Stochastic Differential Equations in Hölder and Sobolev Spaces

Stochastic differential equations (SDEs) provide a mathematical framework for modelling dynamical systems subject to random influences. The regularity of their solutions often hinges on the function-space setting in which drift and diffusion coefficients are posed. Hölder and Sobolev spaces furnish two complementary scales for quantifying spatial and temporal smoothness. Hölder spaces capture pointwise continuity properties through controlled increments, while Sobolev spaces measure integrability and weak differentiability via square-integrable derivatives. In recent years, advances in harmonic analysis and probability theory have led to refined existence, uniqueness and regularity results for SDEs whose coefficients or forcing terms belong to these spaces. This intersection has yielded new gradient estimates, density bounds, and stability criteria under minimal smoothness assumptions, thereby broadening the class of admissible stochastic models in physics, finance and biology.

Central developments include the deployment of Littlewood-Paley theory and heat-kernel techniques to derive Hölder continuity of transition semigroups driven by pure-jump Lévy processes. On the Sobolev side, Bessel potential spaces and Sobolev–Slobodeckij norms have been employed to treat degenerate, discontinuous or non-local operators arising in filtering and integro-differential equations. These methods have also informed numerical schemes for pathwise approximation and Monte Carlo simulation, ensuring convergence rates tied explicitly to the underlying function-space regularity. Collectively, this body of work underscores the global significance of SDEs in irregular regimes, with applications ranging from turbulent transport to signal processing.

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Stochastic Differential Equations in Hölder and Sobolev Spaces publication trend

The graph below shows the total number of articles in stochastic differential equations in hölder and sobolev spaces across all publications each year (not limited to Nature Index journals).

Technical terms

Stochastic differential equation: An equation describing the evolution of a random process driven by deterministic drift and stochastic noise terms, typically formulated in differential form with respect to Brownian motion or jump processes.

Hölder space: A function space C^β in which functions satisfy a uniform bound on the β-power of their increments, measuring pointwise regularity with exponent β∈(0,1].

Sobolev space: A function space W_p^s consisting of functions whose weak derivatives up to order s are p-integrable, quantifying smoothness in an averaged sense.

Bessel potential space: A scale H_p^s obtained by applying fractional powers of the Laplace operator to L_p functions, equivalent to certain Sobolev and Besov spaces in regularity characterisation.

References

1. Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha $-stable processes. Electronic Journal of Probability (2020).
2. On Lp-solvability of stochastic integro-differential equations. Stochastics and Partial Differential Equations: Analysis and Computations (2020).
3. On Solvability of Integro-Differential Equations. Potential Analysis (2020).