Annals of Probability

Nonlinear parabolic P.D.E.\ and additive functionals of superdiffusions

E. B. Dynkin and S. E. Kuznetsov

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Suppose that $E$ is an arbitrary domain in $\mathbb{R}^d$, $L$ is a second order elliptic differential operator in $S = \mathbb{R}_+ \times E$ and $S^e$ is the extremal part of the Martin boundary for the corresponding diffusion $\xi$. Let $1 < \alpha \leq 2$. We investigate a boundary value problem

$$\tag{*}\begin{eqnarray} \frac{\partial u}{\partial r}+Lu−u^\alpha&=&−\eta \quad \textrm{in }S,\\ u&=&v \quad \textrm{on }S^e,\\ u&=&0 \quad \textrm{on }\{\infty\} × E \end{eqnarray}$$

involving two measures $\eta$ and $\nu$. For the existence of a solution, we give sufficient conditions in terms of a Martin capacity and necessary conditions in terms of hitting probabilities for an $(L, \alpha)$-superdiffusion $X$. If a solution exists, then it can be expressed by an explicit formula through an additive functional $A$ of $X$.

An $(L, \alpha)$-superdiffusion is a branching measure-valued process. A natural linear additive (NLA) functional $A$ of $X$ is determined uniquely by its potential $h$ defined by the formula $P_{\mu} A(0, \infty) = \int h(r, x) \mu (dr, dx)$ for all $\mu \in \mathscr{M}^*$ (the determining set of $A$). Every potential $h$ is an exit rule for $\xi$ and it has a unique decomposition into extremal exit rules. If $\eta$ and $\nu$ are measures which appear in this decomposition, then (*)can be replaced by an integral equation

$$\tag{**} u(r, x) + \int p(r, x; t, dy)u(t, y)^{\alpha} ds = h(r, x),$$

where $p(r, x; t, dy)$ is the transition function of $\xi$. We prove that h is the potential of a NLA functional if and only if (**) has a solution $u$. Moreover,

$$u(r, x) = -\log P_{r, x} e^{-A(0, \infty)}.$$

By applying these results to homogeneous functionals of time-homogeneous superdiffusions, we get a stronger version of theorems proved in an earlier publication. The foundation for our present investigation is laid by a general theory developed in the accompanying paper.

Article information

Ann. Probab., Volume 25, Number 2 (1997), 662-701.

First available in Project Euclid: 18 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 31C15: Potentials and capacities 60J25: Continuous-time Markov processes on general state spaces 60J55: Local time and additive functionals 31C45: Other generalizations (nonlinear potential theory, etc.) 35J60: Nonlinear elliptic equations

Measure-valued processes branching natural linear additive functionals probabilistic solutions of PDEs boundary value problems involving measures potentials log-potentials capacities polar sets


Dynkin, E. B.; Kuznetsov, S. E. Nonlinear parabolic P.D.E.\ and additive functionals of superdiffusions. Ann. Probab. 25 (1997), no. 2, 662--701. doi:10.1214/aop/1024404415.

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