# Boundary problems for the fractional and tempered fractional operators

## TL;DR

This paper explores boundary conditions for fractional and tempered fractional PDEs modeling anomalous diffusion, emphasizing the importance of boundary data in the complement of the domain for well-posedness and physical relevance.

## Contribution

It introduces and justifies new boundary conditions for space fractional PDEs based on stochastic and probabilistic insights, ensuring well-posedness and physical interpretability.

## Key findings

- Boundary conditions depend on the solution outside the domain.
- Well-posedness of fractional PDEs with generalized boundary conditions.
- Properties of fractional operators discussed.

## Abstract

For characterizing the Brownian motion in a bounded domain: $\Omega$, it is well-known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; on the contrary, for the L\'evy flights or tempered L\'evy flights in a bounded domain, it involves the information of a solution in the complementary set of $\Omega$, i.e., $\mathbb{R}^n\backslash \Omega$, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous diffusion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial differential equations (PDEs), of specifying `boundary' conditions for space fractional PDEs modeling the anomalous diffusion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved.