Ashok Maitra Memorial Lectures 2025-26

Prof. Felix Otto

MAX PLANCK INSTITUTE, MPI für Mathematik in den Naturwissenschaften Inselstraße 22 • 04103 Leipzig Germany

PROFILE
Speaker Bio

Tentative schedule is given below. Final schedule and details will be updated in January

At Indian Statistical Institute, Delhi Centre


TALK 1
Title: Stochastic partial differential equations and renormalization (I and II)
Date: 27 January 2026 Time: 3:30-4:30 PM
Venue: Auditorium
Abstract: The inclusion of thermal fluctuations into the models of mesoscopic physics typically lead to partial differential equations that are driven by a noise term that is too rough for the non-linearities to be given even a distributional sense. The typical approach from physics is to imagine infinitesimal parameters in front of the non-linearities, leading to a non-linearly coupled hierarchy of linear equations. While the roughness issue persists, this term-by-term approach allows to canonically renormalize the non-linear terms, which amounts to a suitable projection that removes divergent expectations.

Hairer’s notion of a model provides a versatile framework for this strategy. We give this model the informal but intuitive interpretation of charts and transition maps for the non-linear and renormalized solution manifold. This notion of model is more geometric than the combinatorial one indexed by trees.

Constructing the renormalized model necessarily relies on a stochastic estimate. We approach this by taking (Malliavin) derivatives w. r. t. the underlying noise, capitalizing on the fact that e. g. in the Gaussian case, elements of the Cameron-Martin space are less rough than the realizations of the noise. It is then the spectral gap inequality that provides the estimates – modulo the expectation where the renormalization takes place.

Like Lyon’s rough paths, Hairer’s regularity structures provide an actual solution theory by truncating the formal power series from the physics approach, and making the Ansatz that the solution can be written as a suitable modulation of these truncated series. This part of the solution theory is deterministic, i. e. pathwise, an amounts to an augmented Schauder theory. We argue shall explain how this is implemented for our intrinsic model.

This will be presented in two lectures, based on
L. Broux, F. Otto, M. Tempelmayr, Lecture notes on Malliavin calculus in regularity structures. Stochastics and Partial Differential Equations, 2025

L. Broux, F. Otto, R. Steele, Multi-index Based Solution Theory to the Φ 4 Equation in the Full Subcritical Regime, arXiv:2503.01621

At Indian Statistical Institute, Delhi Centre


TALK 2
Title: Stochastic partial differential equations and renormalization (I and II)
Date: 28 January 2026 Time: 3:30-4:30 PM
Venue: Seminar Room 2
Abstract: The inclusion of thermal fluctuations into the models of mesoscopic physics typically lead to partial differential equations that are driven by a noise term that is too rough for the non-linearities to be given even a distributional sense. The typical approach from physics is to imagine infinitesimal parameters in front of the non-linearities, leading to a non-linearly coupled hierarchy of linear equations. While the roughness issue persists, this term-by-term approach allows to canonically renormalize the non-linear terms, which amounts to a suitable projection that removes divergent expectations.

Hairer’s notion of a model provides a versatile framework for this strategy. We give this model the informal but intuitive interpretation of charts and transition maps for the non-linear and renormalized solution manifold. This notion of model is more geometric than the combinatorial one indexed by trees.

Constructing the renormalized model necessarily relies on a stochastic estimate. We approach this by taking (Malliavin) derivatives w. r. t. the underlying noise, capitalizing on the fact that e. g. in the Gaussian case, elements of the Cameron-Martin space are less rough than the realizations of the noise. It is then the spectral gap inequality that provides the estimates – modulo the expectation where the renormalization takes place.

Like Lyon’s rough paths, Hairer’s regularity structures provide an actual solution theory by truncating the formal power series from the physics approach, and making the Ansatz that the solution can be written as a suitable modulation of these truncated series. This part of the solution theory is deterministic, i. e. pathwise, an amounts to an augmented Schauder theory. We argue shall explain how this is implemented for our intrinsic model.

This will be presented in two lectures, based on
L. Broux, F. Otto, M. Tempelmayr, Lecture notes on Malliavin calculus in regularity structures. Stochastics and Partial Differential Equations, 2025

L. Broux, F. Otto, R. Steele, Multi-index Based Solution Theory to the Φ 4 Equation in the Full Subcritical Regime, arXiv:2503.01621

At Indian Statistical Institute, Kolkata


PUBLIC LECTURE
Title: Optimal matching of random point clouds, optimal transportation, and electrostatics
Date: TBA
Venue: TBA

Abstract: The combinatorial optimization problem of matching two large point clouds can be seen as a special case of optimal transportation between measures, a ubiquitous variational problem. In statistics, it is natural to consider random points clouds that arise as empirical measures from sampling from a given distribution, like the uniform distribution and the Poisson point process.

When matching realizations of the Poisson point process (“shot noise”), two space dimension are known to be critical (Ajtai-Koml´os-Tusn´ady). The subtle behavior in the critical case has been predicted by Parisi et. al., relying on the connection between the Monge-Amp`ere equation, which is the first variation of optimal transportation, and its linearization, the Poisson equation from electrostatics. Replacing shot noise by white noise, this provides an explicit Gaussian approximation. Ambrosio et. al. established these predictions rigorously on a macroscopic level.

A variational regularity theory, used as a large-scale regularity theory, allows to establish this connection down to the microscopic level of particle distances. It mimics De Giorgi’s approach to the regularity theory of minimal surfaces in the sense that a harmonic approximation result is at its center.

At Indian Statistical Institute, Kolkata


COLLOQUIUM
Title: Random-field Ising problem, optimal matching, and curves in a Brownian potential
Date: TBA
Venue: TBA

Abstract: Even at vanishing temperature, the presence of a random external field in the Ising model may enforce a unique Gibbs state, up to and including the critical dimension two (Aizenman-Wehr). How does the typical diameter L of the resulting domains scale in the non-dimensional prefactor ϵ ≪ 1 of the field term? Recent work suggests ln L ∼ ϵ −4/3 (Ding-Wirth).

For good reasons, the same exponent came already up when maximizing the “isoperimetric” ratio R Σ ξ |∂Σ| of an integral of white noise ξ over a subset Σ of the plane and the length of its boundary ∂Σ (Leighton-Shor). This problem in turn is dual to an optimal matching problem – and motivated Talagrand to develop a systematic theory for maxima over families of correlated Gaussian variables.

We study a (1+1)-dimensional semi-discrete random variational problem that can be interpreted as a geometrically linearized version of either problem: Σ is restricted to the subgraph of a function h = h(x), the length |∂Ω| is replaced by the Dirichlet integral of h, and R Σ ξ is replaced by R W(x, h(x))dx with {W(x, ·)}x a family of independent two-sided Brownian motions.
Appealing to ideas from Leighton-Shor, we show that at every dyadic scale from the system size down to the lattice spacing, the minimizer h∗ contains at most order-one Dirichlet energy per unit length, naturally leading to the logarithmic scaling of the minimal energy. Using super-additivity in the scales, this allows us to establish a (quantitative) quenched homogenization result in the sense that the leading order of the minimal energy becomes deterministic as the ratio system size / lattice spacing diverges.

This talk relies on F. Otto, M. Palmieri, C. Wagner, On minimizing curves in a Brownian potential, arXiv:2503.12471

At Indian Statistical Institute, Bangalore Centre


TALK 1
Title: Convection-enhanced diffusion, Brownian motion on SL(n), stochastic exponential and intermittency (I and II)
Date: February 09, 2026 (MONDAY) Time: 2:15-3:15 PM
Venue: Auditorium (II floor)
Abstract: In n-dimensional space, we consider a Brownian particle X that is also advected by a drift b, meaning dX = b(X)dt+ √ 2dW; b is assumed to be timeindependent and divergence-free. The latter is known to – often substantially – enhance the effect of bare diffusion, a very ubiquitous phenomenon in convection. We are thus interested in generic b’s of this type, that is, chosen from a stationary ensemble with a scale invariance in law that is consistent with diffusion. This critical case, which requires to remove the small lengthscales from b, leads to the particle X spreading super-diffusively, however just marginally by the square-root of a logarithm.

We tackle this problem, which falls into the class of random motions in random environments, by a homogenization that is incremental in the length scales of b, i. e. by successively replacing a dyadic portion of scales of b by an enhancement of the diffusion coefficient. We so discover that the expected Lagrangian coordinates u of the particle, which keep track of where the particle started, or rather their differential F in a point, which keeps track of how sensitively the position depends on the starting point, are on large scales well-approximated by the (canonical) Brownian motion on the Lie group SL(n) of n × n matrices of determinant one.

This Brownian motion on SL(n), driven by Brownian motion B on the Lie algebra sl(n) of trace-free n × n matrices via dF = F ◦ dB, can be considered a tensorial version of geometric Brownian motion, and thus can be studied fairly explicitly. In particular, F behaves in many ways like a stochastic exponential, and thus ∇u like a Gaussian multiplicative chaos. This transmits to an intermittent behavior of particle X positions over large scales.

We shall present this notions and approaches in two talks, based on the following papers.
G. Chatzigeorgiou, P. Morfe, F. Otto, L. Wang, The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off. Ann. Probab. 53 (2025).

P. Morfe, F. Otto, C. Wagner, A Critical Drift-Diffusion Equation: Intermittent Behavior via Geometric Brownian Motion on SL(n). arXiv:2511.15473

At Indian Statistical Institute, Bangalore Centre


TALK 2
Title: Convection-enhanced diffusion, Brownian motion on SL(n), stochastic exponential and intermittency (I and II)
Date: February 09, 2026 (MONDAY) Time: 3.30-4.30 PM
Venue: Auditorium (II floor)
Abstract: In n-dimensional space, we consider a Brownian particle X that is also advected by a drift b, meaning dX = b(X)dt+ √ 2dW; b is assumed to be timeindependent and divergence-free. The latter is known to – often substantially – enhance the effect of bare diffusion, a very ubiquitous phenomenon in convection. We are thus interested in generic b’s of this type, that is, chosen from a stationary ensemble with a scale invariance in law that is consistent with diffusion. This critical case, which requires to remove the small lengthscales from b, leads to the particle X spreading super-diffusively, however just marginally by the square-root of a logarithm.

We tackle this problem, which falls into the class of random motions in random environments, by a homogenization that is incremental in the length scales of b, i. e. by successively replacing a dyadic portion of scales of b by an enhancement of the diffusion coefficient. We so discover that the expected Lagrangian coordinates u of the particle, which keep track of where the particle started, or rather their differential F in a point, which keeps track of how sensitively the position depends on the starting point, are on large scales well-approximated by the (canonical) Brownian motion on the Lie group SL(n) of n × n matrices of determinant one.

This Brownian motion on SL(n), driven by Brownian motion B on the Lie algebra sl(n) of trace-free n × n matrices via dF = F ◦ dB, can be considered a tensorial version of geometric Brownian motion, and thus can be studied fairly explicitly. In particular, F behaves in many ways like a stochastic exponential, and thus ∇u like a Gaussian multiplicative chaos. This transmits to an intermittent behavior of particle X positions over large scales.

We shall present this notions and approaches in two talks, based on the following papers.
G. Chatzigeorgiou, P. Morfe, F. Otto, L. Wang, The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off. Ann. Probab. 53 (2025).

P. Morfe, F. Otto, C. Wagner, A Critical Drift-Diffusion Equation: Intermittent Behavior via Geometric Brownian Motion on SL(n). arXiv:2511.15473  

About

Ashok Prasad Maitra
AM




Past AMML lectures

Last modified :January 2026