40-minute talks
Amit Apte (IISER Pune)
Title: Stability of particle and Kalman filters for chaotic dynamics
Abstract: The Bayesian formulation of the data assimilation problem in earth sciences leads to non-linear filtering. In many applications, the dynamical models are deterministic and chaotic, in which case most of the classical stability results for nonlinear filtering are not applicable because such systems do not satisfy the assumption of controllability. In this talk, I will discuss both theoretical as well as numerical challenges and methods to overcome these challenges in order to demonstrate stability in this context. (Based on works in collaboration with A.S.Reddy, S.Vadlamani, P.Mandal, and S.K.Roy)
Anup Biswas (IISER Pune)
Title: Recent developments in the nonlocal Hamilton-Jacobi equations.
Abstract: It is well-known that specific forms of Hamilton-Jacobi equations arise in the study of stochastic optimal control. Indeed, these equations are central to deriving optimal Markov controls for the associated problems. In this talk, we will focus on the Hamilton-Jacobi equations linked to ergodic control problems for jump diffusions and present some recent findings in this area.
Debasish Chatterjee (IIT Bombay)
Title: On almost sure detection of the presence of malicious components in networked (cyberphysical) systems
Abstract: Most of the system-theoretic studies on the security of cyber-physical systems assume certain types of adversarial attack models. This essentially means that the designers/defenders already know the plans of the attackers, which does not reflect realistic situations since no reasonable attacker would be inclined to announce their attack strategies to the world. Game-theoretic approaches to studying adversarial attacks are also not quite applicable directly and widely because the designers/defenders cannot possibly know the attackers' objectives unless the dividing line between a designer and an attacker is blurred. Against this backdrop, an approach based purely on the analysis of ‘signals’ — measurements of the systems under consideration — appears to be a natural and general approach to tackling the problem of detecting adversarial attacks. This talk will expose a preliminary result on the detection of malicious attacks in the indicated 'signal-theoretic' spirit.
Tuhin Ghosh (HRI)
Title: Inverse scattering problem for the (fractional) Schrödinger equation
Abstract: In this talk, we will introduce the direct and the inverse scattering problem for the (fractional)
Schrödinger equation. Restricted by the resolvent estimates, in some regime of the nonlocality, we will address those problems.
Harsha Hutridurga (IIT Bombay)
Title: Large time behaviour of solutions to reaction-diffusion systems: quantitative rates
Abstract: This talk aims to analyse degenerate reaction-diffusion systems wherein the rate functions are triangular. Here degeneracies mean that one of the diffusion coefficients vanishes in the reaction-diffusion system. We exploit the mass conservation property, the positivity preserving property and the underlying entropy dissipation structure to show that the solution approaches an equilibrium state for large times. We demonstrate that the decay rate is sub-exponential in time and we obtain explicit constants in the decay estimate. Our method of proof comes under the heading of entropy methods which involves elementary algebraic manipulations and a few functional inequalities. This is a joint work with Saumyajit Das (HRI).
Mathew Joseph (ISI Bengaluru)
Title: Instantaneous blowup for interacting SDEs with superlinear drift.
Abstract: There has been recent interest in blowup questions for stochastic heat equations with superlinear drift. It is expected that blowup occurs when the drift satisfies the finite Osgood condition (as is the case for ODEs), for fairly general initial profiles. We show this is the case for a system of interacting SDEs on the lattice, and in fact blowup occurs instantaneously everywhere. The key idea is to use the splitting method to compare the interacting SDEs to a one-dimensional SDE which blows up. This is based on joint work with Shubham Ovhal.
Chaman Kumar (IIT Roorkee)
Title: Order-one scheme for IPS connected with the McKean–Vlasov SDE with non-differentiable drift.
Abstract: By the randomization of the drift coefficient, we propose an order-one scheme for the IPS associated with the McKean--Vlasov SDE having non-differentiable drift coefficient. The full randomization of the drift allows us to remove its differentiability assumption with respect to both state and measure variables. Our theoretical findings are further demonstrated through numerical experiments. The talk is based on my joint work with S. Biswas, Neelima, G. dos Reis and C. Reisinger and is published in the Annals of Applied Probability.
Chetan D. Pahlajani (IIT Gandhinagar)
Title: Asymptotic analysis of switching diffusion processes with spatially periodic coefficients
Abstract: We study homogenization for a class of switching diffusion processes whose drift and diffusion coefficients, and jump intensities are smooth, spatially periodic functions; we assume full coupling between the continuous and discrete components of the state. Assuming uniform ellipticity of the diffusion matrices and irreducibility of the matrix of switching intensities, we explore the large-scale long-time behavior of the process under a diffusive scaling. Our main result characterizes the limiting fluctuations of the rescaled continuous component about a constant velocity drift by an effective Brownian motion with explicitly computable covariance matrix. In the process of extending classical periodic homogenization techniques for diffusions to the case of switching diffusions, our main quantitative finding is the computation of an extra contribution to the limiting diffusivity stemming from the switching.
Neeraja Sahasrabudhe (IISER Mohali)
Title: Elephant Random Walk with Tampered Memory
Abstract: An Elephant Random Walk (ERW), introduced by Schutz and Trimper in 2004, is a type of memory-based random walk where the walker’s next step depends on all previous steps. In this talk, we will introduce a variation of the classical ERW, obtained by tampering the memory. At any time n > 1, the tampered memory ERW updates as a classical ERW when the uniformly chosen memory point is from a given subset D_{n−1} of the past {1, . . . , n − 1}, but behaves differently when chosen from its complement. We will discuss the limit theorems and their dependence on the nature of the sets D_n. This is a joint work with Vinita Mulay and Dr. Debleena Thacker.
Vivek Tewary (Krea University)
Title: Regularity theory for nonlocal p-Laplace equations
Abstract: Brownian motion is related to the Laplace operator as jump Levy processes are related to the nonlocal Laplace operator. Similarly, the p-Laplace operator has a probabilistic interpretation in a tug-of-war game with noise and the nonlocal p-Laplace operator comes from a nonlocal version of this tug-of-war game. A recent turn is to study these operators using non-probabilistic methods. While the probabilistic interpretations are linked to non-divergence type operators, in this talk, we will focus on one of their divergence type counterparts. In particular, we will discuss the local boundedness and local Holder regularity of solutions of the parabolic divergence type nonlocal p-Laplace equation.
25-minute talks
Arvind Kumar Nath (IIT Kanpur)
Title: Invariant Measure for Linear Stochastic PDEs in the Space of Tempered Distributions
Abstract: We first explore exponential stability by using Monotonicity inequality and use this information to obtain the existence of Invariant measure for linear Stochastic PDEs with potential in the space of tempered distributions. The uniqueness of Invariant Measure follows from Monotonicity inequality.
Mangala Prasad (IIT Kanpur)
Title: Strong convergence of Stochastic generalized Benjamin-Bona-Mahony equation
Abstract: In this talk, we study semi-discrete finite element approximation and full discretization of the Stochastic semilinear generalized BBM equation in a bounded convex polygonal domain driven by additive Wiener noise. We use the finite element method for spatial discretization and the semi-implicit method for time discretization and derive a strong convergence rate with respect to both parameters (spatial and temporal).
Kshitij Sinha (IIT Bombay)
Title: Homogenization of parabolic equations with large lower terms: Quantitative Theory
Abstract: The objective of this talk is to showcase some of our recent results in the quantitative theory of periodic homogenization. We show that the solution to an initial boundary value problem can be factorized in a particular manner. This is accomplished via an associated eigenvalue problem. We shall obtain a quantitative rate for the convergence of one of the factors in the aforementioned factorization to the corresponding homogenized solution.
Simran Soni (IIT Roorkee)
Title: Well-posedness and approximation of McKean-Vlasov SDEs
Abstract: MV-SDE with non-Lipschitz coefficients is considered. Results on well-posedness and convergence properties of the corresponding interacting particle system are discussed. Further, an explicit scheme of order half to approximate the exact solution and its convergence properties are presented.
Raj Karan Gupta (IIT Roorkee)
Title: Regime Switching Stochastic Differential Equations and their Numerical Studies
Abstract: In this work, we study the well-posedness and numerical approximation of regime-switching stochastic differential equations (RS-SDEs) in which the drift coefficient is piecewise Lipschitz continuous, while the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift. To address the analytical challenges, we employ a switching-dependent transformation technique to establish well-posedness of the system. We then develop a switching-adapted Euler scheme designed to handle irregular coefficients and regime changes. Moment bounds are derived, and we prove that the proposed scheme achieves strong convergence with order 1/2. Finally, numerical experiments are provided to validate the theoretical results and confirm the reliability of the method. Joint work with Chaman Kumar, Divyanshu Vashistha and Tejinder Kumar.
Divyanshu Vashishta (IIT Roorkee)
Title: First-order scheme for regime switching SDE driven by Levy noise
Abstract: This is a joint work with Dr. Chaman Kumar (Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India). We develop an Ito’s formula for a regime switching Stochastic Differential Equation driven by Levy noise, which is then used to derive a first-order Milstein scheme. Under Lipschitz continuity of the coefficients and their first derivatives, we analyze the moment stability of the scheme and establish a strong convergence of order 1.0. The analysis addresses the challenges arising from the interplay between the cadlag sample paths with the discontinuous dynamics of the Markov chain. We conclude by discussing the practical implementation of our scheme, highlighting simplifications under commutativity conditions for diffusion and jump operators.
Saumyajit Das (HRI)
Title: Fractional Borg-Levinson problem.
Abstract: In this talk, we will explore how Gel'fand spectral data can be used to determine the perturbation potential associated with the fractional Laplacian on a smooth bounded domain.
Ram Gopal Jaiswal (HRI)
Title: Weak Solutions to the Nonlinear Fragmentation Equation with Mass Transfer
Abstract: Fragmentation processes arise in various phenomena, including certain stages of planetary formation and raindrop breakup. These processes are generally classified into two categories: linear fragmentation, where particles break due to external forces, and nonlinear fragmentation, where particles break due to collisions. The nonlinear fragmentation equation studied in the literature is typically divided into two cases: one where mass transfer after a collision is allowed, and another where it is excluded.
In this talk, we focus on the well-posedness of the nonlinear fragmentation equation with mass transfer. We consider collision rates of the sum/product type and breakage kernels corresponding to scenarios in which each collision produces a finite number of particles. First, we discuss the existence of weak solutions for a restricted class of daughter distributions, assuming that both the initial number of particles and the total mass are finite. We then extend the discussion to a broader class of daughter distributions, which requires additional conditions on the initial data. Finally, we provide a formal proof of the uniqueness of weak solutions.
Sooraj A P (IIT Bombay)
Title: The Eulerian Elephant Random Walk
Abstract: We introduce the Eulerian elephant random walk, a variant of the Elephant random walk on $\mathbb{Z}^2$ where we have distinct probabilities of rotation along the coordinate axes. This is a model inspired by the memory-dependent Elephant random walk and the Eulerian behavior exhibited by several phenomena in Statistical physics. We explore two approaches to this problem - one involves embedding the random walk into a generalized P\'{o}lya urn model, and another in which we tackle the random walk directly via a suitably defined martingale. This is ongoing work with Arijit Haldar, Subhrangshu Sekhar Manna and Parthanil Roy.