arXiv subjects

Salvador Esquivel

Publications and source records attributed to Salvador Esquivel.

3 recordsLinked to original sources

arXiv2026

A priori bounds for the generalized parabolic Anderson model in the full subcritical regime

We show a priori bounds for the generalized Parabolic Anderson Model $(\partial_t - Δ)u = σ(u) \diamond ξ$ in finite volume in the full subcritical regime. The approach is based on the observation that, for constant initial data, the time interval over which the solution remains close to its initial condition can be chosen uniformly in the value of the initial condition. A comparison principle is then used to extend this estimate to general bounded initial data.

math.AP↗

arXiv2026

A Stochastic Flow for the Stochastic Allen-Cahn Equation with Multiplicative Noise

We establish the existence of a stochastic flow on $L^{\infty} (\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise \[ (\partial_t - \partial_x^2) u = u - u^3 + σ(u) ξ\quad \text{on} \quad \mathbb{R}_+ \times \mathbb{T}, \] where $ξ$ is space-time white noise and $σ: \mathbb{R} \rightarrow \mathbb{R}$ is sufficiently smooth, bounded, and has bounded derivatives. Our strategy is to obtain pathwise a priori estimates via regularity structures. In fact, we consider a general singular multiplicative equation with superlinear damping, driven by noises of parabolic regularity $α- 2$, for ${α\in (0, 1)}$, which can be lifted to a weakly admissible model. We show that the required estimates hold whenever \[ m > \frac{2 - α}α \varepsilon_α, \quad \text{where} \quad \varepsilon_α = 1 - α\left( 1 - \frac{2}{3 - α} \right) \in (0, 1) . \] Thus the strength of the damping needs to be chosen only as a function of the regularity of the driving noise. Under an additional smoothness assumption on $σ$, we show that the stochastic flow is differentiable with respect to its initial condition.

math.PR↗

arXiv2024

A priori bounds for the dynamic fractional $Φ^4$ model on $\mathbb{T}^3$ in the full subcritical regime

We show a priori bounds for the dynamic fractional $Φ^4$ model on $\mathbb{T}^3$ in the full subcritical regime using the framework of Hairer's regularity structures theory. Assuming the model bounds our estimates imply global existence of solutions and existence of an invariant measure. We extend the method developed for the usual heat operator by Chandra, Moinat and Weber [CMW23] to the fractional heat operator, thereby treating a more physically relevant model. A key ingredient in this work is the development of localised multilevel Schauder estimates for the fractional heat operator which is not covered by Hairer's original work. Furthermore, the algebraic arguments from [CMW23] are streamlined significantly.

math.AP↗