Gauge Theorem Research Articles

The Hole Argument without the notion of isomorphism

In this paper, I argue that the Hole Argument can be formulated without using the notion of isomorphism, and for this reason it is not threatened by the criticism of Halvorson and Manchak (Br J Philos Sci, 2022. https://doi.org/10.1086/719193). Following Earman and Norton (Br J Philos Sci 38, pp. 515–525, 1987), I divide the Hole Argument into two steps: the proof of the Gauge Theorem and the transition from the Gauge Theorem to the conclusion of radical indeterminism. In the analaysis of the first step, I argue that the Gauge Theorem does not rely on the notion of isomorphism but on the notion of the diffeomorphism-invariance of the equations of local spacetime theories; however, for this approach to work, the definition of local spacetime theories needs certain amendments with respect to Earman and Norton’s formulation. In the analysis of the second step, I postulate that we should use the notion of radical indeterminism instead of indeterminism simpliciter and that we should not decide in advance what kind of maps are to be used in comparing models. Instead, we can tentatively choose some kind of maps for this purpose and check whether a given choice leads to radical indeterminism involving empirically indistinguishable models. In this way, the use of the notion of isomorphism is also avoided in the second step of the Hole Argument. A general picture is that physical equivalence can be established by means of an iterative procedure in which we examine various candidate classes of maps, and, depending on the outcomes, we need to broaden or narrow these classes. The Hole Argument can be viewed as a particular instance of this procedure.

F(R) cosmology via Noether symmetry and Λ-Chaplygin Gas like model

In this work, we consider f(R) alternative theories of gravity with an eye to Noether symmetry through the gauge theorem. For non-vacuum models, one finds Λ like gravity with energy density of Chaplygin Gas. We also obtain the effective equation of state parameter for corresponding cosmology and scale factor behavior with respect to cosmic time which show that the model provides viable EoS and scale factor with respect to observational data.

Boundary Harnack principle for the absolute value of a one-dimensional subordinate Brownian motion killed at 0

We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting 0, when 0 is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0,\infty )$.

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains. Informally the Schrödinger-type operators we consider are of the form L + μ ⋅ ∇ + ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to K d , 1 and ν is a signed measure belonging to K d , 2 . In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C 1 , 1 -domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.

Conditional gauge theorem for non-local Feynman-Kac transforms

Feynman-Kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of Markov processes. General gauge and conditional gauge theorems are established for such transforms. Furthermore, the L 2 -infinitesimal generator of the Schrödinger semigroup given by a non-local Feynman-Kac transform is determined in terms of its associated bilinear form.

Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains

For a symmetric $\alpha$-stable process $X$ on $\mathbf{R}^{n}$ with $0 \lt \alpha \lt 2$, $n \geq 2$ and a domain $D \subset \mathbf{R}^{n}$, let $L^{D}$ be the infinitesimal generator of the subprocess of $X$ killed upon leaving $D$. For a Kato class function $q$, it is shown that $L^{d}+q$ is intrinsic ultracontractive on a Holder domain $D$ of order 0. Then this is used to establish the conditional gauge theorem for $X$ on bounded Lipschitz domains in $\mathbf{R}^{n}$. It is also shown that the conditional lifetimes for symmetric stable process in a Holder domain of order 0 are uniformly bounded.

Boundary Harnack Principle for Symmetric Stable Processes

In this paper we study potential-theoretic properties of the symmetric α-stable processes (0<α<2): establishing the boundary Harnack principle for ratios of α-harmonic functions on any open sets, identifying the Martin boundary with the Euclidean boundary for open sets with a certain interior fatness property, and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets.

The generalized neumann problem for yang-mills connections

The purpose of this paper is defining a new boundary value problem for Yang-Mills connections, which is the most general in the context of Neumann-type problems for forms. We achieve this by reflecting the base manifold across the boundary, and lifting this action non-trivially to the bundle. This way we obtain a twisted boundary value problem in which the boundary conditions are mixed, of Dirichlet type on some of the Lie-algebra components of the connection A, of Neumann type on others. This problem arises naturally and it can be viewed in the context of generalizing non-linear Hodge theory for connections. We prove a good gauge theorem for this problem. We give an application.

Martin Boundary and Integral Representation for Harmonic Functions of Symmetric Stable Processes

Martin boundaries and integral representations of positive functions which are harmonic in a bounded domainDwith respect to Brownian motion are well understood. Unlike the Brownian case, there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process. One kind are functions harmonic inDwith respect to the whole processX, and the other are functions harmonic inDwith respect to the processXDkilled upon leavingD. In this paper we show that for bounded Lipschitz domains, the Martin boundary with respect to the killed stable processXDcan be identified with the Euclidean boundary. We further give integral representations for both kinds of positive harmonic functions. Also given is the conditional gauge theorem conditioned according to Martin kernels and the limiting behaviors of theh-conditional stable process, wherehis a positive harmonic function ofXD. In the case whenDis a boundedC1,1domain, sharp estimate on the Martin kernel ofDis obtained.

Intrinsic Ultracontractivity and Conditional Gauge for Symmetric Stable Processes

It is shown in this paper that the conditional gauge theorem holds for symmetricα-stable processes on boundedC1,1domains inRnwhere 0<α<2 andn⩾2. Two of the major tools used to prove this conditional gauge theorem are logarithmic Sobolev inequality and intrinsic ultracontractivity.

Conditioned Brownian motion in planar domains

We give an upper bound for the Green functions of conditioned Brownian motion in planar domains. A corollary is the conditional gauge theorem in bounded planar domains.

EXACT-VOLUME DIFFERENTIAL FORM DE RHAM COHOMOLOGY AND HAMILTON VARIATIONAL PRINCIPLE

In this paper, we suggested exact-volume differential form (for short: EVDF) and proved four theorems correlative with them: 1. existence theorem, 2. cohomology theorem, 3. constant multiple theorem, and 4. equal gauge theorem. And their application were discussed also. For examlpe, we deduced the particle dynamic equation of the special theory of relativity. At the same time we analyzed and contrasted cohomology theory with Hamilton's variational principle. The contrast showed the superiority of cohomology theory. Moreover, we gave a more complete classification list of differential forms.

The gauge theorem for a class of additive functionals of zero energy

The gauge theorem for a class of additive functionals of zero energy

Probabilistic approach to the Dirichlet problem of perturbed stable processes

LetXt be a symmetric stable process of index α, 0<α<2, inRd (d≧2). In this paper we deal with the perturbation ofXt by a multiplicative functional of the following form: $$M_t = \exp \left\{ {\sum\limits_{s\mathop \leqslant \limits_ - t} {F(X_{s - } ,X_s )} } \right\}$$ withF being a function onRd×Rd satisfying certain conditions. First we prove the following gauge theorem: IfD is a bounded open domain ofRd, then the functiong(x)=Ex{M(τD)} is either identically infinite onD or bounded onD, where τD is the first exit time fromD. Then we formulate the Dirichlet problem associated with the perturbed symmetric stable process by using Dirichlet form theory. Finally we apply the gauge theorem to prove the existence and uniqueness of solutions to the Dirichlet problem mentioned above.

The gauge theorem for a class of additive functionals of zero energy

In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ∫ 0 q(B s )ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures μ in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.

The probabilistic solution of the dirichlet problem for $$\tfrac{1}{2}\Delta + \left\langle {a,\nabla } \right\rangle + b$$ with singular coefficients

We investigate weak solutions ofLu=−p, whereL=\(\tfrac{1}{2}\Delta + \left\langle {a,\nabla } \right\rangle + b\) and |a|2,b andp belong to the Kato classKd (d≥3). We shall characterize the weak solutions by a probabilistic mean value property. This characterization includes a continuity principle. Our second regularity result states that for a weak solutionu, |∇u|2 belongs locally to Kd. Regarding the inhomogeneous Dirichlet problem forL, we shall prove the corresponding gauge theorem and an existence and uniqueness result.

Gauge theorems for resolvents with application to Markov processes

LetV=(Vα)α≧0 be a (not necessarily sub-Markovian) resolvent such that the kernelVα for some α≧0 is compact and irreducible. We prove the following general gauge theorem: If there exists at least oneV-excessive function which is notV-inviriant, thenV0 is bounded.

Gauge and conditional gauge on negatively curved graphs

In this paper we study certain edge-weighted random walks on infinite graphs with bounded vertex degree for which the second smallest eigenvalue of the Laplacian is negative.We find analogues of the Feynman-Kac functional and give several conditions equivalent to the boundeness of the corresponding gauge function.From this result we derive a new formula for the second smallest eigenvalue of the Laplacian and we apply this formula to the theory of graph coverings.An analogue of the conditional gauge theorem is shown to hold for certain Schrodinger operators.

A Conformal Inequality Related to the Conditional Gauge Theorem

A Conformal Inequality Related to the Conditional Gauge Theorem

The probabilistic solution of the third boundary value problem for second order elliptic equations

Using standard reflected Brownian motion (SRBM) and martingales we define (in the spirit of Stroock and Varadhan-see [S-V]) the probabilistic solution of the boundary value problem $$\begin{gathered} \frac{1}{2}\Delta u + qu = 0, in D; \hfill \\ \frac{{\partial u}}{{\partial n}} + cu = - f,on\partial D, \hfill \\ \end{gathered} $$ whereD is a bounded domain withC 3 boundary andn is the inward unit oormal vector on ∂D. The assumptions forq, c andf are quite general. The corresponding Dirichlet problem was studied by Chung, Rao, Zhao and others (see [C-R1] and [Z-M]) and the corresponding Neumann by Pei Hsu in [H2]. Here we show that the probabilistic solution of our problem exists, is unique (unless we hit an eigenvalue), continuous on $$\bar D$$ and equivalent to the weak analytic solution. The method we use is to reduce the problem to an integral equation inD that involves the associated semigroup and, hence, to the study of the properties of this semigroup. In this way we do not have to assume that the spectrum is negative (almost every previous work on these probabilistic solutions makes this assumption). We construct the kernel of this semigroup and we prove certain estimates for it which help us to establish many other results, including the gauge theorem. We also show that, if the boundary functionc is continuous, our semigroup is a uniform limit of Neumann semigroups and, furthermore, that the Dirichlet semigroup is a uniform limit of semigroups of our type. Therefore the Dirichlet spectrum is a “monotone” limit of spectra of mixed problems (see Sect. 5B), a fact which is mentioned without proof in Vol 1, Ch. IV, Sect. 2 of theMethods of Mathematical Physics by Courant and Hilbert. This establishes the interrelation of the three boundary value problems. Finally, we add a drift term to our differential equation, which becomes $$\frac{1}{2}\Delta u + b \cdot \nabla u + qu = 0$$ and we solve the third boundary value problem for this equation probabilistically, with the help of Girsanov's transformation.