Mild solution of the heat equation with a general stochastic measure

Radchenko, V. M.

doi:10.4064/sm194-3-2

Cited by 33 publications

(35 citation statements)

References 11 publications

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“…A similar problem for µ dependent on the spatial variable x was considered in [5]. The stochastic heat equation on fractals was studied in [7], and a review of results on equations driven by SMs is given in [6].…”

Section: Introductionmentioning

confidence: 99%

Heat equation with general stochastic measure colored in time

Radchenko¹

2014

Modern Stoch. Theory Appl.

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Section: Introductionmentioning

confidence: 99%

Heat equation with general stochastic measure colored in time

Radchenko¹

2014

Modern Stoch. Theory Appl.

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“…Equations with sub-Gaussian measures were intensively studied in [6,7]. The articles [10,11] consider equations with general stochastic measures.…”

Section: Introductionmentioning

confidence: 99%

Approximations for a solution to stochastic heat equation with stable noise

Pryhara¹,

Shevchenko²

2016

Modern Stoch. Theory Appl.

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“…The existence and uniqueness of a solution of this equation is proved in [8] under some conditions. It is also proved in [8] that trajectories of a solution satisfy the Hölder condition.…”

Section: Stochastic Heat Equationmentioning

confidence: 99%

“…We extend the definition of h for x < 0 by setting h(x) = 0 for negative arguments. Then condition (8) holds for all x ≤ T and…”

Section: Then There Exists a Version Of The Random Functionmentioning

confidence: 99%

“…In [7] and [8], solutions of some partial differential equations are considered for the case where the randomness comes from an integral with respect to a stochastic measure μ. In this paper, we generalize some of the results of [5,6] to the case of X = R and find a relationship between weak and mild solutions of a stochastic heat equation.…”

Section: If X(x) a ≤ X ≤ B Is A Continuous Square Integrable Martinmentioning

confidence: 99%

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Properties of integrals with respect to a general stochastic measure in a stochastic heat equation

Radchenko

2011

Theor. Probability and Math. Statist.

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Abstract. We prove a theorem on the continuity with respect to a parameter and an analogue of Fubini's theorem for integrals with respect to a general stochastic measure defined on Borel subsets of R. These results are applied to study the stochastic heat equation considered in a mild as well as in a weak form. IntroductionLet X be an arbitrary set, let B be the σ-algebra of subsets of X, and let (Ω, F, P) be a complete probability space. By L 0 = L 0 (Ω, F, P), we denote the set of all random variables (more precisely, the P-equivalent classes of random variables). The convergence in L 0 means the convergence in probability.The measure μ is not assumed to be nonnegative or adapted. In what follows the symbol μ denotes a stochastic measure on B.Below are some examples of stochastic measures.is a stochastic measure defined on Borel subsets of [a, b]. 2. Similarly, the integral with respect to a fractional Brownian motion B H (x) with the Hurst index H > 1/2 defines a stochastic measure (this follows from inequality (1.5) of [1]).3. Other examples and conditions that increments of a stochastic process with independent increments generate a stochastic measure can be found in Sections 7 and 8 of [2].An integral of the form A h(x) dμ(x), A ∈ B, is constructed and its properties are studied in [3] for a measurable nonrandom function h : X → R. The construction is standard and uses an approximation by simple functions. (A similar construction is presented in Section 7 of [2]; also see [4]). In particular, every measurable bounded function h is integrable with respect to an arbitrary stochastic measure μ. An analogue 2010 Mathematics Subject Classification. Primary 60G57, 60H15, 60H05.

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Mild solution of the heat equation with a general stochastic measure

Abstract: Abstract. The stochastic heat equation on [0, T ] × R driven by a general stochastic measure is investigated. Existence and uniqueness of the solution is established. Hölder regularity of the solution in time and space variables is proved.

Cited by 33 publications

References 11 publications

Heat equation with general stochastic measure colored in time

Heat equation with general stochastic measure colored in time

Approximations for a solution to stochastic heat equation with stable noise

Properties of integrals with respect to a general stochastic measure in a stochastic heat equation

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