# Space-time fractional stochastic partial differential equations with L\'evy Noise

## TL;DR

This paper studies non-linear space-time fractional stochastic heat equations driven by Lévy noise, proving existence, uniqueness, and growth behavior of solutions, extending classical results to fractional and jump noise settings.

## Contribution

It introduces new existence and uniqueness results for space-time fractional SPDEs with Lévy noise, extending classical parabolic SPDE theory to fractional derivatives and jump processes.

## Key findings

- Existence and uniqueness of mild solutions are established.
- Solutions exhibit exponential growth under linear noise coefficient growth.
- Nonexistence results are shown when the noise coefficient grows faster than linearly.

## Abstract

We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$ and $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{N }(t,x,h)\bigg]$$ in $(d+1)$ dimensions, where $\alpha\in (0,2]$ and $d<\min\{2,\beta^{-1}\}\alpha$, $\nu>0$, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $I^{1-\beta}_t$ is the fractional integral operator, ${N}(t,x)$ are Poisson random measure with $\tilde{N}(t,x)$ being the compensated Poisson random measure. $\sigma:{\mathbb{R}}\to{\mathbb{R}}$ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in "M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. \emph{ Electron. J. Probab.} {\bf14} (2009), 548--568" and " J. B. Walsh. An Introduction to Stochastic Partial Differential Equations, \'Ecoled'\'et\'e de Probabilit\'es de Saint-Flour, XIV|1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, (1986), 265--439". Under the linear growth of $\sigma$, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when $\sigma$ grows faster than linear.