## The Annals of Probability

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

#### Abstract

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

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

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

Dates
First available in Project Euclid: 18 June 2002

https://projecteuclid.org/euclid.aop/1024404415

Digital Object Identifier
doi:10.1214/aop/1024404415

Mathematical Reviews number (MathSciNet)
MR1443191

Zentralblatt MATH identifier
0880.60080

#### Citation

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. https://projecteuclid.org/euclid.aop/1024404415

#### References

• [1] Adams, D. R. and Hedberg, L. I. (1996). Function Spaces and Potential Theory. Springer, Berlin.
• [2] Baras, P. and Pierre, M. (1984). Singularit´es ´eliminable pour des ´equations semi-lin´eares. Ann. Inst. Fourier (Grenoble) 34 185-206.
• [3] Baras, P. and Pierre, M. (1984). Problems paraboliques semi-lin´eares avec donnees measures. Appl. Anal. 18 111-149.
• [4] Dawson, D. A. (1991). Measure-valued Markov processes. Ecole d'Et´e de Probabilit´es de Saint-Flour XXI Lecture Notes in Math. 1464 1-260. Springer, Berlin.
• [5] Dellacherie, C. and Meyer, P.-A. (1975, 1980, 1983, 1987). Probabilit´es et potentiel. Hermann, Paris.
• [6] Dynkin, E. B. (1965). Markov Processes. Springer, Berlin.
• [7] Dynkin, E. B. (1978). Sufficient statistics and extreme points. Ann. Probab. 6 705-730.
• (Reprinted in E. B. Dynkin. (1982). Markov processes and related problems of analysis. London Math. Soc. Lecture Note Ser. 54. Cambridge Univ.)
• [8] Dynkin, E. B. (1989). Superprocesses and their linear additive functionals. Trans. Amer. Math. Soc. 314 255-282.
• [9] Dynkin, E. B. (1991). Additive functionals of superdiffusion processes. In Random Walks, Brownian Motion and Interacting Particle Systems (R. Durrett and H. Kesten, eds.) 269-281. Birkh¨auser, Boston.
• [10] Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Probab. 19 1157-1194.
• [11] Dynkin, E. B. (1991). Path processes and historical superprocesses. Probab. Theory Related Fields 90 89-115.
• [12] Dynkin, E. B. (1992). Superdiffusions and parabolic nonlinear differential equations. Ann. Probab. 20 942-962.
• [13] Dynkin, E. B. (1993). On regularity of superprocesses. Probab. Theory Related Fields 95 263-281.
• [14] Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185-1262.
• [15] Dynkin, E. B. (1994). An Introduction to Branching Measure-Valued Processes. Amer. Math. Soc., Providence, RI.
• [16] Dynkin, E. B. and Kuznetsov, S. E. (1996). Linear additive functionals of superdiffusions and related nonlinear P.D.E. Trans. Amer. Math. Soc. 348 1959-1987.
• [17] Dynkin, E. B. and Kuznetsov, S. E. (1997). Natural linear additive functionals of superprocesses. Ann. Probab. 25 640-661.
• [18] Dynkin, E. B. and Kuznetsov, S. E. (1996). Solutions of Lu = u dominated by L-harmonic functions. J. Analyse Math. 68 15-37.
• [19] Dynkin, E. B. and Kuznetsov, S. E. (1996). Superdiffusions and removable singularities for quasilinear partial differential equations. Comm. Pure Appl. Math. 49 125-176.
• [20] Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64 337-361.
• [21] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
• [22] Gmira, A. and V´eron, L. (1991). Boundary singularities of solutions of some nonlinear elliptic equations. Duke Math. J. 64 271-324.
• [23] Iscoe, I. (1986). A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields 71 85-116.
• [24] Kuznetsov, S. E. (1974). Construction of Markov processes with random birth and death. Theoret. Probability Appl. 18 571-574.
• [25] Le Gall, J.-F. (1995). The Brownian snake and solutions of u = u2 in a domain. Probab. Theory Related Fields 102 393-432.