Itô-Stratonovich Conversion in Infinite Dimensions for Unbounded, Time-Dependent, Nonlinear Operators

2025/08/05 by Goodair, Daniel
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2508.03424

Abstract

We prove that a solution, in a variational framework, to the Stratonovich stochastic partial differential equation with noise G(t, Ψ_t) ∘ dW_t is given by a solution to the Itô equation with Itô-Stratonovich corrector (1)/(2)∑_i=1^∞ D_uG_i(t, Ψ_t)[G_i(t,Ψ_t)]dt. Here G_i denotes the action of G on the i^th component of the cylindrical noise, and D_uG_i its Fréchet partial derivative in the Hilbert space for which the Itô form is satisfied. The noise operator G may be time-dependent, nonlinear, and unbounded in the sense of differential operators; in the latter case, one must pass to a larger space in order to solve the Stratonovich equation. Our proof relies on martingale techniques, and the results apply to fluid equations with time-dependent and nonlinear transport noise.

Citations

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Related

• Stochastic Calculus in Infinite Dimensions and SPDEs
• Stochastic PDEs with correlated, non-stationary Stratonovich noise of Dean--Kawasaki type
• Stratonovich solution for the wave equation
• Variational solutions to nonlinear stochastic differential equations in Hilbert spaces
• Existence and convergence results for infinite dimensional nonlinear\n stochastic equations with multiplicative noise