arXiv subjects

Shyam Popat

Publications and source records attributed to Shyam Popat.

5 recordsLinked to original sources

arXiv2026

The stochastic Keller--Segel system in critical spaces

We study stochastic, parabolic-parabolic Keller--Segel equations on the $d$-dimensional torus in scaling critical Besov spaces, for $d \geq 3$. Using stochastic maximal regularity estimates, we prove local well-posedness of the equation, i.e., that there exists a unique solution up to a maximal time of existence. We show that the time of existence can be made arbitrarily large with arbitrarily high probability, provided that the initial data is sufficiently small in these critical spaces. Contribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organized by Antonio Agresti and Mark Veraar.

math.PR↗

arXiv2025

Ergodicity for the Dean--Kawasaki Equation with Dirichlet Boundary Conditions: Taming the Square-Root

In this paper, we establish the ergodicity of generalized Dean--Kawasaki equations with correlated noise and Dirichlet boundary conditions. In contrast to the ergodicity results of Fehrman, Gess, and Gvalani arXiv:2206.14789, our analysis accommodates irregular, square-root type noise coefficients. For such irregular coefficients, we prove that the law of the classical Dean--Kawasaki equation converges exponentially fast to equilibrium, while for the porous medium type Dean--Kawasaki equation, the convergence occurs at a polynomial rate. Furthermore, we obtain a regularization by noise effect, showing that the polynomial convergence rate improves to an exponential one whenever the noise coefficient is sufficiently regular, including the case of conservative multiplicative linear noise. Our approach relies on establishing a supercontraction property in a suitably weighted Lebesgue space, achieved through a refined doubling of variables argument. The construction of the weight function crucially exploits the specific structure of the Dean--Kawasaki-type correlated noise.

math.PR↗

arXiv2025

Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions

We establish a central limit theorem and large deviations principle that characterises small noise fluctuations of the generalised Dean--Kawasaki stochastic PDE. The fluctuations agree to first order with fluctuations of certain interacting particle systems, such as the zero range process, about their hydrodynamic limits. Our main contribution is that we are able to consider stochastic PDEs on general $C^2$ bounded domains with Dirichlet boundary conditions. On the level of particles, the boundary condition corresponds to absorption or injection of particles at the boundary.

math.PR↗

arXiv2024

Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions

In this paper we extend the theory of energy solutions for singular SPDEs, focusing on equations driven by highly irregular noise with bilinear nonlinearities, including scaling critical examples. By introducing Gelfand triples and leveraging infinite-dimensional analysis in Hilbert spaces together with an integration by parts formula under the invariant measure, we largely eliminate the need for Fourier series and chaos expansions. This approach broadens the applicability of energy solutions to a wider class of SPDEs, offering a unified treatment of various domains and boundary conditions. Our examples are motivated by recent work on scaling limits of interacting particle systems.

math.PR↗

arXiv2024

Well-Posedness of the generalised Dean-Kawasaki Equation with correlated noise on bounded domains

In this paper, we extend the notion of stochastic kinetic solutions introduced in arXiv:2108.08858 to establish the well-posedness of stochastic kinetic solutions of generalized Dean-Kawasaki equations with correlated noise on bounded, $C^2$-domains with Dirichlet boundary conditions. The results apply to a wide class of non-negative boundary data, which is based on certain a priori estimates for the solutions, that encompasses all non-negative constant functions including zero and all smooth functions bounded away from zero.

math.PR↗