Searcharxiv⌕ Search

arXiv subjects

Michael Salins

Publications and source records attributed to Michael Salins.

At least 19 recordsLinked to original sources
arXiv2026

Uniform large deviation principles for the stochastic heat equation over unbounded sets of initial data

We study small-noise large deviations for the stochastic heat equation (SHE) on the torus with unbounded, multiplicative space-time white noise. We establish uniform large deviation principles (ULDPs) over unbounded sets of initial data, alongside novel well-posedness and regularity results. The ULDPs are obtained for both continuous-in-space and $L^p$ initial data. For these two classes, the ULDP holds uniformly over $L^q$-bounded subsets, with $q<\infty$ and $q<p$ allowed respectively, so these sets are highly unbounded in the state space. The admissible range of $q$ is dictated by the growth of the noise coefficient in the SHE. Crucially, our methods allow us to reach down to $q=1$. This yields ULDPs over $L^1$-bounded subsets, enabling the study of exit times in physical systems with mass conservation.

math.PR↗
arXiv2026

Positive probability of explosion for stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise

This paper studies the finite time explosion of the stochastic heat equation $\frac{\partial u}{\partial t}(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+(u(t,x))^{\beta}+\sigma(u(t,x))\dot{W}(t,x)$. We consider an interval $D=[-\pi,\pi]$ under periodic boundary condition where $\dot{W}(t,x)$ is a space-time white noise and $\sigma(u)\approx u^{\gamma}$ near $\infty$. Our results refine existing results by identifying behavior in a previously less understood regime, where we show that if $\beta\in(1,3),\gamma\in(\frac{\beta}{2},\frac{\beta+3}{4})$ or $\beta>1,\gamma\in(0,\frac{\beta}{2}]$ then mild solutions can explode with positive probability. This paper provides a partial characterization of the explosion behavior in an intermediate parameter regime, and contribute to the understanding of the interplay between the drift and diffusion terms.

math.PR↗
arXiv2026

Global in time solutions to stochastic reaction-diffusion systems with superlinear reactions satisfying a triangular control of mass

We study systems of reaction-diffusion equations perturbed by multiplicative noise, where the reaction terms satisfy quasipositivity, a triangular mass-control structure, and polynomial growth. Our results apply to a broad class of reaction-diffusion systems arising, most notably, in chemistry and biology. In the deterministic setting these assumptions are known to guarantee the global existence of solutions. In the stochastic setting, however, reaction-diffusion systems have typically been analyzed under different assumptions on the reactions that preclude many natural models, such as reversible chemical reaction networks modeled by the mass-action law, and the question of global existence and uniqueness under a mass-control structure has remained open. In this work, we show that stochastically perturbing reaction-diffusion systems with triangular control of mass by suitable multiplicative noise leads to solutions that exist for all time.

math.PR↗
arXiv2025

Gaussian fluctuations for the nonlinear stochastic heat equation with drift

In this article, we prove the Quantitative Central Limit Theorem (QCLT) for the spatial average of the solution of the nonlinear stochastic heat equation with constant initial condition, driven by space-time Gaussian white noise in dimension 1. The novelty is that the equation contains a drift term. We assume that the drift and diffusion coefficients are twice differentiable with bounded first and second order derivatives. For the proof, we use Malliavin calculus, and the second-order Poincar\'e inequality due to Vidotto (2020). To estimate the moment of the second Malliavin derivative of the solution, we develop a novel estimate for the product of two heat kernels, which is of independent interest. Finally, we provide the functional result corresponding to this CLT.

math.PR↗
arXiv2025

Nonexplosion for a large class of superlinear stochastic parabolic equations, in arbitrary spatial dimension

This paper explores the finite time explosion of the stochastic parabolic equation $\frac{\partial u}{\partial t}(t,x)=Au(t,x)+\sigma(u(t,x))\dot{W}(t,x)$ in arbitrary bounded spatial domain with a large class of space-time colored noise under Neumann, periodic or Dirichlet boundary conditions where $A$ is second-order self-adjoint elliptic operator and $\sigma$ grows like $\sigma(u)\approx C(1+|u|^{\chi})$ where $\chi=1+\frac{1-\eta}{2\beta}$ with $\eta$ and $\beta$ are the parameters related to the singularities of heat kernel and noise covariance kernel. We improve upon previous results by proving the theory in arbitrary spatial dimension, general elliptic operator, general space-time colored noise, a larger class of boundary conditions and proves that $\chi$ can reach the level $1+\frac{1-\eta}{2\beta}$.

math.PR↗
arXiv2025

Uniform attraction and exit problems for stochastic damped wave equations

We consider a class of wave equations with constant damping and polynomial nonlinearities that are perturbed by small, multiplicative, space-time white noise. The equations are defined on a one-dimensional bounded interval with Dirichlet boundary conditions, continuous initial position and distributional initial velocity. In the first part of this work, we study the corresponding deterministic dynamics and prove that certain neighborhoods of asymptotically stable equilibria are uniformly attracting in the topology of uniform convergence. Then, we consider exit problems for local solutions of the stochastic damped wave equations from bounded domains $D$ of uniform attraction. Using tools from large deviations along with novel controllability results, we obtain logarithmic asymptotics for exit times and exit places, in the vanishing noise limit, that are expressed in terms of the corresponding quasipotential. In doing so, we develop arguments that take into account the lack of both smoothing and exact controllability that are inherent to the problem at hand. Moreover, our exit time results provide asymptotic lower bounds for the mean explosion time of local solutions. We introduce a novel notion of "regular" boundary points allowing to avoid the question of boundary smoothness in infinite dimensions and leading to the proof of a large deviations lower bound for the exit place. We illustrate this notion by providing explicit examples for different classes of domains $D$. Conditions under which lower and upper bounds for exit time and exit place logarithmic asymptotic hold, are also presented. In addition, we obtain deterministic stability results for linear damped wave equations that are of independent interest.

math.PR↗
arXiv2024

Global solutions to the stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise

We prove that mild solutions to the stochastic heat equation with superlinear accretive forcing and polynomially growing multiplicative noise cannot explode under two sets of assumptions. The first set of assumptions allows both the deterministic forcing and multiplicative noise terms to grow polynomially, as long as the multiplicative noise is sufficiently larger. The second set of assumptions imposes an Osgood condition on the deterministic forcing and allows the multiplicative noise to grow polynomially. In both cases, the multiplicative noise cannot grow faster than $u^{\frac{3}{2}}$, as this would cause explosion.

math.PR↗
arXiv2023

Importance sampling for stochastic reaction-diffusion equations in the moderate deviation regime

We develop a provably efficient importance sampling scheme that estimates exit probabilities of solutions to small-noise stochastic reaction-diffusion equations from scaled neighborhoods of a stable equilibrium. The moderate deviation scaling allows for a local approximation of the nonlinear dynamics by their linearized version. In addition, we identify a finite-dimensional subspace where exits take place with high probability. Using stochastic control and variational methods we show that our scheme performs well both in the zero noise limit and pre-asymptotically. Simulation studies for stochastically perturbed bistable dynamics illustrate the theoretical results.

math.PR↗
arXiv2023

Global solution for superlinear stochastic heat equation on $\mathbb{R}^d$ under Osgood-type conditions

We study the \textit{stochastic heat equation} (SHE) on $\R^d$ subject to a centered Gaussian noise that is white in time and colored in space.The drift term is assumed to satisfy an Osgood-type condition and the diffusion coefficient may have certain related growth. We show that there exists random field solution which do not explode in finite time. This complements and improves upon recent results on blow-up of solutions to stochastic partial differential equations.

math.PR↗
arXiv2023

Solutions to the stochastic heat equation with polynomially growing multiplicative noise do not explode in the critical regime

We investigate the finite time explosion of the stochastic heat equation $\frac{\partial u}{\partial t} = Δu(t,x) + σ(u(t,x))\dot{W}(t,x)$ in the critical setting where $σ$ grows like $σ(u) \approx C(1 + |u|^γ)$ and $γ= \frac{3}{2}$. Mueller previously identified $γ=\frac{3}{2}$ as the critical growth rate for explosion and proved that solutions cannot explode in finite time if $γ< \frac{3}{2}$ and solutions will explode with positive probability if $γ>\frac{3}{2}$. This paper proves that explosion does not occur in the critical $γ=\frac{3}{2}$ setting.

math.PR↗
arXiv2022

Existence and uniqueness of global solutions to the stochastic heat equation with super-linear drift on an unbounded spatial domain

We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$ along with additional restrictions. For example, consider the forcing $f(u) = u \log(e^e + |u|)\log(\log(e^e+|u|))$. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.

math.PR↗
arXiv2022

Moderate deviations for systems of slow-fast stochastic reaction-diffusion equations

The goal of this paper is to study the Moderate Deviation Principle (MDP) for a system of stochastic reaction-diffusion equations with a time-scale separation in slow and fast components and small noise in the slow component. Based on weak convergence methods in infinite dimensions and related stochastic control arguments, we obtain an exact form for the moderate deviations rate function in different regimes as the small noise and time-scale separation parameters vanish. Many issues that appear due to the infinite dimensionality of the problem are completely absent in their finite-dimensional counterpart. In comparison to corresponding Large Deviation Principles, the moderate deviation scaling necessitates a more delicate approach to establishing tightness and properly identifying the limiting behavior of the underlying controlled problem. The latter involves regularity properties of a solution of an associated elliptic Kolmogorov equation on Hilbert space along with a finite-dimensional approximation argument.

math.PR↗
arXiv2021

Global solutions for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

A condition is identified that implies that solutions to the stochastic reaction-diffusion equation $\frac{\partial u}{\partial t} = \mathcal{A} u + f(u) + σ(u) \dot{W}$ on a bounded spatial domain never explode. We consider the case where $σ$ grows polynomially and $f$ is polynomially dissipative, meaning that $f$ strongly forces solutions toward finite values. This result demonstrates the role that the deterministic forcing term $f$ plays in preventing explosion.

math.PR↗
arXiv2021

Systems of small-noise stochastic reaction-diffusion equations satisfy a large deviations principle that is uniform over all initial data

Large deviations principles characterize the exponential decay rates of the probabilities of rare events. Cerrai and Rockner [13] proved that systems of stochastic reaction-diffusion equations satisfy a large deviations principle that is uniform over bounded sets of initial data. This paper proves uniform large deviations results for a system of stochastic reaction--diffusion equations in a more general setting than Cerrai and Rockner. Furthermore, this paper identifies two common situations where the large deviations principle is uniform over unbounded sets of initial data, enabling the characterization of Freidlin-Wentzell exit time and exit shape asymptotics from unbounded sets.

math.PR↗
arXiv2021

Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain

We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution.

math.PR↗