Mathematics / research
Stochastic partial differential equations
Formulate infinite-dimensional stochastic evolution and examine noise regularity, solution concepts, and singular products.
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Module 01 research
Stochastic partial differential equations
Formulate infinite-dimensional stochastic evolution and examine noise regularity, solution concepts, and singular products.
- 01
Cylindrical Wiener processes and Hilbert-space noise
Distinguish a family of scalar Brownian coordinates from a genuine Hilbert-valued Gaussian process through covariance traces.
37 min - 02
Stochastic integrals in Hilbert spaces
Use predictable Hilbert–Schmidt integrands and the covariance-weighted Itô isometry to construct infinite-dimensional stochastic integrals.
38 min - 03
The stochastic heat equation and convolutions
Derive spectral covariance and Sobolev thresholds from the Hilbert–Schmidt norm of a heat semigroup acting on cylindrical noise.
40 min - 04
Mild, weak, and variational solutions
State which operator domain, dual tests and energy space each solution formulation uses, without confusing analytic weakness with a weak probabilistic solution.
40 min - 05
Existence, uniqueness, and monotonicity
State a controlled additive-noise variational theorem and use coercivity and one-sided monotonicity to explain its energy and uniqueness estimates.
42 min - 06
Invariant measures and ergodicity
Construct stationary Gaussian laws from an infinite-time covariance trace and distinguish existence of a law from contraction toward it.
40 min - 07
Stochastic wave equations and regularity
Calculate displacement and velocity covariances and separate oscillatory wave propagation from heat-semigroup smoothing.
40 min - 08
Space-time white noise and scaling
Define white noise by test-function covariance and use parabolic power counting without mistaking a subcriticality calculation for a solution theorem.
41 min - 09
Renormalization of singular stochastic products
Check finite Gaussian Wick corrections and explain why field-level renormalization requires distributional convergence and equation-dependent counterterms.
42 min - 10
Regularity structures and paracontrolled overviews
Compare enhanced local expansions and paraproduct expansions, with explicit reconstruction hypotheses and an honest account of the subcritical scope.
43 min