Stochastic damped wave and Euler-Bernoulli equations with singular locally Hölder continuous drift
Abstract
Let U and H be two separable Hilbert spaces and T>0. We consider a stochastic differential equation which evolves in the Hilbert space H of the form labelSDEa dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), t∈[0,T], X(0)=x ∈ H, where A:D(A)⊆ H→ H is the infinitesimal generator of a strongly continuous semigroup (etA)t≥0, W=(W(t))t≥0 is a U-cylindrical Wiener process defined on a normal filtered probability space (Ω,ℱ,{ℱt}t∈ [0,T],ℙ), G:U→ H is a linear bounded operator and B:[0,T]× H→ Hβ is a locally θ-Hölder continuous function with respect to the second variable, uniformly with respect to the first one, for some suitable θ∈(0,1). Here, Hβ is a Hilbert space which contains H with continuous embedding, so that the drift term is singular: it is not well-defined from the ambient space H into itself. The coefficient β≥0 measures the order of such a singularity and the case β=0 corresponds to a drift term with values in H. We prove that, under suitable assumptions on the coefficients, weak and pathwise uniqueness hold true for equation \eqref{SDEa}. 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 up to dimension $3$, even in the hyperbolic case.
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BibTeX
@misc{addona2026stochastic,
title = {{Stochastic damped wave and Euler-Bernoulli equations with singular locally Hölder continuous drift}},
author = {Davide Addona and Davide Augusto Bignamini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03819},
eprint = {2609.03819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03819},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Stochastic damped wave and Euler-Bernoulli equations with singular locally Hölder continuous drift
AU - Davide Addona
AU - Davide Augusto Bignamini
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03819
UR - https://arxiv.org/abs/2609.03819
AB - Let U and H be two separable Hilbert spaces and T>0. We consider a stochastic differential equation which evolves in the Hilbert space H of the form labelSDEa dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), t∈[0,T], X(0)=x ∈ H, where A:D(A)⊆ H→ H is the infinitesimal generator of a strongly continuous semigroup (e^tA)_t≥0, W=(W(t))_t≥0 is a U-cylindrical Wiener process defined on a normal filtered probability space (Ω,ℱ,{ℱ_t}_t∈ [0,T],ℙ), G:U→ H is a linear bounded operator and B:[0,T]× H→ H_β is a locally θ-Hölder continuous function with respect to the second variable, uniformly with respect to the first one, for some suitable θ∈(0,1). Here, H_β is a Hilbert space which contains H with continuous embedding, so that the drift term is singular: it is not well-defined from the ambient space H into itself. The coefficient β≥0 measures the order of such a singularity and the case β=0 corresponds to a drift term with values in H. We prove that, under suitable assumptions on the coefficients, weak and pathwise uniqueness hold true for equation \eqref{SDEa}. 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 up to dimension $3$, even in the hyperbolic case.
ER -