From Singular Memory Laws to Abstract Volterra Equations

Space-time fractional stochastic Burgers-type equation

We establish the existence and uniqueness of solutions to a $(1+1)$-dimensional fractional stochastic Burgers-type equation featuring a Caputo fractional time derivative, a fractional Laplacian, and space-time white noise forcing modified by a Riemann-Liouville fractional integral. From a physical perspective, equation...

Upper and Lower Solutions for Non-convolution Fractional Volterra Integral Equations

Long-time behavior of McKean--Vlasov stochastic systems with singular coefficients

We develop a quantitative framework for the long-time behavior of McKean--Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition wi...

Inverse source problems for time-fractional diffusion with a singular involution operator

Affine pure-jump Volterra fields

We study a class of non-negative spatio-temporal random fields that exhibit self-exciting clustering, which we refer to as affine pure-jump Volterra fields. They are defined via stochastic integration of a Volterra kernel against a thinned Poisson random measure and the thinning is such that the field admits a represen...

Configurational nonlocal Hamilton-Jacobi framework for interface propagation

We introduce a class of nonlocal Hamilton–Jacobi equations in which the classical gradient is replaced by a finite-horizon interaction operator, thereby embedding an intrinsic length scale directly into first-order nonlinear evolution. In contrast with integro-differential formulations where nonlocality appears as an a...