Itô-Stratonovich Conversion in Infinite Dimensions for Unbounded, Time-Dependent, Nonlinear Operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912521376497664 |
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| author | Goodair, Daniel |
| author_facet | Goodair, Daniel |
| contents | We prove that a solution, in a variational framework, to the Stratonovich stochastic partial differential equation with noise $G\left(t, Ψ_t\right) \circ dW_t$ is given by a solution to the Itô equation with Itô-Stratonovich corrector $\frac{1}{2}\sum_{i=1}^\infty D_uG_i\left(t, Ψ_t\right)\left[G_i(t,Ψ_t)\right]dt$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03424 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Itô-Stratonovich Conversion in Infinite Dimensions for Unbounded, Time-Dependent, Nonlinear Operators Goodair, Daniel Probability Analysis of PDEs We prove that a solution, in a variational framework, to the Stratonovich stochastic partial differential equation with noise $G\left(t, Ψ_t\right) \circ dW_t$ is given by a solution to the Itô equation with Itô-Stratonovich corrector $\frac{1}{2}\sum_{i=1}^\infty D_uG_i\left(t, Ψ_t\right)\left[G_i(t,Ψ_t)\right]dt$. 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. |
| title | Itô-Stratonovich Conversion in Infinite Dimensions for Unbounded, Time-Dependent, Nonlinear Operators |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2508.03424 |