A Signed Measure on Path Space Related to Wiener Measure

Hochberg, Kenneth J.

doi:10.1214/aop/1176995529

Cited by 125 publications

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“…The return probability is defined as the spatial average of P , i.e., 8 Parabolic equations of the type (87) were studied in Refs. [86,[104][105][106][110][111][112] and generalized to higher-order operators ∇ n x and several other forms. the trace of the heat kernel per unit volume,…”

Section: G Spectral and Walk Dimensionsmentioning

confidence: 99%

Diffusion in multiscale spacetimes

Calcagni

2013

Phys. Rev. E

140

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We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples, and the most general spectral-dimension profile of multifractional spaces is constructed.

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Section: G Spectral and Walk Dimensionsmentioning

confidence: 99%

Diffusion in multiscale spacetimes

Calcagni

2013

Phys. Rev. E

140

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“…Introduction* In recent years some papers have appeared related to signed measures on function spaces (see [3] and [4]). This paper extends certain results for probability measures on function spaces to finite signed measures (FSM's) on such spaces.…”

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Finite signed measures on function spaces

Haagen¹

1981

Pacific J. Math.

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Some results for probability measures on function spaces are extended to finite signed measures (FSM's). In particular FSM's on the space of continuous functions and right-continuous functions with left-hand limits are patched together by a procedure of Stroock and Varadhan. Given an increasing sequence of stopping times the procedure is carried out repeatedly. A sequence of transition functions and, an extension result for the linear maps associated with these transition functions are obtained.Introduction* In recent years some papers have appeared related to signed measures on function spaces (see [3] and [4]). This paper extends certain results for probability measures on function spaces to finite signed measures (FSM's) on such spaces. A more detailed discussion can be found in [8].In part I we introduce conditional FSM's and consider the existence of a regular conditional distribution (ROD) of an FSM on a standard measurable space mimicking Chapter V of [5]. Further, the Jordan decomposition of an RCD is investigated. We then consider a sequence of transition functions and associate linear maps between Banach spaces of FSM's with these transition functions. An extension result for the linear maps is obtained.In part II FSM's on Ω = C([0, oo); S) and Ω = D ([0, <»); S) with S a separable metric space are patched together by the procedure used in [6] for probability measures on C([0, oo); E d ). In fact, if * is the σ-field on Ω generated by the coordinate projections {X u t ^> s} and τ an s-stopping time with respect to the (/-fieldŝ f t

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“…Specifically, several authors have extended such results to signed measures. For example, Hochberg [4] derives generalizations of the Brownian motion or Wiener process to higher-order signed stochastic processes, an analogue of the Itô stochastic calculus for these signed processes, and a probabilistic-style analysis of the spectral properties of higherorder elliptic operators.…”

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Central limit theorem for signed distributions

Hochberg¹

1980

Proc. Amer. Math. Soc.

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Abstract. This paper contains an improved version of existing generalized central limit theorems for convergence of normalized sums of independent random variables distributed by a signed measure. It is shown that under reasonable conditions, the normalized sums converge in distribution to "higher-order" analogues of the standard normal random variable, in the sense that the density of the limiting signed distribution is the fundamental solution of a higher-order parabolic partial differential equation that is a generalization of the heat equation.1. Introduction. Probability theory is often thought of as the study of nonnegative bounded measures. However, several theorems usually considered to be purely probabilistic are, in fact, true in more general contexts. Specifically, several authors have extended such results to signed measures. For example, Hochberg [4] derives generalizations of the Brownian motion or Wiener process to higher-order signed stochastic processes, an analogue of the Itô stochastic calculus for these signed processes, and a probabilistic-style analysis of the spectral properties of higherorder elliptic operators.Similarly, the central limit theorem for convergence of normalized sums of random variables to the Gaussian distribution has been extended to the case where the random functions are distributed by a signed measure. Zukov [8] derives such a theorem for difference operators in a study of a problem in numerical analysis. Studnev [6] states, without proof, other generalizations, with an application to probability theory itself. Hersh [3] gives a more complete study of generalized central limit theorems, though his results do not apply to the cases n = 0 (mod 2) in what follows; that is, Hersh's theorems exclude the cases where the signed distribution has a density function which is the fundamental solution of the parabolic partial differential equation (2) below whose order is a multiple of four. Our techniques include these cases.In an earlier paper (Hochberg [4]), we derived two such central limit theorems, which we restate in the following two propositions. Here

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A Signed Measure on Path Space Related to Wiener Measure

Cited by 125 publications

References 0 publications

Diffusion in multiscale spacetimes

Diffusion in multiscale spacetimes

Finite signed measures on function spaces

Central limit theorem for signed distributions

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