Modern Stochastics: Theory and Applications

This paper introduces a novel stochastic partial differential equation (SPDE) driven by space-only fractional Lévy noise ${\dot{L}_{x}^{d}}$ with $d\in (0,1/2)$, governed by a second-order elliptic operator with piecewise constant coefficients. We establish a new framework for analyzing diffusion in heterogeneous media, proving the existence, uniqueness, and equivalence of mild and weak solutions for the resulting non-Gaussian random field. A key original contribution is the explicit characterization of the covariance kernel in terms of transmission-reflection paths induced by the interface. We provide sharp estimates for the sample path regularity, proving that the solution is almost surely Hölder continuous of order $\gamma \lt d$ in space and $\beta \lt \frac{2d+1}{4}$ in time. Our results demonstrate that the fractional order of the noise robustly dictates path regularity, which remains preserved despite the singular transmission conditions and the jump structure of the driving noise.

Multidimensional generalized backward stochastic differential equations (GBSDEs) are studied within a general filtration that supports a Brownian motion under weak assumptions on the associated data. The existence and uniqueness of solutions in ${\mathbb{L}^{p}}$ for $p\in (1,2)$ are established. The results apply to generators that are stochastic monotone in the y-variable, stochastic Lipschitz in the z-variable, and satisfy a general stochastic linear growth condition.