Regularity Structures
Organized by Prof. Ajay Chandra
Summer 2026
My research areas in probability theory include the sample path properties of random fields, rough path theory, and stochastic partial differential equations (SPDEs). During my Ph.D., I studied geometric measure theory and probabilistic potential theory, and used these tools to establish sample path results for highly irregular Gaussian processes — processes whose modulus of continuity can be rougher than any power function, i.e. beyond the classical Hölder scale. This included results on the Hausdorff dimension of the image, graph, and level sets of such processes, as well as their hitting probabilities and local times.
As a postdoc, my research program has expanded along two main directions, described below.
Stochastic calculus for processes indexed by the plane — “sheets” rather than paths — is significantly more delicate than the one-dimensional theory: the change-of-variables formula for a planar increment picks up an extra cross-derivative term that has no classical analogue. Together with Samy Tindel, I am developing a “Rough–Young” framework for this setting. Rather than relying on the large combinatorial signatures used in earlier approaches, we work in an asymmetric Hölder regularity regime and give a controlled, Taylor-type representation for the integrands, from which the relevant 2D rough integrals can be rigorously defined as limits of Riemann sums and the classical change-of-variables formula recovered. The goal is to use this pathwise calculus to make sense of singular hyperbolic SPDEs, in particular the stochastic wave equation driven by fractional noise — a regime where the noise is rougher than space-time white noise and existing frameworks such as regularity structures, built primarily for parabolic equations, do not directly apply. Our preprint “On Itô–Stratonovich formula for rough sheets” (see the publications page) presents this framework as a first step: it develops the theory in the asymmetric Rough–Young regime described above. Extending the construction to the fully rough regime — where both directions can have Hölder regularity below 1/3 — is the subject of ongoing work with my collaborators.
A recurring theme in my work is the fine sample path and potential-theoretic behavior of Gaussian processes and random fields: their Hausdorff dimension, hitting probabilities, and the polarity of points. With Mohamed Erraoui, I have studied fractional Brownian motion perturbed by a deterministic drift, establishing fractal codimension formulae for its range and level sets, determining how the regularity of the drift governs hitting probabilities, and showing that its image can have positive Lebesgue measure and non-empty interior. With Frederi Viens, I studied irregularity scales for general Gaussian processes, relating their local irregularity to Hausdorff dimension and hitting probability estimates. Building on these foundations, my current work with Cheuk Yin Lee and Yimin Xiao turns to the delicate critical-dimension regime: whether a broad class of Gaussian random fields with stationary increments can hit a fixed point at all. Extending a covering argument of Talagrand based on sojourn-time estimates, we obtain sharp, essentially optimal conditions for the polarity of points, in analogy with Kesten’s classical result connecting polarity to the existence of local times for Lévy processes; we conjecture — and are working toward — a similar characterization for a broad class of non-Markovian Gaussian random fields. See the publications page for these papers.
Fractional Brownian motion and fractional Brownian sheets · rough path theory and the sewing lemma · regularity structures · Hausdorff dimension and hitting probabilities · local times · stochastic wave and heat equations · stochastic Navier–Stokes equations.