A Stochastic Calculus Approach for the Brownian Snake

2000 ◽  
Vol 52 (1) ◽  
pp. 92-118 ◽  
Author(s):  
Jean-Stéphane Dhersin ◽  
Laurent Serlet
Keyword(s):  
Brownian Motion ◽  
Wide Class ◽  
Stochastic Equation ◽  
Martingale Problem ◽  
Stochastic Calculus ◽  
Brownian Snake ◽  
Super Brownian Motion ◽  
Path Dependent ◽  
Definition Of ◽  
Well Posed

AbstractWe study the “Brownian snake” introduced by Le Gall, and also studied by Dynkin, Kuznetsov, Watanabe. We prove that Itô’s formula holds for a wide class of functionals. As a consequence, we give a new proof of the connections between the Brownian snake and super-Brownian motion. We also give a new definition of the Brownian snake as the solution of a well-posed martingale problem. Finally, we construct a modified Brownian snake whose lifetime is driven by a path-dependent stochastic equation. This process gives a representation of some super-processes.

On a stochastic nonclassical diffusion equation with standard and fractional Brownian motion

2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Stochastic Integrals ◽  
Well Posedness ◽  
Time Data ◽  
Final Time ◽  
Fractional Brownian Motions ◽  
Nonclassical Diffusion Equation ◽  
Definition Of ◽  
Well Posed

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

Wiener's test for super-Brownian motion and the Brownian snake

1997 ◽  
Vol 108 (1) ◽  
pp. 103-129 ◽  
Author(s):  
Jean-Stéphane Dhersin ◽  
Jean-François Le Gall
Keyword(s):  
Brownian Motion ◽  
Brownian Snake ◽  
Super Brownian Motion

Singular Spacetime Itô's Integral and a Class of Singular Interacting Branching Particle Systems

2003 ◽  
Vol 06 (02) ◽  
pp. 321-335 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Brownian Motion ◽  
Unique Solution ◽  
Martingale Problem ◽  
Particle Systems ◽  
Dual Process ◽  
Singular Function ◽  
Singular Coefficient ◽  
Duality Method ◽  
Super Brownian Motion ◽  
Branching Diffusions

In Wang,8 a class of interacting measure-valued branching diffusions [Formula: see text] with singular coefficient were constructed and characterized as a unique solution to ℒε-martingale problem by a limiting duality method since in this case the dual process does not exist. In this paper, we prove that for any ε ≠ 0 the superprocess with singular motion coefficient is just the super-Brownian motion. The singular motion coefficient is handled as a sequential limit motivated by Antosik et al.1 Thus, the limiting superprocess is investigated and identified as the motion coefficient converges to a singular function. The representation of the singular spacetime Itô's integral is derived.

A Schilder Type Theorem for Super-Brownian Motion

1996 ◽  
Vol 48 (3) ◽  
pp. 542-568 ◽  
Author(s):  
Klaus Fleischmann ◽  
Jürgen Gärtner ◽  
Ingemar Kaj
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Stochastic Equation ◽  
Large Deviation ◽  
Type Theorem ◽  
Space Time ◽  
Path Space ◽  
Super Brownian Motion ◽  
Large Deviation Probabilities

AbstractLet X be a d-dimensional continuous super-Brownian motion with branching rate ε, which might be described symbolically by the "stochastic equation" a space-time white noise. A Schilder type theorem is established concerning large deviation probabilities of X on path space as ε → 0, with a representation of the rate functional via an L2 -functional on a generalized "Cameron-Martin space" of measure-valued paths.

scholarly journals A note on symmetries of diffusions within a martingale problem approach

2019 ◽  
Vol 19 (02) ◽  
pp. 1950011 ◽  
Author(s):  
Francesco C. De Vecchi ◽  
Paola Morando ◽  
Stefania Ugolini
Keyword(s):  
Diffusion Processes ◽  
Martingale Problem ◽  
Second Order ◽  
Lie Symmetries ◽  
Definition Of

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second-order geometry and Itô integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.

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