}a_{m}(x^{(m)}_{k})+m^{-\frac{1}{\alpha}}b_{m}(x^{(m)}_{k})v(y_{k}),\ y_{k+1}&=T(y_{k}),\end{aligned}\right

where T is the Thaler map and v is the observable constructed in Section 3 .

The following result shows that stochastic dynamics emerges in the above family of deterministic dynamical systems in the limit m\to\infty . Moreover, the limiting dynamics are described by a stochastic differential equation (SDE) driven by a heavy-tailed Lévy-stable process.

Let \alpha\in(1,2) and assume that the coefficients a_{m} , b_{m} and the initial condition x_{0}^{(m)} of ( 9 ) are well-behaved as m\to\infty . Then, under certain regularity conditions on the a_{m} , b_{m} , the process (x_{\lfloor mt\rfloor}^{(m)})_{t\geq 0} converges in distribution to the solution (X_{t})_{t\geq 0} of the SDE:

\mathrm{d}X_{t}=a(X_{t})\mathrm{d}t+b(X_{t})\diamond\mathrm{d}\mathrm{L}_{t}^{\alpha,\beta} with X_{0}=x_{0}\in\mathbb{R}^{d},

as m\to\infty , where a=\lim\limits_{m\rightarrow\infty}a_{m} , b=\lim\limits_{m\rightarrow\infty}b_{m} , x_{0}=\lim\limits_{m\to\infty}x_{0}^{(m)} , \diamond denotes the Marcus integration, defined in Section B.2.1 , and (\mathrm{L}^{\alpha,\beta}_{t})_{t\geq 0} is a Lévy process with the characteristic exponent

\Psi(\xi)=|\xi|^{\alpha}(1-i\beta\,\mathrm{sign}(\xi)\tan(\alpha\pi/2))

In particular,
Theorem 4.1 implies convergence in distribution of x^{(m)}_{\lfloor mt\rfloor} to X_{t} at fixed t .
It is a special case of Theorem C.1 , which covers a large class of the maps T and observables v , and extends the results in [ 16 , 6 ] to the case where the coefficients are dependent on m . We provide technical details on the topology used for the convergence and the proof in Appendix B - C .

Specializing Theorem 4.1 (extended to the case of q=2 ) to the MPGD recursion ( 8 ), we see that the rescaled MPGD iterate process, (x_{\lfloor mt\rfloor}^{(m)})_{t\geq 0} , converges in distribution to the solution (X_{t})_{t\geq 0} of the following SDE:

\mathrm{d}X_{t}=- abla\hat{\mathcal{R}}(X_{t},S_{n})\mathrm{d}t-\mu\ {\rm{diag}}(X_{t})\diamond\mathrm{d}\mathrm{L}_{t}^{\alpha_{1},\beta_{1}}+\sigma\mathrm{d}\tilde{\mathrm{L}}_{t}^{\alpha_{2},\beta_{2}},\ \ X_{0}=\lim_{m\to\infty}x_{0}^{(m)},

in the limit m\to\infty , which is equivalent to sending the step-size, \eta:=1/m , in the MPGD to zero. Here (\mathrm{L}_{t}^{\alpha_{1},\beta_{1}})_{t\geq 0} and (\tilde{\mathrm{L}}_{t}^{\alpha_{2},\beta_{2}})_{t\geq 0} are independent Lévy processes with characteristic exponent ( 10 ), and \beta_{1} , \beta_{2}\in[-1,1] are the parameters appearing in the construction of the observables v_{1} and v_{2} respectively. Therefore, we see that MPGD converges to a gradient flow driven by heavy-tailed Lévy-stable processes in the considered scaling limit. The driving processes contain both additive and multiplicative components. The multiplicative component can be interpreted as a dynamical version of the weight decay, whereas the additive component can be viewed as gradient noise [ 27 ] .