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Arnab Sen is a mathematician working in probability theory and mathematical physics. His own biography identifies a Berkeley doctorate completed in 2010, subsequent research at Cambridge, and work at the University of Minnesota. His research includes random graphs, random matrices and disordered models in statistical physics.^[ref]

With Nathanaël Berestycki and Christophe Garban, he developed a new state space and topology for the coalescing Brownian flow and proved an probabilistic invariance principle for coalescing random walks under a finite-variance assumption. The theorem connects discrete particle systems with a continuum stochastic flow.^[ref]

The physical picture is simple even though convergence of an entire flow is subtle: particles wander separately until meeting, then continue as one. Tracking only the position of one particle misses when two histories merge. A framework for the whole collection must retain that interaction.

A probabilistic invariance principle is a limit theorem saying that a suitably rescaled random process converges to a universal limiting process. A basic instance is the functional central limit theorem: many different microscopic increment laws produce the same limiting Brownian motion after centering and variance normalization.

Let $X_1,X_2,\dots$ be independent identically distributed random variables with mean zero and finite variance ${\sigma }^2>0$. Put $S_k={\sum }_{j=1}^k{X}_j$, and interpolate the values $S_k/(\sigma \sqrt{n})$ linearly at times $k/n$. The resulting random continuous function $W_n$ converges in distribution in $C([0,1])$, with its uniform norm, to standard Brownian motion.^[ref]

This is stronger than convergence at each fixed time: it controls continuous functionals of the whole path. For a simple symmetric random walk, the maximum satisfies, for $a>0$,using the Brownian reflection principle and the standard normal distribution function $\Phi$. For $a=1$ this is about $0.3173$. The maximum here is one-sided, rather than the maximum absolute displacement. Convergence of finite-dimensional distributions alone would not establish a path-space theorem; control of path fluctuations, expressed through tightness of probability measures, is also needed.