Weak uniqueness for stochastic partial differential equations in Hilbert spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929733819695104 |
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| author | Addona, Davide Bignamini, Davide Augusto |
| author_facet | Addona, Davide Bignamini, Davide Augusto |
| contents | Let $U,H$ be two separable Hilbert spaces. The main goal of this paper is to study the weak uniqueness of the Stochastic Differential Equation evolving in $H$ \begin{align*} dX(t)=AX(t)dt+\mathcal{V}B(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x \in H, \end{align*} where $\{W(t)\}_{t\geq 0}$ is a $U$-cylindrical Wiener process, $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup, $\mathcal{V},G:U\rightarrow H$ are linear bounded operators and $B:H\rightarrow U$ is a uniformly continuous function. The abstract result in this paper gives the weak uniqueness for large classes of heat and damped equations in any dimension without any Hölder continuity assumption on $B$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_19572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak uniqueness for stochastic partial differential equations in Hilbert spaces Addona, Davide Bignamini, Davide Augusto Probability Analysis of PDEs Let $U,H$ be two separable Hilbert spaces. The main goal of this paper is to study the weak uniqueness of the Stochastic Differential Equation evolving in $H$ \begin{align*} dX(t)=AX(t)dt+\mathcal{V}B(X(t))dt+GdW(t), \quad t>0, \quad X(0)=x \in H, \end{align*} where $\{W(t)\}_{t\geq 0}$ is a $U$-cylindrical Wiener process, $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup, $\mathcal{V},G:U\rightarrow H$ are linear bounded operators and $B:H\rightarrow U$ is a uniformly continuous function. The abstract result in this paper gives the weak uniqueness for large classes of heat and damped equations in any dimension without any Hölder continuity assumption on $B$. |
| title | Weak uniqueness for stochastic partial differential equations in Hilbert spaces |
| topic | Probability Analysis of PDEs |
| url | https://arxiv.org/abs/2502.19572 |