Existence of Hamiltonians for some singular interactions on manifolds

Çağlar Doǧan; Fatih Erman; O. Teoman Turgut

doi:10.1063/1.4705291

CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150600160x ◽

2008 ◽

Vol 51 (2) ◽

pp. 337-362 ◽

Cited By ~ 3

Author(s):

Torben Fattler ◽

Martin Grothaus

Keyword(s):

Boundary Condition ◽

Markov Process ◽

Diffusion Process ◽

Lebesgue Measure ◽

Dirichlet Form ◽

Measure Zero ◽

Particle Systems ◽

Lebesgue Measure Zero ◽

Compact Set ◽

Singular Interactions

AbstractWe give a Dirichlet form approach for the construction and analysis of elliptic diffusions in $\bar{\varOmega}\subset\mathbb{R}^n$ with reflecting boundary condition. The problem is formulated in an $L^2$-setting with respect to a reference measure $\mu$ on $\bar{\varOmega}$ having an integrable, $\mathrm{d} x$-almost everywhere (a.e.) positive density $\varrho$ with respect to the Lebesgue measure. The symmetric Dirichlet forms $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ we consider are the closure of the symmetric bilinear forms\begin{gather*} \mathcal{E}^{\varrho,a}(f,g)=\sum_{i,j=1}^n\int_{\varOmega}\partial_ifa_{ij} \partial_jg\,\mathrm{d}\mu,\quad f,g\in\mathcal{D}, \\ \mathcal{D}=\{f\in C(\bar{\varOmega})\mid f\in W^{1,1}_{\mathrm{loc}}(\varOmega),\ \mathcal{E}^{\varrho,a}(f,f)\lt\infty\}, \end{gather*}in $L^2(\bar{\varOmega},\mu)$, where $a$ is a symmetric, elliptic, $n\times n$-matrix-valued measurable function on $\bar{\varOmega}$. Assuming that $\varOmega$ is an open, relatively compact set with boundary $\partial\varOmega$ of Lebesgue measure zero and that $\varrho$ satisfies the Hamza condition, we can show that $(\mathcal{E}^{\varrho,a},D(\mathcal{E}^{\varrho,a}))$ is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and diffusion process $\bm{M}^{\varrho,a}$ (i.e. an associated strong Markov process with continuous sample paths). Furthermore, since $1\in D(\mathcal{E}^{\varrho,a})$ (due to the Neumann boundary condition) and $\mathcal{E}^{\varrho,a}(1,1)=0$, we obtain a conservative process $\bm{M}^{\varrho,a}$ (i.e. $\bm{M}^{\varrho,a}$ has infinite lifetime). Additionally, assuming that $\sqrt{\varrho}\in W^{1,2}(\varOmega)\cap C(\bar{\varOmega})$ or that $\varrho$ is bounded, $\varOmega$ is convex and $\{\varrho=0\}$ has codimension at least 2, we can show that the set $\{\varrho=0\}$ has $\mathcal{E}^{\varrho,a}$-capacity zero. Therefore, in this case we can even construct an associated conservative diffusion process in $\{\varrho>0\}$. This is essential for our application to continuous $N$-particle systems with singular interactions. Note that for the construction of the self-adjoint generator $(L^{\varrho,a},D(L^{\varrho,a}))$ and the Markov process $\bm{M}^{\varrho,a}$ we do not need to assume any differentiability condition on $\varrho$ and $a$. We obtain the following explicit representation of the generator for $\sqrt{\varrho}\in W^{1,2}(\varOmega)$ and $a\in W^{1,\infty}(\varOmega)$:$$ L^{\varrho,a}=\sum_{i,j=1}^n\partial_i(a_{ij}\partial_j)+\partial_i(\log\varrho)a_{ij}\partial_j. $$Note that the drift term can be singular, because we allow $\varrho$ to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift that is not integrable with respect to the Lebesgue measure.

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Boundary triples for Schrödinger operators with singular interactions on hypersurfaces

Nanosystems Physics Chemistry Mathematics ◽

10.17586/2220-8054-2016-7-2-290-302 ◽

2016 ◽

pp. 290-302 ◽

Cited By ~ 1

Author(s):

J. Behrndt ◽

M. Langer ◽

V. Lotoreichik

Keyword(s):

Schrödinger Operators ◽

Schrodinger Operators ◽

Singular Interactions

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Myosin 1b is an actin depolymerase

Nature Communications ◽

10.1038/s41467-019-13160-y ◽

2019 ◽

Vol 10 (1) ◽

Cited By ~ 5

Author(s):

Julien Pernier ◽

Remy Kusters ◽

Hugo Bousquet ◽

Thibaut Lagny ◽

Antoine Morchain ◽

...

Keyword(s):

Myosin Ii ◽

Actin Dynamics ◽

Myosin Motor ◽

Actin Depolymerization ◽

Cellular Processes ◽

Myosin Motors ◽

Singular Interactions ◽

Sliding Motility

AbstractThe regulation of actin dynamics is essential for various cellular processes. Former evidence suggests a correlation between the function of non-conventional myosin motors and actin dynamics. Here we investigate the contribution of myosin 1b to actin dynamics using sliding motility assays. We observe that sliding on myosin 1b immobilized or bound to a fluid bilayer enhances actin depolymerization at the barbed end, while sliding on myosin II, although 5 times faster, has no effect. This work reveals a non-conventional myosin motor as another type of depolymerase and points to its singular interactions with the actin barbed end.

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