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Stochastic analysis develops calculus and analytic tools for stochastic processes, especially when their sample paths are too irregular for ordinary differentiation. Brownian motion is the central example: its increments have size of order $\sqrt{\Delta t}$ rather than $\Delta t$, and its paths are almost surely nowhere differentiable.

The Itô integral is defined through an appropriate limit of sums whose integrands use information available before each increment. Its Itô isometry gives $\mathbb{E}({\int }_0^t{H}_{s}d{B}_s{)}^2=\mathbb{E}{\int }_0^t{H}_s^{2}ds$ for square-integrable adapted integrands. The resulting Itô formula includes a second-derivative correction:Taking expectations recovers $\mathbb{E}{B}_t^2=t$. Ordinary calculus would miss the extra term and predict the wrong variance. This is why stochastic calculus is needed for stochastic differential equations and their stochastic partial differential equation counterparts, rather than treating random trajectories as differentiable curves.^[ref]

Vitalii Konarovskyi is a mathematician working in probability theory working on stochastic analysis and interacting particle systems. In his paper on coalescing heavy diffusion particles, he constructed a modified Arratia flow in which particles carry conserved mass and their diffusion rate is inversely proportional to that mass, starting from uniformly distributed mass on an interval.^[ref]

The mechanism is easy to picture: after particles merge, the heavier cluster diffuses more slowly. A rigorous construction must accommodate a continuum of initial particles, rather than only a finite collection of independent paths. His work develops a stochastic integral for the resulting flow and describes total local time through an occupation measure. His authored notes on the large deviation principle also explain how unlikely fluctuations acquire quantitative path costs.^[ref]