Stochastic Analysis of Heat Kernels and Diffusion Processes

Stochastic analysis of heat kernels and diffusion processes encompasses the study of probabilistic representations of solutions to heat-type equations and the dynamical behaviour of associated random motions. At its core is the heat kernel, a fundamental solution that describes how heat or probability mass propagates on manifolds, domains or discrete structures over time. This framework unites techniques from partial differential equations, probability theory and differential geometry to address questions of regularity, asymptotics and long‐time behaviour. Recent advances have focused on precise short‐time expansions near singularities or boundaries, global two‐sided estimates capturing exponential decay and novel convolution schemes that link path‐integral formulations with analytic approximations. Simultaneously, the theory of diffusion processes—stochastic processes with continuous paths governed by infinitesimal generators such as the Laplacian or its fractional counterparts—has flourished. Attention has turned to anomalous transports modelled by stable processes, boundary‐driven reflections in polygonal domains and stochastic partial differential equations driven by jump noise. These developments have yielded sharper analytic bounds, deeper insight into geometric influences and broader applications ranging from materials science to quantitative finance.

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Stochastic Analysis of Heat Kernels and Diffusion Processes publication trend

The graph below shows the total number of articles in stochastic analysis of heat kernels and diffusion processes across all publications each year (not limited to Nature Index journals).

Technical terms

Heat kernel: The fundamental solution of the heat equation that gives the transition density for a diffusion process over time.

Diffusion process: A continuous‐path stochastic process whose dynamics are governed by a second‐order differential operator, typically the Laplacian or its extensions.

Asymptotic expansion: A series representation describing the behaviour of a function or kernel in the limit of small or large time, often capturing leading geometric or probabilistic features.

Spectral heat content: A quantity measuring the integral of the heat kernel over a domain, reflecting how much “heat” remains in the region as time tends to zero.

Stochastic partial differential equation (SPDE): A differential equation in which randomness enters through noise terms, modelling phenomena such as heat flow with random forcing.

Lévy noise: A type of discontinuous stochastic input characterised by jumps, generalising Gaussian noise and leading to anomalous diffusion behaviour.

References

1. Strong short-time asymptotics and convolution approximation of the heat kernel. Annals of Global Analysis and Geometry (2018).
2. On sharp heat kernel estimates in the context of Fourier–Dini expansions. Journal of Approximation Theory (2024).
3. Spectral heat content for α-stable processes in C1,1 open sets. Electronic Journal of Probability (2022).
4. A Graphical Representation of the Truncated Moment of the Solution of a Nonlinear SPDE. International Journal of Analysis and Applications (2023).
5. Heat Flow in Polygons with Reflecting Edges. Integral Equations and Operator Theory (2023).