Absolute Continuity of Measures Corresponding to Diffusion Processes in Banach Space

Absolute Continuity of Symmetric Markov Processes

We study Girsanov’s theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of “gradient type”. We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the cas...

Banach Space of Absolute Summable Complex Sequences

The subset the set of l1-complex sequences of the linear space of complex sequences is defined by the condition (Def. 1). (Def. 1) Let x be a set. Then x ∈ the set of l1-complex sequences if and only if x ∈ the set of complex sequences and idseq(x) is absolutely summable. The following proposition is true (1) Let c be a Complex, s1 be a complex sequence, and r1 be a sequence of real numbers. Su...

Banach Space of Absolute Summable Real Sequences

The subset the set of l1-real sequences of the linear space of real sequences is defined by the condition (Def. 1). (Def. 1) Let x be a set. Then x ∈ the set of l1-real sequences if and only if x ∈ the set of real sequences and idseq(x) is absolutely summable. Let us observe that the set of l1-real sequences is non empty. One can prove the following two propositions: (1) The set of l1-real sequ...

Absolute continuity for some one-dimensional processes

We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with Hölder continuous coefficients. Fu...

Absolute Continuity and Singularity of Probability Measures in Functional Spaces