# On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups

## Abstract

We consider the semilinear Cauchy problem for a class of pseudo-differential operators generating sub-Markovian semigroups. Solutions of such problems with negative definite nonlinearity play an important role in constructing branching measure-valued processes. We establish local existence and uniqueness of solutions in the context of the Dirichlet space associated to the problem. Comparison and global properties of solutions are also studied.

- Authors:

- Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales (United Kingdom), E-mail: N.Jacob@swansea.ac.uk
- Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW (United Kingdom), E-mail: V.Moroz@bristol.ac.uk

- Publication Date:

- OSTI Identifier:
- 21067492

- Resource Type:
- Journal Article

- Journal Name:
- Applied Mathematics and Optimization

- Additional Journal Information:
- Journal Volume: 45; Journal Issue: 2; Other Information: DOI: 10.1007/s00245-001-0037-3; Copyright (c) Inc. 2001 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CAUCHY PROBLEM; DIRICHLET PROBLEM; GROUP THEORY; LAPLACE EQUATION; MARKOV PROCESS; MATHEMATICAL OPERATORS; MATHEMATICAL SOLUTIONS; MATHEMATICAL SPACE; MEASURE THEORY; NONLINEAR PROBLEMS

### Citation Formats

```
Jacob, N., and Moroz, V.
```*On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups*. United States: N. p., 2002.
Web. doi:10.1007/S00245-001-0037-3.

```
Jacob, N., & Moroz, V.
```*On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups*. United States. doi:10.1007/S00245-001-0037-3.

```
Jacob, N., and Moroz, V. Mon .
"On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups". United States. doi:10.1007/S00245-001-0037-3.
```

```
@article{osti_21067492,
```

title = {On the Log-Laplace Equation for Nonlocal Operators Generating Sub-Markovian Semigroups},

author = {Jacob, N. and Moroz, V.},

abstractNote = {We consider the semilinear Cauchy problem for a class of pseudo-differential operators generating sub-Markovian semigroups. Solutions of such problems with negative definite nonlinearity play an important role in constructing branching measure-valued processes. We establish local existence and uniqueness of solutions in the context of the Dirichlet space associated to the problem. Comparison and global properties of solutions are also studied.},

doi = {10.1007/S00245-001-0037-3},

journal = {Applied Mathematics and Optimization},

issn = {0095-4616},

number = 2,

volume = 45,

place = {United States},

year = {2002},

month = {7}

}

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