Pathwise uniqueness in infinite dimension under weak structure conditions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909543906148352 |
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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 and $T>0$. We consider an SDE which evolves in the Hilbert space $H$ of the form \begin{align} dX(t)=AX(t)dt+\widetilde{\mathscr L}B(X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(Ω,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $B:H\to H$ is a bounded and $θ$-Hölder continuous function, for some suitable $θ\in(0,1)$, and $\widetilde{\mathscr L}:H\to H$ and $G:U\to H$ are linear bounded operators. We prove that, under suitable assumptions on the coefficients, the weak mild solution to the equation depends on the initial datum in a Lipschitz way. This implies that pathwise uniqueness holds true. Here, the presence of the operator $Λ$ plays a crucial role. In particular the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler--Bernoulli Beam equation upto dimension $3$ even in the hyperbolic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pathwise uniqueness in infinite dimension under weak structure conditions Addona, Davide Bignamini, Davide Augusto Probability 60H15, 60H50 Let $U,H$ be two separable Hilbert spaces and $T>0$. We consider an SDE which evolves in the Hilbert space $H$ of the form \begin{align} dX(t)=AX(t)dt+\widetilde{\mathscr L}B(X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(Ω,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $B:H\to H$ is a bounded and $θ$-Hölder continuous function, for some suitable $θ\in(0,1)$, and $\widetilde{\mathscr L}:H\to H$ and $G:U\to H$ are linear bounded operators. We prove that, under suitable assumptions on the coefficients, the weak mild solution to the equation depends on the initial datum in a Lipschitz way. This implies that pathwise uniqueness holds true. Here, the presence of the operator $Λ$ plays a crucial role. In particular the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler--Bernoulli Beam equation upto dimension $3$ even in the hyperbolic case. |
| title | Pathwise uniqueness in infinite dimension under weak structure conditions |
| topic | Probability 60H15, 60H50 |
| url | https://arxiv.org/abs/2405.14819 |