Probability theory and stochastic processes Theorem

Fukushima correspondence between Dirichlet forms and processes

Builds on Dirichlet form Hunt process Radon measure Locally compact space Contraction semigroup Beurling–Deny formula

Statement

Let $X$ be a locally compact separable metric space and let $m$ be a positive Radon measure on $X$ with full support. Let $(\mathcal{E}, \mathcal{F})$ be a regular Dirichlet form on $L^2(X, m)$ , and let $(T_t)_{t \ge 0}$ be the strongly continuous symmetric Markovian contraction semigroup on $L^2(X, m)$ associated with it. Then there is an $m$ -symmetric Hunt process $M$ on $X$ whose transition function $(p_t)_{t \ge 0}$ satisfies, for every $t > 0$ and every bounded Borel function $f \in L^2(X, m)$ : $p_t f = T_t f \quad m\text{-almost everywhere}.$ Such a process is unique up to equivalence. That is, two $m$ -symmetric Hunt processes with this property have the same transition functions outside a set of zero capacity for $\mathcal{E}$ . This exceptional set can be chosen so that neither process enters it.

Remarks

The process is built from the semigroup, but the semigroup only acts on classes of functions defined almost everywhere. The difficulty is to choose good pointwise versions, and this is done with the capacity of $\mathcal{E}$ . Every element of $\mathcal{F}$ has a quasi-continuous version, which is continuous outside open sets of small capacity. Regularity of the form provides enough continuous functions in $\mathcal{F}$ to separate points and to build the process on $X$ itself. Moreover $p_t f$ is a quasi-continuous version of $T_t f$ . Example: the form $\tfrac12 \int_{\R^n} \nabla u \cdot \nabla v \, dx$ on $W^{1,2}(\R^n)$ corresponds to Brownian motion on $\R^n$ . Ma and Röckner extended the correspondence to quasi-regular Dirichlet forms, which include forms on infinite-dimensional spaces.

Going deeper

For a student

The theorem is a central result of the monograph of Fukushima, Oshima and Takeda on Dirichlet forms. Together with the Beurling–Deny formula, it gives a meaning to the three parts of the form. The local part, the jumping measure and the killing measure describe the continuous motion, the jumps and the killing of the process. A regular Dirichlet form has the local property exactly when the process is a diffusion, that is, has continuous paths up to its lifetime. A nearly Borel set has zero capacity if and only if the process, started from $m$ -almost every point, never hits it, which gives a probabilistic meaning to capacity.

Reading path

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Contraction semigroup concept
Dirichlet form concept
Hunt process concept
Locally compact space concept
Radon measure concept
Beurling–Deny formula statement
Fukushima correspondence between Dirichlet forms and processes statement

Tag	`5PJF1NDCTT`
URI	`m2://5PJF1NDCTT`
Kind	statement, theorem
Writing	W0 stub
MSC	60J46, 31C25, 60J25
Status	This copy of the site has no build data. Run `m2 status 5PJF1NDCTT`.
About	Dirichlet form, Hunt process
Uses	Dirichlet form, Hunt process, Radon measure, Locally compact space, Contraction semigroup, Beurling–Deny formula